Methods, tiles and boards for playing games that schematize competitive yet non-combative ecological processes, including multi-generation games of strategy and territory occupation played with progressively sized tiles on geometric grids
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Strategy board games that schematize competitive, yet non-combative, ecological processes, including preferred embodiments comprising multi-generation games of strategy and territory occupation played on a grid. During a first generation, players alternate placing game pieces, generally one grid unit in size, into unoccupied spaces on the grid, until substantially all territory is occupied. During successive generations, progressively larger game pieces are used and players alternate three-part moves consisting of. a. removing a smaller piece of one own's color; b. placing as many larger pieces as possible of one own's color; and, c. removing a smaller piece of one own's color, until no more smaller pieces are on the board. After the final generation, the player with the most pieces/territory wins.

Geshwind, David Michael (New York, NY, US)
Application Number:
Publication Date:
Filing Date:
Primary Class:
International Classes:
A63F3/00; A63F3/02; G06F17/00; A63F3/04; A63F7/04; A63F; (IPC1-7): A63F3/00
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Primary Examiner:
Attorney, Agent or Firm:
Anne C Avellone (New York, NY, US)
1. -92. (Cancelled)

93. -106. (Cancelled)

107. A method for playing a board game comprising a competitive yet non-combative structure based on a schematic version of an ecological or biological process.

108. A method as in claim 107, wherein said process is evolution.

109. A method as in claim 108, wherein said process comprises competition for territory.

110. A method as in claim 109, wherein for each player game pieces of that player are not removed by any other player.

111. A method as in claim 110, wherein said game progresses in a multiplicity of generations and the configuration of pieces progresses from generation to generation.

112. A method as in claim 111, wherein said game is played on a square grid.

113. A method as in claim 112, wherein said configuration of pieces progresses by doubling in area at each generation and alternates between squares and 2×1 rectangles.

114. A method for playing a board game comprising a competitive yet non-combative structure based on a societal or cultural process.

115. A method as in claim 114, wherein said process is completion of complementary groupings.

116. A method as in claim 115, wherein said process comprises competition for territory.

117. A method as in claim 116, wherein for at least one generation a move for a particular player consists of: a. removing four pieces of a first size comprising said complementary grouping; and, b. placing, if possible, at least one of said particular player's pieces of a second size, four times the area of said first size.

118. A method for playing a board game, wherein for at least one generation a move for a particular player consists of: a. removing one of said player's pieces of a first size; b. placing, if possible, at least one of said player's pieces of a second size, double that of said first size; and, c. if possible, removing one of said player's pieces of a first size.

119. A method as in claim 118, comprising, in addition, a preliminary generation where moves consist of players alternating placing one of their own unit square tiles within said squares of said board grid until substantially all of said squares are occupied.

120. A method as in claim 119, comprising three generations played on a 7×7 grid.

121. A method as in claim 120, comprising two players each utilizing tiles of a particular color.

122. A method as in claim 112, comprising, in addition, dead zones.

123. A method as in claim 111, wherein said game is played on an equilateral triangular grid.

124. A method as in claim 111, wherein said game is played on a grid comprising squares and right isosceles triangles.

125. A method as in claim 111, wherein said game is played on grid of cubes.

126. A method as in claim 111, wherein said game is played on grid of tetrahedrons.

127. A method as in claim 111, wherein said game is played on grid of squares of at least two designations.

128. A method as in claim 119, comprising five generations played on a grid of a size between 12×12 and 15×15.

129. A method as in claim 119, comprising seven generations played on a grid of a size between 24×24 and 31×31.

130. A method as in claim 119, comprising 2N+1 generations, played with tiles between a single square and tiles 2N on a side, and played on a grid of a size between 3×2N and (2N+2)−1 on a side.

131. A method utilizing in some combination the tiles of claim 113 and tiles of three squares in a row and tiles of three squares in an L configuration.

132. A method utilizing in some combination the tiles of claim 131 and tiles of four squares in at least some orthogonally adjacent configurations.

133. A method utilizing in some combination tiles of claim 113 and tiles comprising combinations of unit squares half unit squares comprising right isosceles triangles.

134. A method as in claim 133 comprising, in addition, at least some 45° markings limiting where certain pieces are permitted to be placed.

135. A method as in claim 134 wherein said markings are interpreted as being enabling.

136. A method as in claim 134 wherein said markings are interpreted as being disabling.

137. A method for playing the game of claim 119 comprising, in substitution, distinct markings at the centers, edges, and corners of said squares, and play with arbitrary game pieces at each generation.

138. A method as in claim 119 played on a playing field that is arbitrarily configured yet composed of grid units.

139. A method as in claim 119 played on a grid of unit squares, but where the tile unit is three squares in an L configuration, odd generations use tile in that configuration increasing in area by a factor of four, and tiles in the even generations are composed of orthogonally adjacent squares and are intermediate factors of two in area.

140. A method as in claim 123, utilizing tiles comprising alternating generations of equilateral triangles and rhombuses with angles of 60° and 120°.

141. A method as in claim 123, utilizing tiles comprising some combination of equilateral triangles, rhombuses with angles of 60° and 120°, parallelograms of 60° and 120°, trapezoids with angles of 60° and 120°, and regular hexagons.

142. A method as in claim 110 wherein, for at least some game pieces, if a particular game piece is not removed during a turn by a player within a specified time period said particular piece disappears vacating territory for potential use by an opponent.

143. A method as in claim 110, comprising in addition a ‘mutation’ mechanism where it is possible that some pieces spontaneously change.

144. A method as in claim 110, comprising at least in part moves where a single player: a. removes at least one piece; b. is permitted to place at least one piece; and, c. removes at least one additional piece.

145. A method as in clam 144 wherein said removed pieces are of a first size and said at least one placed piece is of a second larger size.

146. A method as in claim 111, wherein said game is played on a multiplicity of superimposed grids each of a different resolution.

147. A method as in claim 111, wherein said game is played on grid of hyper-cubes.

148. A method as in claim 119, comprising three generations, utilizing pieces of size 1×1, 2×1 and 2×2, played on a 7×7 grid.

149. A method as in claim 119, played on a non-rectangular board.

150. A method as in claim 119, played on an M×N rectangular grid.

151. A method as in claim 150, comprising two players each utilizing tiles of a particular color.

152. A method for playing a board game, wherein a multiplicity of players each have a unique color of playing tiles and: a. play occurs on a board comprising a 7×7 grid of squares; b. for a first generation moves consist of players alternating placing one of their own 1×1 tiles within an open space of said board grid until all players have placed and equal number of tiles and the number of open squares is less than the number of players; c. for a second generation moves consist of: i. removing one 1×1 tile of the player's own color; ii. placing as many 2×1 tiles as possible, either vertically or horizontally, in adjacent pairs of open grid squares; and, iii. removing another 1×1 tile of the players own color, if present; d. for a third generation moves consist of: i. removing one 2×1 tile of the player's own color; ii. placing as many 2×2 tiles as possible, in contiguous 2×2 cells of open grid squares; iii. removing another 2×1 tile of the players own color, if present; and, iv. play continues until no player has any 2×1 tiles on the board and players without any 2×1 tiles on the board cannot make a partial move; and, e. the player(s) with the most 2×2 tiles at the end of play win(s).

153. A computer program comprising instructions to play the game of claim 109.

154. A computer program comprising instructions to play the game of claim 111.

155. A computer program comprising instructions to play the game of claim 130.

156. A computer program comprising instructions to play the game of claim 142.

157. A computer program comprising instructions to play the game of claim 152.

158. A method as in claim 130, wherein said computer program plays against at least one human player.

159. A method as in claim 130, wherein said computer program mediates between at least two human players.

160. A method as in claim 159, wherein said at least two players each play at distinct computers via a network.

161. A product comprising the method of claim 109 supplied as instructions intended for human comprehension.

162. A product comprising the method of claim 111 supplied as instructions intended for human comprehension.

163. A product comprising the method of claim 130 supplied as instructions intended for human comprehension.

164. A product comprising the method of claim 142 supplied as instructions intended for human comprehension.

165. A product comprising the method of claim 152 supplied as instructions intended for human comprehension.

166. A product comprising: a. a board comprising a square 7×7 grid; b. at least two distinct sets of 1×1 square pieces which, in toto, comprise enough to cover substantially the entire board; c. at least two distinct sets of 2×1 pieces each sufficient to cover a substantial portion of the board; and, d. at least two distinct sets of 2×2 pieces each sufficient to cover a substantial portion of the board.

167. A product as in claim 166 comprising: e. in substitution or addition a board comprising a square grid between 12×12 and 15×15; f. in addition at least two distinct sets of 2×4 pieces each sufficient to cover a substantial portion of the board; and, g. at least two distinct sets of 4×4 pieces each sufficient to cover a substantial portion of the board.

168. A product as in claim 166 comprising: h. in substitution for each of said two sets of pieces in element c. a single set with distinct designations on each side; and, i. in substitution for each of said two sets of pieces in element d. a single set with distinct designations on each side.

169. A product as in claim 168 comprising: j. in substitution for each of said two sets of pieces in element b. a single set with distinct designations on each side.

170. A product as in claim 167 comprising: k. in substitution for each of said two sets of pieces in element f. a single set with distinct designations on each side; and, l. in substitution for each of said two sets of pieces in element g. a single set with distinct designations on each side.

171. A method as in claim 110, comprising, in addition, pieces of different thicknesses for different piece sizes.

172. A device for presenting game information comprising a container of beads wherein: a. a multiplicity of beads distinguishable by color, marking or otherwise; b. a large end of said container into which said beads flow as said container is held with said large end down, and within which said beads are randomized via mechanical agitation; c. a closed tubular end of said container into which said beads flow as said container is held with said tubular end down; d. said tubular end further being of sufficient a diameter that said bead may pass through it unimpeded yet not of sufficient diameter to permit two beads to reside laterally in said tube such that at least some of said beads will reside in an order within said tube when said container is turned tube side down; e. said tubular end further being sufficiently transparent so that said beads are identifiable with said tube.

173. A device as in claim 172, comprising in addition an openable hatch or removable cap to permit the complement of beads to be modified.

174. A device as in claim 172, comprising in addition tapered section connecting said large end and said tubular end.

175. A device as in claim 172, comprising a tube of sufficient length such that all beads can be accommodated in said tube at once.

176. A device as in claim 172, comprising sufficient beads such that all beads can not be accommodated in said tube at once.



1. Field of the Invention

Generally, the instant invention relates to strategy board games that schematize competitive, yet non-combative, ecological or biological processes, including a preferred embodiment which comprises a multi-generation game of strategy and territory occupation played with tokens on a gridded board. Here generation means a level or phase of play where a particular set or sets of tokens are utilized.

2. Description of Related Art

The instant invention has been made in the general realm of games; however, it is a game of strategy, as opposed to games such as sports (e.g., baseball, basketball, football or tennis) video action games or games of skill (e.g., Doom or Pacman) physical games (e.g., jacks or Twister) or role playing games (e.g., Pokemon or Dungeons and Dragons).

Further, it is a board game, as opposed to other strategy games such as those utilizing cards (e.g., Gin, Poker or Mille Bornes) or tiles (e.g., Mah Jong). Note that as used in conjunction with Mah Jong, the word “tiles” refers to solid pieces with symbols that are comparable to playing cards. However, as used in conjunction with this application, the word “files” is generally synonymous with game pieces, markers or tokens, such as those used in chess, checkers, etc.

More particularly, the instant invention is a board game of territory occupation, as opposed to theme games (e.g., Monopoly or Careers) or games of position and rearranging pieces (e.g., checkers, chess or backgammon).

However, unlike games such as Risk, or other tactic & strategy war simulation games, the instant invention is generally played on a geometric grid and, thus, has most in common with games such as ‘go’ and Othello (Reversi).

There are also some similarities with the game Cathedral, in as much as that game does use pieces of several shapes; however, that game only has a single generation, and the pieces are not used in the same way as with the instant invention.

Unlike ‘go’, in which stones are placed at grid intersections, with the instant invention, tiles are placed within the grid's squares (or, whatever grid units are used). (Note, it is possible to construct versions of the instant invention, where the game pieces are played at intersections, which are the equivalent or ‘dual’—in the sense of graph theory—of the embodiments described herein. However, for clarity, these variations will not be further described herein.)

Further, unlike any game currently known to inventor, with the instant invention, pieces (also called tiles, tokens or markers) at different generations of play are of different sizes and/or configurations—generally, progressively larger—and are replaced by each other. That is, the tiles are geometrically distinct and successively played, as opposed to different types of pieces in other games which are generally played during the same phase of the game. Different pieces in games like chess have completely distinct functions, and are not replaced by larger pieces; nor is ‘kinging’ a checker like the use of alternative tiles in the instant invention. Even with those games that do use pieces of different sizes or values, the pieces are not used as in the instant invention. For example, in Risk—in order to save space on the board and the number of pieces needed to play—10 small cube-like pieces representing 1 army each can be replaced by a single loaf-like piece, approximately twice the volume, representing 10 armies. Similarly, in Monopoly, after purchasing four houses on a property, you can trade them (and additional cash) in for one slightly larger hotel piece. These represent different amounts of military strength or monetary value, not geometric territory and, as will be seen, the configuration and use of distinct types of tiles in the instant invention is quite different.

The intended practitioner of the present invention is someone who is skilled in designing, implementing, building, creating, printing or publishing board games; or, programming computer versions of such board games. That is, one skilled in the art required to practice the instant invention is capable of one or more of the following: design, graphics production, printing, publishing and/or construction of game boards, pieces and/or packaging; or, programming computer simulations of such games.

The details of accomplishing such standard tasks are well known and within the ken of those skilled in those arts; are not (in and of themselves, except where noted) within the scope of the instant invention; and, if mentioned at all, will be referred to but not described in detail in the instant disclosure.

Rather, what will be disclosed are novel configurations of boards and pieces, and move algorithms or rules of play.

In summary, the disclosure of the instant invention will focus on what is new and novel and will not repeat the details of what is known in the art.


As stated, the instant invention has most in common with the extant games ‘go’ and its simplified cousin Othello (itself a commercial version of the classic Reversi). However, those games, as well as chess and many other games, are metaphors for, or schematics of, war; and, play is combative, with opponents attacking or capturing each other's pieces or positions.

In contrast, games based upon the instant invention are competitive, yet not combative. The mechanism for success, generally (a few specific embodiments aside), is not battle with, or decimation of, the enemy but, rather, fitness (expressed as strategy and tactics of taking, releasing and re-taking space) to expand into unoccupied areas better, or faster, or more stably, than the competition.

Further, the basic idea of a main class of embodiments is to provide a schematic version of what happens as single-celled organisms, over multiple generations, become larger and more complex, and compete with each other for biological niches land resources (SPACE).

Briefly, in a preferred embodiment of a version of the game called 2vo—short for BINARY (base 2) EVOLUTION—a three (or more) generation (or phase, or level) game of strategy and territory occupation is played on a 7×7 (or larger, for more than three generations) grid of squares. During a first generation, players (usually two but, optionally, more) alternate placing 1×1 unit-square game pieces (of a different color for each player) into unoccupied spaces on the grid, until substantially all territory is occupied; in the two-player game, one space is left open.

During the second generation, order of play is reversed. The pieces put into play are now larger—2×1—and are placed on any two adjacent unoccupied squares, either horizontally or vertically. Each player, in turn: a) removes one 1×1 piece of their own color; b) places as many larger 2×1 pieces as possible of their own color into adjacent pairs of unoccupied spaces; and, c) removes a smaller 1×1 piece of their own color. Players alternate these three-step moves until no more smaller pieces are on the board.

During the third generation, order of play is again reversed. The pieces put into play are now larger still—2×2—and are placed on any 2×2 cell of adjacent unoccupied squares. Each player, in turn: a) removes one 2×1 piece of their own color; b) places as many larger 2×2 pieces as possible of their own color into unoccupied 2×2 cells; and, c) removes a smaller 2×1 piece of their own color. Players alternate these three-step moves until no more smaller pieces are on the board. In the embodiment just described, at most nine 2×2 pieces can be fit on the board (usually, it is nine, but some placements of pieces can lower this amount) and, thus, with two players a tie, while possible, is rare.

Optionally, additional generations are played with progressively larger tiles alternating between the ‘brick’ and square configurations.

After the final generation, the player with the most pieces/territory wins.

FIG. 16 depicts a more general flow diagram of the preceding algorithm, comprising a functional specification from which to program the algorithmic control portion of a computer simulation of the game. Methods for creating other portions of such a program, for example display of graphic representations and GUI implementation, are well developed and well known to those skilled in the programming arts.


FIG. 1 depicts a 7×7 square grid board suitable for playing some embodiments of the instant invention.

FIG. 2 depicts seven progressively sized pieces suitable for playing some embodiments of the instant invention and, in particular, 2vo.

FIG. 3 depicts additional size-3 pieces suitable for playing additional embodiments of the instant invention.

FIG. 4 depicts additional size-4 pieces suitable for playing additional embodiments of the instant invention.

FIG. 5 depicts additional preferred 45° angled pieces suitable for playing additional embodiments of the instant invention.

FIG. 6 depicts additional ‘oddity’ 45° angled pieces.

FIG. 7 depicts use of additional preferred 45° angled pieces.

FIG. 8 depicts board layouts with markings suitable for playing additional embodiments of the instant invention.

FIG. 9 depicts marked tiles and grid squares suitable for playing additional embodiments of the instant invention.

FIG. 10 depicts two-colored pieces, and alternatively shaped pieces, suitable for playing additional embodiments of the instant invention.

FIG. 11 depicts a board on which the first three generations of the basic game 2vo are played by placing markers at the center, edges and corners of the grid's unit-squares. Such markings permit playing with (nearly) uniformly sized tokens.

FIG. 12 depicts how the scheme of FIG. 11 can be generalized to a fourth generation in three dimensions; and to four and one dimensions as well.

FIG. 13 depicts a board with irregular and discontinuous areas suitable for playing additional embodiments of the instant invention.

FIG. 14 depicts a board inscribed with square grids of three resolutions.

FIG. 15 depicts board and tiles with examples of ‘colonies’.

FIG. 16 depicts a flow diagram of the moves for the first and subsequent generations for the basic preferred embodiment of the instant invention.

FIG. 17 depicts an additional tile set with “L” units.

FIG. 18 depicts a board with grid units composed of equilateral triangles.

FIG. 19 depicts tiles suitable for playing on the board of FIG. 18.

FIG. 20 depicts a grid with two types of unit squares.

FIG. 21 depicts tiles suitable for playing on the board of FIG. 20.

FIG. 22 depicts grids comprising multiple shapes, and appropriate game tiles.

FIG. 23 depicts a three-dimensional board and pieces for an alternative embodiment.

FIG. 24 depicts a more general three-dimensional board and pieces.

FIG. 25 depicts a device to randomize player order.

FIG. 26 depicts a single set of elements to construct both board and tiles.


BASIC GAME: The instant invention, generally, relates to strategy board games that schematize competitive, yet non-combative, ecological or biological (or even societal or cultural, including economic or political) processes, including but not limited to, a preferred embodiment comprising a multi-generation game of strategy and territory occupation, played on a grid. During a first generation, players alternate placing game pieces, generally one grid unit in size, into unoccupied spaces on the grid, until substantially all territory is occupied. During successive generations progressively larger game pieces are used and players alternate moves consisting of: a) removing a smaller piece of one own's color; b) placing as many larger pieces as possible of one own's color; and, c) removing a smaller piece of one own's color. Players alternate these three-step moves until no more smaller pieces are on the board. After each generation: a) the order of play is reversed (1608, 1631); and, b) the pieces that were put down in the previous generation become the ‘small’ pieces to be picked up in the upcoming generation, and still larger pieces are selected to be the ones to be put down (1631). After the final generation, the player with the most pieces on the board, or territory, wins.

STRATEGIC STRUCTURE: A key element of this preferred embodiment that enhances ‘playability’, is the structure of the three-part move. In an alternative embodiment, in generation two and beyond, the game is played by picking up a small piece and then, if possible, putting down a large piece. Since the default opening of generation two (with two players) is one space open, the first player will pick up an adjacent small piece, thereby create a 2×1 hole, and then take it. There will then be no open space. The second player will pick up one small piece and have no space to place a large piece. This set of circumstances will repeat, almost always, and the first player will obtain virtually all territory in generation two. Therefore (at least on the average) two small pieces must be removed for each turn permitting a large piece to be placed.

The previously disclosed three-part move will be referred to as “up down up” (“UDU”). Two alternatives are: both removals precede the placement (“UUD”); or, both follow (“DUU”). Although any of the three will work, UDU is preferable because it provides a good balance between offense and defense, while UUD is primarily offensive and DUU is primarily defensive. That is, a removal prior to placement is offensive in that the player attempts to open a (best) hole for themselves to occupy; and, a removal subsequent to a placement is defensive in that the player attempts to avoid providing any similar (or, at least, only to provide the strategically worst) opportunity for their opponent(s). With UDU each move comprises both elements.

Further, with UUD many moves will comprise picking up two adjacent pieces of one's own color and immediately filling the vacated space. The players do not fully interact strategically until a relatively few scattered small pieces remain. Additionally, if a situation develops where a player has no alternative but to pick up two non-adjacent pieces, because there are no two adjacent pieces of their own color, then it is highly likely that the opponent will be able to pick up two of their own pieces, one next to each just vacated, and take two larger pieces of territory. The first player is then in the same position on the next turn. This is an unstable situation that will then to lead to lopsided and, thus, unsatisfying games.

Similarly, with DUU many moves will comprise picking up two pieces from within occupied territory so that no holes develop that are large enough for the opponent(s) to occupy with a large piece. It is only after the board becomes swiss cheese that the players fully interact strategically, and that large enough holes are open to take territory with large pieces. Again, at that point, the game tends to become unstable and cascade in favor of a first player, when their opponent is forced to vacate pieces that connect individual holes into a large size area. The first player fills that area and, likely, can also create a situation where they can (more) safely perform their removals. The opponent, then, is often put in the same bad position repeatedly.

Thus, UUD and DUU each, in their own way, provide games that tend to spiral out of control for one player or the other; and, the winning strategy is based on factors that are often tiny and/or hard to comprehend (a'la the butterfly effect of chaos theory) and, most likely, not intended on the part of the winner. Such games may be exciting. However, this inventor believes that games which are stable and balanced, and are won by carefully considered strategy and astutely executed tactics, are ultimately much more satisfying. That is, this is especially so as one gains understanding (for example, the strategic significance of edges, corners, and ‘safe’ positions where a tile of one's own color is surrounded left, right, top, and bottom by one's own color or edges), develops skill and sees their game improve. Further, such well-balanced games exercise logical thinking, attention, visualization, planning and imagination. Providing these experiences as an absorbing and open-ended challenge (especially in a face-to-face physical format) provides educational and social benefits to children too often exposed to solitary electronic pastimes. Recently, chess has been offered to some students as a way to develop cognitive skills and self-esteem. However, chess is a fairly complex game with a substantial learning curve; and, in some circles, it has a bit of an ‘egghead’ taint. 2vo has neither of these problems, and has been tested with children as young as six, who are able to play and comprehend the rules and basic strategy of the game by the second game they play. Finally, it is suggested that the “competitive, yet non-combative” paradigm of games employing the instant invention, provides an ethically distinct, and arguably preferable, model for children to emulate, when compared to the schematic “war” that characterizes games such as chess, checkers, ‘go’, Othello, Stratego, Battleships, Risk etc.

Other optional variations on these strategic structures are practiced, to take into account other playing sequences of tiles, or for other reasons. For example, if (201) tiles are used in generation one and (302) tiles in generation two, an UUDU or UDUU structure will balance the 1:3 area ratio of these two tiles. Similarly, if (201), (203) and (205) square tiles are used in the first three generations, structures with 4 Us and 1D, for example UUDUU, will balance the 1:4 area ratio between successive square tiles. In general any structure of the form Xuydzu with removals on both sides of a potential placement will exhibit the balance between offense and defense described above. Each of the three phases, but especially the D phase may, optionally, consist of 0, 1, N, ‘as many as possible’, etc., in different circumstances, and are either specifically required, or at the player's option. Additionally, move structures such as UUUD and DUUU can also be used, but are less preferred because they do not balance well offense and defense in a single move. However, for some embodiments, such structures are necessary.

Longer sequences are also optional, for example UDUDU . . . , where each D is of a single piece only (generally, a player may place as many pieces as possible during a single D phase) so long as the alternation can be maintained; i.e., until there are no more spaces opened by an U phase that permit a larger piece to be placed in a D phase.

In the D phase, generally, as many pieces as possible may be placed if the player desires. However, under some circumstances or in some embodiments: the number is restricted to only one piece; the player is required to place as many as possible; or, the player may at his discretion place fewer than that maximum possible, even refraining from placing any.

In addition, there are optional relaxations of the strict requirement that a first generation of tiles is completely replaced by a second generation of tiles before a third generation of tiles comes into play. For example, during generation two of the basic game, it is possible (although, more likely toward the end of a generation) that a hole will develop that is big enough to place two adjacent (202) tiles in a 2×2 open area and, with this variation, it is permitted to place a (203) instead, even though it is still generation two. Optionally, if a (203) tile is being placed in generation two, the comparable U phase(s) removal must be a (202) tile, or two (201) tiles.

ALTERNATiVE LAST MOVE TO GENERATION ONE: With two, three, four or six players, the last move of of the first generation (phase or level of play) of 2vo leaves open a single empty grid space on the board. Particularly with two players, the second player (who now opens generation two) may be perceived to have too great an advantage (as will be explained, below).

In the standard game of two players, player one goes first in generation one and generation three, and second in generation two.

1. Generally, going first in generation one is arguably a (slight) advantage in that the first player is the first to have an opportunity to grab a corner, for example, and the second player may feel (if not actually be) at a disadvantage—constantly on the defensive.

2. Going first in generation three may, actually, often be a disadvantage. Since there is no compelentary 2×1 space intentionally left open at the end of generation two, player one (who, again, goes first in generation three) may need to remove two small pieces without being able to put down a large piece. This can give player two a lot of opportunities early on in generation three.

The above two elements can be thought to approximately balance out; however, in generation two, player two starts and will almost certainly be able to take the only open grid space by removing a 1×1 tile next to it, and placing a 1×2 brick tile in the larger space opened up. Player two then, by removal of another 1×1 tile, opens another grid space, but it is of their own choice. And that choice may make it impossible for player one to make a good move; they may be forced to remove two 1×1 tiles without being able to play a 1×2 brick tile. If player one has not prepared adequately, by arranging two ‘safe haven’ configurations, then player two will again have at least one good move. From this point, the game will likely proceed fairly evenly with players of similar skill (mostly depending upon consolidation of position and ‘safe haven’ configurations). However, player two may well be up by a critical piece or two.

A slight modification to the end of generation one is suggested as an alternative that will tend to lessen the imbalance described above.

In the standard game, in the last move of generation one, player two takes one of the last two open grid spaces, leaving one space open.

In the alternative last move of generation one, with two grid spaces open player two, instead, removes one 1×1 tile (presumably next to one of the two open grid spaces) and (most likely) places a 2×1 brick tire. Then, because player two did not get to place their 24TH tile, they do not pick up a 1×1 tile to end this move.

Play proceeds normally from that point on, with player one having a single open grid space to work with.

Alternatively—especially for seven players, where there are no open spots at the end of the first phase of play—for at least a first round in generation two have moves proceed UUD so that moves can generally be made. However, as that will still not guarantee open spaces, another alternative for the first one or few round(s) is to utilize UUDU moves that will both, generally, permit moves and leave at least one grid space open. ***

COMPONENTS FOR BASIC GAME: FIG. 1 shows a board (100) suitable for playing the particular preferred embodiment, comprising three generations and played on a 7×7 grid of squares, as described in the sections entitled BRIEF SUMMARY OF INVENTION and BASIC GAME, above. A decorative edge, or the physical border of the playing board, is indicated by the double line (101). The edge of the active playing area is depicted by the single line (102). One of 49 unit-squares is designated as (103). Playing tiles are to be placed within the boundaries of the grid's squares.

FIG. 2 shows game pieces, or tiles, suitable for playing the version of the game described thus far. A 1×1 tile is shown at (201), a 2×1 tile is shown at (202), and a 2×2 tile is shown at (203). The double lines (208) show the tile outlines; the single lines (209) are drawn to show the number of unit-squares involved and, although such lines might, optionally, be drawn on the tiles, are not meant to show physical divisions or other features of the tiles.

FIG. 16 depicts a general flow diagram of the algorithm for the basic game, played by any number of players, and played for any number of generations. It comprises a functional specification from which to program the algorithmic control portion of a computer simulation of the game. Methods for creating other portions of such a program, for example display of graphic representations and GUI implementation, are well developed and well known to those skilled in the programming arts. The flow diagram, in concert with the instant specification, is essentially self-explanatory; but, several comments, following, will elucidate.

The loop of elements (1601) through (1606) comprise the first generation, ellipsis (1603) indicates steps for additional players between 1 and N, if N>2 (1607) results in branching to a later generation(s), first passing through element (1608) which reverses order of play; looped passes through elements (1609) through (1628) comprise a later generation, ellipsis (1619-1621) indicates steps for additional players between F(irst) and L(ast), if N>2 (1629) through (1633) determine whether to perform an additional generation, or not (1634); and, (1635-1636) are performed after play is over.

When it is decided (1630) that an additional generation is to be played: a) the order of play is reversed (1608, 1631); and, b) the pieces that were put down in the previous generation become the ‘small’ pieces to be picked up in the upcoming generation, and still larger pieces are selected to be the ones to be put down (1631). In generation one the players are described as 1 through N. In later generations they are referred to as F(irst) and L(ast) because the order of play is reversed in alternate generations.

BOARD SIZE AND SHAPE: In the preferred embodiment described thus far, with three generations played on a 7×7 grid, three types of pieces (201-203) are used. However, two additional generations are, optionally, played with 2×4 tiles (204) and 4×4 tiles (205) resulting in only a single 4×4 tile fitting on the board at the last move. However, such play ‘to the bitter end’ would be anticlimactic, and too dependent upon who was going first in the last two generations, and it is recommended that play end when nine square tiles can be fit onto the board. With the three-generation preferred embodiment described, a 7×7 grid was chosen because this was the maximum size grid that fit this criteria. With a 5×5 board only four 2×2 tiles would fit. With a 6×6 board nine tiles would fit, but there would be no ‘wiggle room’; that is, labeling both rows and columns from 0 through 6 with the upper-left corner labeled (0,0), if any tiles were not put with their upper-left corner on a square with both X and Y being even, fewer than nine tiles would fit. Put another way: nine 2×2 tiles cover 36 squares; and, a 6×6 board is exactly 36 squares. On the other hand, a 7×7 board permits some of the 2×2 tiles to be offset by one grid square, in X and/or Y, and yet still have nine tiles fit on the board. (Note that with some offsets in the placement of 2×2 tiles, fewer than 9 will fit on the board, with 4 as a minimum.) If an 8×8 board is used, than up to 16 2×2 tiles can be fit on the board. Thus, in this case, the only number that is greater than 6 and less than 8 is 7; so a 7×7 grid is used.

Using the same criteria, if it is desired to increase the number of generations to 5—using tiles of size 1×1 (201), 2×1 (202), 2×2 (203), 4×2 (204) and 4×4 (205)—then the size of a square grid would need to be more than 12×12 (where there is no ‘wiggle room’) and less than 16×16 which would permit 16 4×4 tiles to fit. Thus, acceptable values are 13, 14 or 15. Two resulting elements trade off as the size of the board is increased. With a 13×13 board only 168 moves need to be made during the first generation, but a minimum of ‘wiggle room’ is available. With a 15×15 board 224 moves need to be made during the first generation, but there is a maximum of ‘wiggle room’ permitting more variation in moves and strategy. With a 14×14 board these two elements are both intermediate; however, with an even number of squares, when playing with the most usual number of players—two—either 0 or 2 spaces will be left open after the first generation; thus, odd-numbered boards are not necessary, but preferred.

On the other hand, if dual (or more) resolutions are to be inscribed on a single board, such as is shown in FIG. 14, then the 7×7 board leads to 14×14 and 28×28 higher resolutions as unit squares are halved and quartered, in each direction.

Similarly, for a game of 7 generations, the board size would need to be greater than 24×24 (if at least a single row and column of ‘wiggle room’ were made available, exactly 24×24 if no ‘wiggle room’ were made available) and less than 32×32 (or up to 16 8×8 tiles would fit). Again, a 25×25 board would make for the fastest game; and 31×31 board would make for the most flexible placement of tiles and, thus, the most complex strategy and tactics.

Even larger boards are, optionally, used and, with the embodiment using generations alternating between tiles that are squares and those that are 2:1 ratio ‘bricks’ on a square board, the following algorithm holds. For N=1, 2, 3, etc.: the number of generations=(N×2)+1; the minimum tile is 1×1 and the maximum tile is a square of 2N on a side. In order to have nine tiles in the last generation, the minimum sized board (permitting at least some ‘wiggle room’) is (3×2N)+1 on a side, and the maximum sized board is (2N+2)−1 on a side. However, for physical board games, grids much beyond 25×25 may not be practical; for example, ‘go’ is typically played on (the intersections of) a 19×19 grid, and is a long game of one generation only.

Nevertheless, with a computer-mediated version of 2vo, or of other variations on or embodiments of the instant invention, larger board sizes are practical. Played over a network, in particular, many players may conveniently collaborate on a large game; or, long games with players making moves asynchronously are also practicable. Also, on a large computer-mediated board, several games may go on simultaneously, where pieces from other games block by taking up space, but do not otherwise interact. For example, in one embodiment, several games, each going on in its own area of a very large board, are generally disjoint, but overlap somewhat with other games at the edges of areas. Such abutting games are played synchronously or asynchronously. Also, a board on a computer screen may be zoomed into, or out of, permitting larger board sizes to be conveniently viewed. Further, it is possible to have the computer fill in some or all of the tiles in the initial generation, at random or with some pattern, in order to speed along the game. Similarly one or more generations are, optionally, skipped (or the computer fills in from a later generation). However, if two generations are skipped, this is just equivalent to starting with a 2×2 square as the first generation; or, starting with a board of (roughly) N/2×N/2.

FIG. 14 shows how a single board is inscribed with three (as shown, or even more) grids of different resolutions, so that play with several sizes of unit tiles may be played on the same board. In practice, different colors of lines are, optionally, used; and, lines which overlap may be inscribed with adjacent lines of all colors present. However, in FIG. 14, which shows only the upper-left corner of a board, grids of the largest size, or lowest resolution, are shown as thick lines; lines of the intermediate resolution are shown as medium lines; and, lines of the smallest size, or highest resolution, are shown as thin lines. Only the lowest resolution line is shown where they overlap. A game of three generations would utilize the largest squares; a game of five generations would utilize the intermediate squares; and, a game of seven generations would utilize the small squares.

An example of an alternative embodiment playable, for example, on a board with both 7×7 and 14×14 resolutions is as follows. The basic three generations are played with tiles suitable for the 7×7 grid. However, it is permitted to place the tiles on the lines of the 14×14 grid. Thus, it is possible to place the unit square 1×1 tiles on half-unit line increments. Therefore, by placing a tile one half unit from an edge, 1.5 units are blocked or controlled; and, by placing a tile one half unit out in both directions from a corner, 2.25 units are blocked or controlled. Similarly, being able to place tiles on half unit increments opens additional strategic and tactical techniques throughout the entire game. Another way to think of this particular example is as a game of five generations played on a 14×14 grid, but started at the third generation.

Further, as will be shown later, the units on the grid need not be squares (e.g., see FIG. 8, diagrams (801) & (802), and FIGS. 12 and 18), or even uniform (e.g., see FIG. 8, diagrams (801) & (802), and FIG. 20). However, even with a square grid, the board need not conform to the N×N sizes described above. The board is alternatively an N×M rectangle; or, optionally, is not even rectangular, regular, convex or even contiguous. While there may be certain advantages of strategic comprehensibility when using a square board as described above, and in using the same size or type of board at all times, the instant invention, as generally described in its various embodiments, can be played on an arbitrarily sized and shaped board. For example, FIG. 13 shows a playing field (1300) which comprises: two discontiguous sections (1301 &1302); and a ‘hole’ (1303); as well as concave (1304) and convex (1302) sections. Further, neither the tiles need be rectangular (e.g., see FIGS. 3, 4, 5, 6, 17, 19, 21 and 22) nor the grids square (e.g., see FIG. 8, diagrams (801) & (802), and FIGS. 12, 18 and 22).

NUMBER OF PLAYERS: Returning now to the 7×7 board, the number of players will be discussed. With the standard default of two players, in the first generation each player puts down 24 1×1 tiles so that 48 squares are covered, leaving one square open to begin tie second generation. With three players, each puts down 16 1×1 tiles so that, again, 48 squares are covered and one is left open for generation two. With four player, each places 12 tiles and, again, 48 squares are covered and one is left open at the end of generation one. Similarly for 6 players each placing 8 tiles and 8 players each placing 6 tiles. With the 7×7 board, even more than eight players is possible, but more than about six are probably not very practical, strategically.

With 7 players, the board is full at the end of generation one, each player having placed 7 1×1 tiles. Thus, when generation two is started the first player in generation two (who was the last player in generation one) will have to pick up two 1×1 tiles without being able to put any down. Alternatives to avoid this are: each player only places 6 tiles, leaving 7 spaces open to begin generation two; or, each player picks up one piece before generation two starts, which is mathematically equivalent, but not strategically, because the first tile picked up may not be the same as the last one put down (or the one left open) by any particular player. Similarly, with five players, each would place nine tiles, leaving three squares open for generation two.

For other board sizes, and numbers of players, similar situations develop. In general, the algorithm for generation one is that N players alternate placing 1×1 tiles, until there are N or fewer open squares. However, there are optional variations on this rule, and these and other such variations are within the scope of the instant invention. For example, in a game where several open squares are present between the first and second generations, the number of larger pieces put down is limited to one for each player, during the first round (or two, or more) of turns for that generation. Alternatively, if there are several spaces open, all but one are filled with null pieces (e.g., of a color not used by any player, or specially marked as in (901)). These are placed: by players taking turns before, after, or anytime during play of the first generation; by some published rule (e.g., symmetrically equivalent, any corner for 1, any 2 diagonal corners for 2, any 3 corners for 3, all corners for four); or, by chance (e.g., by throwing two special dice with more than 6 sides displaying (columns) 1-7 and (rows) At, as per FIG. 15). The pieces are: left throughout the second generation only and then removed before generation three; left throughout the game as ‘dead zones’; picked up algorithmically during generation two (e.g., a null is removed with every fifth regular unit tile picked up); any one picked up as the first, second or on both removes, by players until null pieces are gone; any one picked up by a player as an open strategic option at any time instead of, or in addition to, a piece of their own color; etc.

Tournament structures of any kind are practiced to score multiple games, particularly for multiple players. Scoring over multiple games optionally counts or accumulates: number of games won; total area or number of pieces; highest scores; or, other ranking method. Timing is optionally made an element. Total game time, or the time of individual moves is optionally limited, or scoring optionally takes time expended into account.

Any known or custom tournament structure and rules are optionally applied. For example, with three players, games of only two players are used and a ‘round robin’ tournament is performed with three games consisting of A vs B, B vs C, and C vs A. The player with the best accumulated score wins; or, the best two players then go head-to-head in one game or a series. Such a series, between several or even just two players, is decided based upon: winning N out of M games (e.g., 2 out of 3); total score after N games; play as many games as necessary until one (or more) player reaches a score of N, at which time play is ended immediately or the current game is completed before scoring; or, otherwise.

When playing with several people, in one variation, play starts with all playing together, and proceeds to eliminate players until only two are left to play a final game or series. One or more worst scorer is eliminated at each round; and, how many are eliminated depends, optionally, upon the particular scores. For example, consider four players A, B, C & D and a final round with nine tiles. A number of example scores and possible outcomes follow:

    • A=1 B=2 C-2 D=4: D wins, or drop A only.
    • A=1 B=1 C=3 D=4: D wins, or drop both A and B.
    • A=0 B=1 C=3 D=4: D wins, or drop A, or drop both A and B.
    • A=0 B=1 C=3 D=3: Drop A, or drop both A and B.
    • A=1 B=1 C=3 D=3: C and D play run-off game or series.

In most games of strategy, there is a real or perceived strategic advantage (or disadvantage) to going first. Thus, the standard embodiment of the instant invention reverses order of play at each generation, and reverses starting order in the first generation in alternate games. However, when playing with more than two people, the situation is not symmetrical because there will be one or more players ‘in the middle’ who never go first or last. Further, with two players, A follows B, and B follows A. However, with three players, for example, A follows B, B follows C, and C follows A, again strategically asymmetrical. While knowledge of the strengths and weaknesses of who precedes and follows a player can be used to strategic advantage, it nevertheless may be desired to eliminate or, at least, randomize such relationships after each game or generation, or even as often as after each round of moves. This is accomplished by rolling dice or cutting cards for rank, or any other standard selection mechanism but, in particular, two example embodiments follow.

There are six ways three players may be ordered, 24 ways for four players, 120 ways for five players, and so on. A single standard die shows six sides; two such distinguishable (e.g., one red and one blue) dice provide 36 combinations; two distinguishable 12-sided dice provide 144 combinations, etc. A simple printed table is provided to convert dice outcomes to player order so that player order may be quickly enough established so as to be practicable to perform even for each move. For rolls beyond the range needed, table entries will repeat some entries or specify, ‘roll again’ ‘reverse last order’ ‘use last order’ or some other instruction. For three players the table data constitutes:

    • for a die roll of 1, player order is A B C;
    • for a die roll of 2, player order is A C B;
    • for a die roll of 3, player order is B C A;
    • for a die roll of 4, player order is B A C;
    • for a die roll of 5, player order is C A B; and,
    • for a die roll of 6, player order is C B A.

For more than four (and certainly beyond five) players, such a print table is, perhaps, too cumbersome. In that case, a device similar to an electronic calculator is provided into which the number of player is initially entered. Than at each move a randomize button is hit and a random player sequence is displayed. Such a function is trivially included in a computer-mediated embodiment. However, for physical embodiments, an inexpensive alternative to the electronic calculator is shown in FIG. 25.

A number of uniform small balls or beads (2501-2508) typically of plastic, are provided and are distinguishable by their color, a number or letter, or some other marking. Enough are supplied for the maximum number of players and are sealed within the supplied container (2500), or the container is supplied with a hatch (2510), or a removable cap on tube end (2530), into which the appropriate beads are loaded. If the device is sealed and contains more beads than needed, each player selects a color, letter, etc. and any not selected are ignored when reading a player sequence. One end of the container is preferably bulbous (2515), but of any shape with sufficiently large volume, and preferably opaque. The other end tapers or funnels (2520) into, or just ends in, a tube (2525) that is transparent, closed at the other end (2530) and of sufficient diameter to accommodate a single bead without friction, but narrow enough to prevent two beads from being positioned laterally to each other. Thus, the container is held with the tube up and the beads fall into the bulb; the bulb is shaken, randomizing the beads; the container is then turned and held with the tube downward; the beads accumulate in the tube in an ordered column; and, the beads are read, from the top or bottom, to determine player order. Further, this device performs other functions with appropriate sets of beads.

TILE CONSTRUCTION: With the basic 7×7 board, three generations, and two players, for each color: 24 1×1 tiles (24 units in area) are required, always; 24 1×2 tiles (48 units in area) are required at an absolute maximum, although this is a highly unlikely, strategically lopsided situation in the middle game (12 are the average, but not sufficient, and 18 each would probably cover well more than 95% of situations, but what do you do in those few other situations?); and, 9 2×2 tiles (36 units in area) are required at a maximum, but a shutout is a much more likely possibility in the end game. Thus, for each color, 57 pieces, totaling 108 units in area are required. If one color, pieces can comprise inexpensive reversible tokens like wooden checkers or plastic tiddlywinks; or, they may be more expensive weighted pieces with a felted bottom, like pieces supplied with deluxe versions of other games.

An elegant alternative is to produce dual-sided pieces, as are used with the game Othello. These are one color (nominally black) on one side and a second color (nominally white) on the other. They are turned one side up, or the other, depending upon which player places them in play. In addition to elegance, an advantage is that only 24 1×2 pieces are needed in total, not 24 for each color; and, similarly only 9 2×2 pieces. If the pieces are painted with two different colors, or imprinted with two symbols, the material (but not necessarily the manufacturing) cost is cut in half for the middle and larger pieces. However, if a standard black piece and a standard white piece are sandwiched, then the cost of sandwiching is added to the materials cost. Further, for the 1×1 tiles, only 24 of each single color are needed. If these are sandwiched, then only 24 dual-sided 1×1 pieces result, and 24 more are needed, doubling materials. So, an alternative is to have 24 single-sided (a geometric misnomer) 1×1 pieces of each color, and dual-sided pieces for the larger sizes. Dual-sided pieces are painted, stained, coated or printed; or, different colors of material (wood, plastic, metal, foam or otherwise) may be sandwiched; etc. If a single layer, or sandwiched materials, or a middle layer between a sandwich, is magnetic or metallic, and the playing board is complementary, then pieces adhere to the board and a ‘travel’ style game is produced.

Additionally, if pieces of a third color are added, and these are definitely designated as the third color (or dual-sided third/fourth color), then only 16 1×1's are needed; but, to cover all possibilities, 24 1×2's and 9 2×2's are still needed. Similarly, the designated fourth color requires only 12 1×1's. However, limiting color choice for primary colors may not be worth the corner cutting; and, providing full complements of all colors is preferred to maximize customer satisfaction.

On a computer screen, the tiles will, generally, be the same size as the grid elements they are placed upon. However, with a physical game, it is necessary to be able to place and remove tiles from the board without disturbing the other tiles on the board. Thus, the tiles will, generally, be somewhat smaller than the territory they are meant to occupy. For example, if the unit grid is a 1×1 square, the unit tile would be a ¾×¾ square, leaving ¼ unit between tiles for fingers to grasp the tiles. However, for later generations and larger tiles, the ratio will need to be adjusted. At the fifth generation, tiles are 4×4 unit squares. If the ¾ factor were applied, this would result in a 3×3 unit tile that could be exactly fit into a 3×3 space, when a 4×4 space is what is required strategically. Therefore, rather than making tiles a uniform size that is ¾ of the linear distance of the space they are to occupy, leaving a uniform (or, just slightly progressively larger) border around the tile will produce tiles that are both handlable and unambiguously fill the required space. For example, the 4×4 unit tile would be produced as 3-½×3-½ units.

Another alternative that will help with distinguishing, and physical handling of, the tiles is to make them of different heights. For example, for a three generation game the 1×1 tiles are made 1 unit high, creating a 1×1×1 cube; the 2×1 rectangles are made ½ unit high, making a fairly standard ‘brick’ shape; and the 2×2 squares are made ¼ unit high, making a shape similar to ceramic wall tiles.

Further, if the pieces are made thicker, e.g., as 1×1×1 unit cubes, then up to six different colors (one on each side), for six different players, are presented. For the 1×1×2 ‘bricks’ only four long sides are available (the piece is not useful for the game, as described thus far, if stood on either of the two 1×1 ends) so four colors for four players is a practical limit. Thus, two sides of the unit cube are available for markings such as ‘dead zones’ (901). Similarly, a large 2×2×2 cube can be painted for four players on four of its six sides. Of course ‘painted’ is meant loosely, and such pieces are, optionally, constructed of different colored plastic (such as the Instant Insanity cubes), or other materials, or by other methods.

Creating a tool for plastic injection molding is expensive, so limiting the number of molds is desired. FIG. 26 shows how a small set of molded pieces are used to construct both pieces and boards. Element (2610) shows a piece that has interlocking jigsaw puzzle-like protrusions at top and left, and indentations at bottom and right. Alternatively (2615) shows a tile with tongue at top and left, and groove at bottom and right. Either may be used (in one color for each player) alone for a unit square tile (201), interlocked in a 2×1 or 1×2 tile configuration (202), interlocked in a 2×2 tile configuration (203), etc.; or, (in a different color) in a 7×7 board configuration (100).

Alternatively element (2620), shown from top and side, is a tile that is grooved on all edges, and element (2625), shown from top and end, is a double-tongued ‘grout strip’ that has a visible divider (2626) that is optionally of zero width, and which is cut into appropriate lengths. The tongue and grove are, optionally, keystoned (exaggerated as shown) so that, once snapped in place, they will not slip apart. Assuming just two players, these pieces are fabricated in gray for the board (with (2625) optionally contrasting), and black and white for the players. The board is made up of 49 gray tiles (2620); six horizontal strips of (2625) that, in length, are seven times the width of (2620) plus six times the width of (2626); and, 42 (6 each for 7 rows) vertical spacers of (2625) that are each the length of the width of (2620). A single (2620) is comparable to (201); two (2620) tiles with a short spacer connecting them are comparable to a (202); etc. For the tiles, the tiles and grout are matching for each player; or, the contrasting grout may be used to enhance visibility between adjacent larger tiles of the same color, which have internal but no bordering grout strips. Additionally, an optional protrusion or knob (2621) on one side of each piece will serve as a handle to assist in placing and removing tiles and, when turned upside down, one of many feet on the board. Alternately, if not turned upside down, and optional indentation (2622) is present, their combination will ensure proper registration of the tiles as placed on the board. Lastly single-tongued (2631) and notched (2632) frame strips, such as (2630), not drawn to the same scale, are supplied to finish the board. Fabrication cost and shipping space may be saved by supplying the game in pieces to be assembled by the user.

ADDITIONAL TILES COMPOSED OF GRID UNITS: Thus far, with the embodiment called 2vo: playing pieces start as a single grid unit (a square in the basic game) in the first generation; double in one dimension to a 2×1 unit cell in the second generation; and, double in the other dimension to a 2×2 unit cell, returning to a larger version of the original configuration, in the third generation; and so on. At each generation after the first, the larger piece is twice the area of the smaller and, thus, the three-part move—remove a small piece, add a (usually one, strategic or tactical advantage aside) larger piece, and remove a second small piece—yields neither a net gain or loss of territory.

In an alternative embodiment, pieces grow linearly, by one unit at each generation, instead of doubling in size. (Such an embodiment can be titled 1vo″ or Lvo″ in keeping with the typographical convention of the 2vo name, or LINEAR 2vo″) Thus, when two small pieces are removed and one larger piece deposited, territory is changed by (N+1)−2N; or, N−1 additional grid units are left open. Once such ‘holes’ accumulate, it is very likely a player will be able to deposit more than one larger piece on a given move. Alternatively, a move structure of UD or DU results in (N+1)−N; or, an increase by one unit of occupied territory on the average. Thus, players will not be able to place a larger piece in every move.

With the linear growth embodiment the first two generations are still played with pieces (201) and (202) respectively. However, in generation three, while pieces (202) are picked up, pieces of size-3 are put down. There are two possible configurations of size-3 and these are shown as (301) and (302) in FIG. 3. Since the occupiable space on the basic board comprises, nominally, 48 squares it is possible to place, at most, 16 size-3 pieces on the board. This is an average of 8 each for two players assuming equal strategic performance. Thus, it must be decided how many of each of the two possible configurations to supply and how they are to be played. For example, if dual-sided pieces are supplied with one player's color on each side, eight of each configuration will be enough to cover the board, and (since (301) and (302) are symmetrical) each player may choose the configuration they desire until pieces of one type run out for both players. Alternatively, each player may get 8 of each configuration in their own color and, on the average, have enough of either (301) or (302) but, only in toto, have enough of (301) and (302) taken together, under the most extreme circumstances, to fill all but one square of the board. In still another option, 16 dual-sided pieces of each configuration will permit each player to choose either (301) or (302), at each turn, without the possibility of running out of either.

Similarly, at generation four, (301) and/or (302) are picked up, and pieces comprising four grid units are put down. FIG. 4 shows the six (401-406) possible configurations (in addition to 203) of size-4 with square grid units. Note that two pairs of pieces, (402 &403) and (405 &406), are mirror reflections of each other. If pieces are of a single color, and sufficiently uniform on both sides as to be reversible (such as checkers), then only one type of piece for each pair need be manufactured and is used reversibly. However, if the pieces have a bottom (e.g., lined with felt, such as with some chess pieces) or are dual-sided with two colors (such as with Othello) the pieces of both polarities will need to be manufactured. In particular, with dual-sided pieces, a black (402) is a white (403) and vise versa; and, similarly with (405) and (406). Again, these are supplied according to any of several alternative plans: sufficient number of each configuration in each color are provided to completely cover the board; or, two of each configuration are supplied to each player totalling 56 squares in toto, more than 48; or, some larger number of (203) and just one or two each of (401-406); and, if pieces are reversible and of a single color than ‘each’ may include both (402) and (403) or just one reversible type, and similarly for (405) and (406); etc.

Other game variations and embodiments include, without limitation, any combination of rules and/or pieces for the geometric growth (e.g., doubling) or linear growth embodiments. For example, the first two generations are played as per the rules of 2vo. The third generation is as well, going to pieces of four units in area, not three, except that some combination of standard pieces (203) and other pieces (401-406) are used instead of just the (203) pieces.

Other game embodiments include, for example, using (201), (202) and (203) tiles, as well as the tiles of FIGS. 3 and 4, to play a ‘continuous’ game, as opposed to a game with distinct generations. Complements of tiles other than just described are, optionally, used. However, the rules and strategy are critically dependent on, and adjusted depending upon, which tiles are chosen and how many of each are available. In one such embodiment there are two phases of play.

During a first phase the empty board is filled. On each move: a player places a (201) of their own color in an empty square; or, a player removes any tile of their own color of size-N and places a tile of size-N+1 in the same space plus any one adjacent square. (To generalize, the first type of move may be considered as a tile of size-0 being removed and replaced with a size-1 tile.) This continues until all space is covered.

During a second ‘attrition’ or ‘consolidation’ phase, during each move, a player removes a (201) of their own color and places one, or more, larger tile, if possible. This continues until there are no (201) files left. Optionally, during this phase, during a move, any tile next to open space may be removed and replaced by a larger (or, at least not a smaller) tile. Play continues until no more such moves are possible, or until each player is happy with their position.

Scoring options include: area occupied for each player is simply counted; (201) tiles do not count; only the largest pieces count; larger pieces count disproportionately, for example, tiles score as N−1 each or N2 each; etc.

ADDITIONAL 45° ANGLED TILES: A further alternative to the game tiles already described are those constructed out of unit squares and half squares (in this case, right isosceles triangles) used in combination. FIG. 5 shows a grouping of tiles that are used as substitutions for, or in combination with, those of FIG. 2 and/or FIGS. 3 and 4. Tiles (501-503) each comprise one unit square in area, but span two half grid units; they are comparable to piece (201). Pieces (501) and (503) are mirror images of each other and the manufacturing and supply principles discussed earlier regarding reversible and dual-sided pieces apply here as well. (The same also applies to (504) and (508), for which only one of two mirror images are shown.) Each of these three pieces may be turned by multiples of 90° (but not 45° as the tile side lengths would not then coincide with sides and diagonals of the grid squares, even though the angles would align) for placement on the standard square grid. These pieces are, perhaps, more interesting when used in conjunction with (201) tiles in generation one and, particularly, when at least some rectangular (202) pieces are used in generation two. Pieces (501-503) contribute to, or block, more than one grid unit; but, they do not entirely vacate either of those spaces upon removal. For example, in configuration (701) of FIG. 7, five (501) tiles are laid in a horizontal sequence (A, B, C, D and E). Even if any two internal (B, C and D) adjacent (501) tiles are removed, vacating two unit squares in area, a (202) piece cannot be put down in generation two. However, a (504) tile, if available, can be placed in such a circumstance. Similarly, if (B) and (D) are removed, then no two unit area piece can be placed; but, when (C) is removed, (202), (505) or (504) (but not its mirror image) can be put in the space so vacated.

Tiles comparable to the two unit (202) tiles are asymmetrical (504) (and its mirror image, not shown) as well as symmetrical tiles (505-507). Each can, again, be rotated by multiples of 90°, some with no effect. Tiles (508), its mirror image (not shown), and (509) are all 4 units in area, are similar in shape to, but larger than, (501), (503) and (502) respectively, and are comparable to (203) and its relation to (201). The alternating enlarging progression, comparable to that of the (2xx) tiles, can continue indefinitely (within reason) with the (5xx) tiles as well. What are called the asymmetrical tiles (501), (504) and (508) (and their mirror images) are skewed (to the right, or left, on top) versions of (201), (202) and (203) respectively.

Embodiments of the instant invention are, optionally, played with the FIG. 2 tiles, the FIG. 5 tiles, or any combination at each generation; and can be played in any combination with FIG. 3 and/or FIG. 4 tiles (or other tiles explicitly described or depicted herein, or otherwise) as well. Again, some embodiments provide effectively unlimited numbers of tiles (e.g., for each of two colors, 24 each of (201), (501), (502) and (503)) permitting choice without restriction for each player. Alternatively, for example, a set of 24 tiles may be composed of six of each type; or, eight each of types (201), a reversible (501/503), and (502); or, 12 of (201), and an additional 12 equally divided between the two or three other shapes. In an intermediate approach, more than 24 unit area tiles are provided in total, but not 24 of each; for example, 24 of (201) and 8 each of the two or three other shapes, providing some but not maximal flexibility of which shapes to play. And so on. Similar options for mixing tiles are applicable to the number of tiles provided of the two and four unit area sizes.

Additional tiles are possible and optionally supplied to be used instead of, or in addition to, any of those disclosed herein. These optionally incorporate elements embodying: other shapes; other angles; partial or half-cells of rectangles, or other shapes, instead of triangles; protrusions and/or indentations that are complementary, interlocking or otherwise; are symmetrical or asymmetrical, uniform or non-uniform; blank or colored or marked in a number of ways; etc. Further, some or all tiles are supplied in limited quantities (for any particular type, or any combination, or in toto) so that, optionally, the entire board cannot be covered by a single player, some combination of players, or all player together, during some phase(s) of play.

It is suggested that for pieces with 45° sections, the foregoing are the most strategically comprehensible and, thus, more interesting. However, in the interests of thoroughness, FIG. 6 shows two additional symmetrical one unit area tiles (601 &602) each composed of two triangles connected at vertices. (These are most practical in computer-mediated embodiments, because such vertex connections are physically fragile.) The other tiles on this page (603-608) are symmetrical (606 &608) and asymmetrical (i.e., requiring mirror image versions or reversing) tiles of two unit areas. Some of these (particularly 606 and 608) are more likely to be strategically comprehensible and useful when optionally provided in an embodiment.

FIG. 7 shows several combinations (701), (702) and (704) of one unit area pieces; and one combination (703) of two unit area pieces. It should be noted that skewed pieces of the same mirror polarity are abutted linearly (701); and that pieces of opposite skews (703) are used to ‘turn the corner’. Configuration (704) shows an example of how skewed, triangular and square pieces, each one unit in area, are combined to tile the grid of squares without leaving gaps.

Some elements of FIG. 9 (depicted in single-lined boxes) show markings that are, optionally, inscribed within the squares of the grid. These alter the grid to comprise, at least in part, right isosceles triangles as well as squares; and are used to restrict which pieces are permitted to be placed on areas of the grid. The four inscriptions are: empty (917), slash (918), backslash (919) and X (920). These are, optionally, interpreted in either an enabling or disabling fashion.

With the enabling interpretation: only tiles or tile sections that comprise full squares can be placed in, or atop, an empty square (917); tile sections that proceed from lower-left to upper-right may be placed in either slash (918) or X (920) inscribed squares; and, tile sections that proceed from lower-right to upper-left may be placed in either backslash (919) or X (920) inscribed squares. Square sections can be placed anywhere, as the square grid lines enable them.

With the disabling interpretation, a tile may not ‘cut’ an inscribed line. Thus, with the disabling interpretation: tile sections that proceed from lower-left to upper-right are prohibited from being placed in either backslash (919) or X (920) inscribed squares; tile sections that proceed from lower-right to upper-left are prohibited from being placed in either slash (918) or X (920) inscribed squares; square tile sections can be placed upon any square (in one sub-variation) or only upon empty (917) squares (in a second sub-variation).

(Such embodiments which include angled tiles can be titled 45vo″ or Avo″ in keeping with convention of the 2vo name, or ANGLED 2vo″)

MARKINGS ON BOARDS: Two schematic examples of how the markings (917-920) are applied to a grid, to control how angled pieces may be placed, are shown in layouts (801) and (802) of FIG. 8. In (802) the grid comprises right isosceles triangles, one quarter unit square in area, with their bases orthogonal to the sides of the square, as well as squares; in (801) right isosceles triangles, one half unit square in area, with their bases diagonal to the sides of the square, are added as well.

Markings are also placed directly on the board for additional purposes. In computer-mediated versions, this is more flexible because the markings are not permanent. Diagram (810) of FIG. 8 shows several alternative markings on a 7×7 board, but these are only exemplary and any type of marking is used to incorporate additional game features into various embodiments of the instant invention. Elements (811) and (812) each show an ‘X’ indicating a dead zone. The position of (812) is as far out from the corner as possible without permitting a 2×2 tile to be placed behind it. Two such tiles placed in diagonally opposite corners would reduce the maximum number of possible tiles at the end of the game from 9 to 7; and, that number works well regarding the avoidance of ties. If (811) is the lone dead spot then eight 2×2 tiles are possible and ties, with the most usual number of players (i.e., 2) are likely.

The other example markings shown are used in scoring at the end generation. Tiles or unit squares are counted at the end, in order to arrive at a final score to determine the winner. Element (813) indicates times 2′ and will multiply the value of the tile (its area, or if tiles have marked values, like Scrabble tiles) or the entire tile or territory count, if a player manages to cover this square at the end of the game. Similarly (814) and (815) ‘add 4’ or ‘subtract 8’ points, respectively, from the final score of the player who occupies those squares. Since, optionally, as many tiles as possible must be placed, it is possible to intentionally lose space, to advantage, if it will cost an opponent a net loss in their score. Lastly, for the 3D boards depicted in FIGS. 23 and 24, each level is, optionally, assigned a different value for scoring purposes.

ALTERNATIVE FIRST GENERATIONS: A number of the embodiments described, particularly those following with regard to marked tiles, work only, or better, with the first generation, using unit tiles. A use of these embodiments is to add variety to the first generation, and to have the game proceed from there, to later generations, as usual. (Alternatively, the same variations are applied, as appropriate, to any of the second or later generations as well.) Other alternative first generations used in this way include any game algorithm, now known or later developed, that will assign units of two or more colors to the spaces of the game board. Random placement of tiles will add an interesting element to the game. For example, a random-number generator is used in a computer-mediated embodiment. Or, grid spaces are specified by repeatedly throwing two special dice, with more than 6 sides, displaying numbers and letters representing coordinates of columns 1-7 and rows A-G, as shown in FIG. 15. Twelve-sided dice are available and, if used, the other five faces are blank, or have options such as ‘player's choice’, ‘roll again’, ‘same as other die’, etc. In standard embodimants, strategy in the first generation is reflected throughout the game; with a random first generation, that strategic ability is removed, and the skills needed for the second and subsequent generations are contemplated in isolation. Also, this can serve as an ‘equalizer’ by taking the multi-level strategic impact of the first generation play out of the hands of experienced players. Alternatively, embodiments of the invention, which comprise those disclosed variations suitable for alternative first generation play are, optionally, played as a first and only generation, and pieces or territory are tallied at the end of that generation.

MARKED TILES: FIG. 9 shows various marking that are, optionally, inscribed upon, or otherwise affixed to, tiles (depicted in double-lined boxes). Only unit square tiles are shown and, generally, these are placed down in generation one. However, markings are, optionally, made on any tiles as desired, and are played in any generation as appropriate. Further, these are examples; other marking are optionally used for similar or other purposes to incorporate other gaming elements into various embodiments of the instant invention.

Through the marking of an X (901) on tiles, or the use of tiles of a different color than any player is using, a ‘dead zone’ is indicated. As discussed, above, some dead zones may be temporary, being removed during play. Generally, however, one or more dead zone pieces are placed: prior to play according to some rule or diagram; as a requirement or by the players at their choice (if and/or where) before or during generation one play; with, or in lieu of, a standard move; or, otherwise. With a 7×7 board, if more than a few dead pieces are placed (say up to five or six at a practical maximum) the number of 2×2 tiles placable in generation three will possibly be severely reduced. Alternatively some dead pieces may be placed or others removed during generation two or three play. However placed or removed, a dead zone prevents any player's pieces from being placed and, usually, is permanent for the game. Thus, depending on how many and where placed, even a few single square dead zones may prevent 2×2 (or larger) pieces from being placed on the board. Such may be used to advantage by a player, for example as follows. Consider that during generation one Black has three out of four in a corner. If White takes the fourth, it is still likely that Black will be able to eventually get a black 2×2 tile in the corner at generation three. However, if White plays an optional one of a limited number of permanent dead zone unit tiles as the fourth in that corner, then neither player will be able to get their 2×2 tile there in generation three. Given that it is unlikely White will occupy the corner in the end game, it is to White's advantage to play such a spoiler tile.

Dead zones may also be ‘provisional’. For example, dead zone pieces may be colored like other tiles and also have an X on them. Each player will then get one or more X tiles of their own color and, optionally be required or permitted to, place them in generation one. These colored X tiles are dead zones for the other player(s); but, they are ‘free space’ for the player who placed them, able to be removed, and the vacated space moved into, at any convenient time. Other alternatives include placing one or more ‘provisional’ dead zone piece for your opponent, instead of, or in addition to, one, or more, for yourself.

Elements (902-908) depict a set of tile markings used to enclose a colony that, once consolidated, becomes the territory of the player completing it. Two basic ways for playing such tiles are described, following, but other variations are within the scope of the invention. In a first main embodiment such tiles are provided for both (all) players in their own colors. Then, when a player creates a properly enclosed and filled colony (or, just an enclosing border, with optional automatic fill-in of the enclosed empty space), the colony is complete and the player may proceed to start another colony. In this main embodiment, colonies are, generally, limited to tiles of a single color. An option permits colors to be mixed until a colony is complete, but this may be confusing and, is similar to the second main embodiment which is clearer. Another option permits removal of opponents' pieces within the completed enclosure as is done in ‘go’. Another option permits more than one simultaneous colony to be worked on by a particular player. In a second main embodiment, only one set of such tiles is supplied (in a neutral color, say white with black dots, and with players using tiles with true hues, say red and blue) and all players use them to grow one (or more) colonies. Once a colony is completed, the player who completed it replaces all black and white tiles in the colony with colored tiles of their own. Other options include the various contiguous placement alternatives as described elsewhere in this application.

In any event, the primary algorithm for using this set of tile markings to enclose colonies is as follows. A set of tiles is enclosed when all external sides: 1. have a dot; 2. abut an edge of the board; 3. abut an edge of another tile that has a dot on it; or, 4. abut a colored tile that has been converted to a particular player's consolidated territory. Several examples follow in diagram (1500) of FIG. 15.

In section (1501): tile A5 is of type (904); tiles A6 and A7 are either empty or, filled with type (908) as shown or, equivalently, with unmarked type (902, 201), depending upon the requirements of whether the consolidated area just needs to be bordered or filled; tiles B5 and B6 are of type (903); and, tile B7 is a tile that has already been consolidated and converted to a standard colored player's tile (201). Tiles A5, A6, A7 and B6 comprise the surrounded area just completed and which is, optionally converted and consolidated, or just left as is in favor of starting a new colony. Tile A5 has its own dots below and to the left, a board edge above, and an internal colony edge on its right; tile A6 has a board edge above, and internal colony edges on the other three sides; Tile A7 has board edges above and right, an internal colony edge on the left, and abuts a previously consolidated tile below; and, tile B6 has the dot of tile B5 to the left, its own dot below, in internal colony edge above, and abuts the previously consolidated tile on the right. Tile B5 is not part of the consolidated area in that, while it contributes to the border with its dot, it is not entirely within that border; it is vulnerable or open on its left and below. Now, if tile B5 had instead been a (904) turned with its dots down and to the left, it would (potentially) have been included in the consolidated colony. Reasons this was not done include: the (903) tile may have been placed early before the plan to consolidate the colony was formed; placed by another player in an attempt to block formation of the colony; another (904) may not have been available due to tile type scarcity; a strategic decision not to include tile B5 in the colony may have been made; or, this action would have put the lower dot in A5 within the colony and, in one optional rule about colony formation, internal dots are not permitted.

In configuration (1502), at C2: a (903) placed with its dot down would complete a colony consisting of the one cell D2; but, if the (903) is placed with its dot up, cells C1, C2 and C3 join D2 in the colony; if a (906) is placed with its open side up, the colony consist of C1, D2 and C3, but not C2 itself which is open at the top; if a (907) is placed there, it will make the colony of C1, C2, C3 and D2, or be disallowed because of the three internal dots thus created, depending upon which optional rule is applied. Alternatively, at B2: placing a (906) with dots up, left and right completes a colony of five cells (B2 and the previous four) unless C2 requires filling with a (902) or (980) first; and, placing a (905) with dots left and right sets the stage for a colony of six cells by extending to A2 with a (906) with dots up, left and right. And so on.

Configuration (1503) is complete and tiles D6, E6 and E7 are ready to be consolidated.

A (908) tile is, optionally, used to create a single-cell stand-alone colony; or, these are optionally used in place of (902) to depict internal cells of a colony. Similarly (907) is, optionally, considered a single-cell stand-alone colony. It may also be used to connect up to four partially completed colonies at once. In such a circumstance, so long as all four dots are used to complete lobes of the colony, even if and of the four dots of the (907) become internal at that move, it is, optionally, permitted (with a conceptual opening in the middle of the tile between the four dots making them external). Thus, along row G (1504), four single tiles are shown that each, in at least some alternatives, comprise one-celled colonies. As just explained, the (908) at G1 and (907) at G3 are, optionally, considered stand-alone colonies. The (906) at GS has three sides dotted and, against an edge, is completed; the (904) at G7 has two sides dotted and, against a corner, is completed.

In general, particularly if internal dots are not permitted, laying down border dots has two conflicting effects: the border protects the enclosed cells; but, it also limits the growth of the colony.

Markings (909) and (910) are the symbols for “male” and “female” respectively. Any two symbols could be used, but these are particularly ‘biological’. Alternatively, different shapes of tiles are used, for example (1009) and (1010) shown in FIG. 10. Each player is given 12 of each to place in generation one. These may be placed: without restriction; all the males first; all the females first; alternating; or, otherwise. As two pieces must be removed during a move in generation two (or later), for any particular move one tile removed must be male and the other female. The order required may be male first, female first, alternating, player's option, or otherwise. Or, on any move both must be either male or female, and this may be at player's option, or alternating. Larger tiles are, optionally, also supplied in marked, or rounded and rectangular, form, and played similarly.

Markings (911) and (912) show a “1” and “2” respectively, but these are examples. In one embodiment all 24 tiles are marked uniquely from 1 to 24. In another embodiment there are several sets with numbers; for example, four each of 1, 2, 3, 4, 5 and 6; etc. In any case, the pieces must be picked up, during generation two, in order: 1, 2, 3 . . . 24 in the first case; and, all the 1's before any 2, and so on, in the other cases. In an optional further restriction, the pieces must also be placed in the same order during generation one. The first of these restrictions makes the playing of generation two (but not three) determined, to at least some degree, at the end of generation one. However, the mechanical nature of generation two is offset by the additional attention to strategy that must be applied during the play of generation one.

Marking (913) shows an “A” but this is an example. Each player may be given 12 each of A and B; 8 each of A, B and C; 6 each of A, B, C and D; 12 of A, 6 of B, 3 each of C and D; etc. More than 24 pieces may also be given; for example, 12 each of A, B, C, D, F and F, etc. Grouping symbols, other than letters, may also be used. A set of lettered pieces must be placed on the board contiguously. That is, once a first “A” piece (of a given color separately, or for all players together) is placed on the board, a subsequent “A” piece can only be played in an open grid location that is also adjacent (just the four orthogonal positions or, optionally, any of the eight orthogonal and diagonal positions) to some other “A” piece. In a further optional restriction, the new piece must be adjacent to the last such piece played. ‘Colonies’ or ‘cultures’ of cells are, thus, built up. Options for continued play include: at any time a new letter may be played without the adjacency requirement, starting a new colony; a new colony may only be started when a (all) player(s) run(s) out of tiles of a given letter, and/or if a colony is ‘boxed in’ with no more adjacent open space; the remainder of tiles from a boxed in colony are then not used or, alternatively, (if there are not sufficient total tiles) they may be used to start a separate colony; if two, or more, lettered colonies (e.g., “A” and “B”) touch (again, options for just orthogonal, or both orthogonal and diagonal), then they combine and a tile of any involved letter may be played adjacent to another tile of any involved letter. Again, options include that a particular letter of a particular color is distinct; or colonies may include tiles of all colors with a particular letter. Also, larger pieces may have the same markings and similar placement restrictions; and/or, have the additional restriction that they may only replace smaller tiles of the same letter.

Another example, using lettered (or similarly marked in groups) tiles, embodying a process that is more societal or cultural (specifically economic or political) than biological or ecological, follows. This specific example employs: a 13×13 board; three generations utilizing tiles (201), (203) and (205); equal numbers of tiles marked with letters A, B, C & D and, four players; but, this type of process is also practiced in games with other elements and options. In particular, the rules defining the complementary groupings optionally increase in complexity (in concert with, or instead of, the geometric size or complexity of the tiles themselves) from generation to generation.

In generation one, each of four players has ten each of A, B, C and D tiles, as well as two N wild (201) tiles of their own color. These are placed as normally during generation one, leaving one open space. In generation two, each player must pick up four tiles, comprising a complementary grouping or ‘deal’, that consists of one each of A, B, C and D, where an N may substitute for one of the other lettered tiles. Optionally, these tiles must also consist of: all the player's own color; all of any one player's color; one each of each player's color; some other fixed or variable rule. After such a pick up of four tiles the player places one or more (203) tiles, if possible, which are similarly lettered and colored (where the move is, optionally, structured as UUDUU, DUUUU or otherwise, instead of UUUUD). At most 36 (203) tiles are placed, so it is recommended that each player have sufficient tiles of each letter and, optionally, a few N's as well. Generation three proceeds quite similarly to generation two except that the (205) tiles are colored, but need not be lettered. Alternately, the (205) tiles are also lettered, and only complementary groups of four count at scoring time.

Marking (914) is an example of a schematic ‘biomarking’ (a single celled organism with cilia) and may be either decorative or functional. Purely decorative markings will progress in complexity of organism depicted, or other pattern, as the size of the tiles progress; and are, optionally, different for each player. As an example of functional markings: some tiles (say half) are marked as cocci and the others as amoebas. Amoebas may be played singly as with the standard game; but, cocci can only be placed in a contiguous culture or colony, as with any of the options for the lettered groups described, above.

Marking (915) shows a clock. This is probably only playable on a computer display with the clock counting down; but, may also be implemented using the 24 numbered tiles on a physical board. In either case, the organisms represented by the tiles have a limited life that starts when they are placed down. Thus, although a player is not forced to pick up tiles in the order they were placed, he must pick them up fairly soon (say, within one or two moves of their assigned order) or they will ‘die’ at the end of that turn and vacate the board giving additional empty space for the opponent(s) to occupy. As an aid to tracking which tiles will die, a counter (such as beads or pegs used in pool or cribbage, or a mechanical ‘clicker’ for head counting, or an electronic timer's 1cd display) will indicate which numbered tile (plus any with lower numbers) is in danger. The count is incremented at each move, or at each round of moves, or after a timed period. Alternatively, a rack for removed tiles has a number printed next to each tile position so, as tiles are removed, moves are counted, and the adjacent number represents which numbered tile is in danger at the current move. If the first slot is labeled “1”, then no flexibility is permitted. However, if the first N are blank, and the N+first is labeled “1”, then there is some flexibility when removing tiles, and other tiles may be strategically favored for N moves. Placing the number labels on a movable slide permits the degree of delay/flexibility to be adjusted; and, providing separate slides for each player permits the delay to be used as a handicapping mechanism.

In an embodiment similar to the colony groupings described with the grouping letters, players must put down unmarked tiles in contiguous groupings and: may start a new colony at will; only after a colony reaches a certain minimum size; or, is forced to at maximum a size; and/or if a colony is ‘boxed in’ (perhaps with a missed turn as a penalty for poor strategy). With any of those alternative embodiments, the arrow tiles (916) are optionally used as follows. A further restriction is optionally applied that says, “not only must the new piece be adjacent to the colony, but to the last piece played.” In that case, the arrow on the previous piece played is pointed to the next open square (if not boxed in) which is chosen, as a way of reserving it so no other opponent will take it. They may however, take ‘next over’ spaces to force a block in, etc. If this ‘add on to last piece and reserve the space rule is at the players’ option, pointing the arrow ‘back the way you came’ indicates that the next move will be discontiguous (but, otherwise secret) and no next space is reserved. Alternatively, just a circle, or a face, or some other symbol (not shown) is used at the head of such a growing path, to memorialize the last tile added, which is now the position to be extended from. Then when the next move is made, the head marker is moved into the new space, and a blank tile put in its place.

MUTATIONS ON A THEME: The principle of evolution as manifested in many embodiments of the instant invention is that, in concert with growth and succession, what happens early on enables what happens in the middle game and that, in turn, determines what happens in the end game. This may be conceived of and enacted upon: as a simple gestalt visualization geared toward ‘clumping’ smaller pieces into sufficiently large masses that they will persist until the end; or, as a set of analytically strategic configurations implemented at each level that are intended to affect similarly particular configurations at each later level, with the goal of maneuvering your opponent into attrition; or, any number of other ways. However, these have in common that two (or more) players are competing, and that the players' actions are deterministic of the outcome.

What follows are a number of optional or alternative embodiments where a player's own actions are not entirely deterministic of the disposition of the tiles he places or removes. Rather, other players' actions, or random forces comparable to ‘mutation’, interfere with a player's game plan in unpredictable ways. These uncertainties must be taken into account when playing, even if they cannot be predicted precisely. The examples below, generally, will assume two players, black and white, on a 7×7 board, playing three generations, using (201), (202) and (203) tiles, respectively; but, are applicable to other embodiments as well.

In the most straightforward variation, player one plays white in generation one, black in generation two, and white again in generation three; and, vice versa for player two. Thus, at generation two, each player will attempt to ‘trash’ the strategic advantage of the set of pieces they are removing, and create as little strategic advantage for the pieces they are putting down. Consequently, at generation one, the goal is changed, from setting up a position that is maximally advantageous if managed correctly, to setting up a position that is maximally robust if mis-managed with extreme prejudice. Similarly, at generation two, the goal is changed to placing pieces in a way that they cannot be managed at all well. Optionally, the switching is done only at one generation or the other; and, for longer games, switching occurs at any combination of generations.

A variation on the above is for player one to place white pieces at generation one; and, to remove white pieces but place black pieces at generation two; and, vice versa for player two. A similar arrangement, between generation two and three, etc., can optionally be performed instead of, or in addition to, the one just described.

Another slightly different variation on the above is for player one to place white pieces at generation one; and, to remove black pieces but place white pieces at generation two; and, vice versa for player two. A similar arrangement, between generation two and three, etc., can optionally be performed instead of, or in addition to, the one just described.

These elements of these variations can be assembled in other combinations in the various generations as well.

The classes of embodiments just described are ‘all or nothing’ at each generation. Alternatively, in generation one, instead of giving ‘white’ 24 white pieces and ‘black’ 24 black pieces, each player may be given 12 of each, or some other combination such as 20 of your own and 4 of your opponent's pieces to place. Various optional rules will specify how much choice each player has to place the two types of pieces: your opponents first, last, alternating with your own, at your discretion, etc. At later generations, such partial ownership of tiles will be more confusing, and harder to manage, because it is not pre-determined how many tiles each player will be placing. Consequently, next will be described special sets of tiles that will enhance and enable such embodiments.

FIG. 10 depicts alternative rectangular tiles, each with a border area (1001, 1003, 1005 &1007) and central area (1002, 1004, 1006 &1008) of different colors. Two tiles, (1001/1002) and (1003/1004), are both of unit size but two different color schemes, which are shown by the four different patterns. More than two color schemes are supplied as needed, for additional players or other embodiments. For larger tiles, only the first color scheme is depicted but both (or more) are supplied. Further, the particular shape of the border and central areas are exemplary, other shapes or symbols (such as are used in FIG. 9) are used for the same purposes in alternative embodiments.

These tiles are put down according to the color of the border, and picked up according to the color of the central area Now, it is possible to use black and white for both color sets; however, for black/black and white/white tiles will look solid and black/white and white/black tiles will visibly exhibit two areas. If this is confusing, two sets of colors are assigned to each player: for example, the first player plays white and blue and the second player plays black and red. The unit tiles (1001/1002) and (1003/1004) are then colored black (1001) and white (1003) in the border area and red or blue (depending) in either of the central areas (1002 &1004). Thus, there are now four types of unit tiles: black/red, black/blue, white/red and white/blue. The first player puts down all the white tiles in generation one, and picks up all the blue tiles in generation two; the second player puts down all the black tiles in generation one, and picks up all the red tiles in generation two. The four color combinations may be supplied in equal numbers of 12 each; then each player in generation one will play 12 tiles for themselves and 12 for their opponent, when it is considered who will play those tiles in generation two. Alternatively, any ratio can be embodied, e.g., 20 of your own and 4 of your opponent, etc. The two types of unit tiles may be placed in generation one: in any order as desired by player, one per turn; alternating on odd and even turns; one of each at each turn; etc.

Similarly, with three, four or more players. However, in that case there are probably not enough colors for each player to have two. For example, if there were four colors black, white, red and blue, each player would get 12 tiles in generation one. Of the 12 tiles for the black player all 12 would have black border areas, and 3 of each central area would be black, white, red and blue; thus, three of black's tiles would be black/black and appear solid. A thin white border between the border and central areas would fix this anomaly. Similarly for the other three players. Alternatively, for black, four of each would have central areas in red, white and blue. In that case, in the first generation, each player plays four pieces for each opponent and none for themselves. Other ratios and variations are also within the scope of the instant invention.

Returning, now, to the embodiment with two players and four colors, the next sized 2×1 pieces (1605/1006), comparable to (202), would be colored red or blue in the border area and black or white in the central area. Thus, in a move where two 1×1 tiles with blue central areas are picked up, one 2×1 tile with a blue outer area is put down; and, it will have either a black or white central area. Since in the second generation it is not guaranteed that each player will be placing exactly 12 tiles, some other mechanism is needed for distributing among ‘your’ tiles and ‘their’ tiles. One such mechanism is to alternate, or do one out of 3 or 4 for your opponent, etc.; or, each player may place only their own or only their opponpnt's pieces in generation two; or the border area colors of the tile(s) vacating is deterministic of whose piece is placed. In the third (and finally, for this example) generation the 2×1 tiles are picked up by their inner colors and single colored 2×2 pieces, of white for player one and black for player two, are placed.

Alternatively, these tiles may be used like (909) and (910). The outer area determines the player (white or black) the inner area determines the gender (red or blue) at all generations. Players, at each generation have equal numbers of each gender, and the placement and pick up rules are any of those discussed earlier. More than two inner colors are used for similar purposes. For example, four color ‘genders’ are put down and/or picked up, in groups, or cyclic order on moves alternating in cycles of four.

Shapes, several colors, and symbols can be combined on any of the tiles to create other variations for additional embodiments.

An embodiment employing both different colors and different shapes (or related sets of colors, or markings) permits complete or partial (depending on whether scoring is individual or by team) cooperative team play as follows.

Two teams of two players each will be described, but this embodiment is optionally practiced with more teams (basic colors) and/or more members per team (distinguished by additional shades, shapes or markings on tiles, for example).

The members of a team will share a basic color (e.g. redish or bluish) but have distinct shades (e.g., for three each: brick red, pink and orange vs. royal blue, sky blue and green); or, will have identical colors but two (or more) distinct shapes (e.g., 1×1 circles, 2×1 ovals and 2×2 circles vs. 1×1 squares, 2×1 rectangles and 2×2 squares).

In this example, player 1 is red round, player 2 is blue round, player 3 is red square, and player 4 is blue square.

Play progresses as usual for four players. However, for generations two and three (and beyond) players may optionally, or are required to, adhere to rules for removing pieces which include, without limitation, for example:

    • pick up two of their own pieces;
    • pick up two of their partners pieces;
    • pick up two of the same, either their's or their partner's;
    • pick up one of each, partner's first;
    • pick up one of each, partner's last; or,
    • pick up one of each, in either order.
      The choice is alternatively fixed and determined before play begins, entirely at players' option, based on the roll of a die or other random choice device, or otherwise.

Generally, like with bridge, communication between partners as to strategy is forbidden or, at least, sub rosa; but, open verbal cooperation and planning is optionally permitted.

Scoring is done on a team basis (complete team play) or per player (cooperative but still competitive partial team play).

Thus far, it is the other player(s) who are interfering, or cooperating, with a particular player's strategy. Next will be described how random elements, more akin to ‘mutations’, are incorporated into the instant invention. It is practical, with computer-mediated versions, to apply these features as entirely random events that may happen at any time to any tile, and the use of random number generators to select the tile involved, and the probabilistic distribution of events that happen, are well known and within the ken of those skilled in programming, in general, and in programming games of chance in particular. Therefore, what follows, is an example of how to implement such elements as components of a physical board game.

If a ‘mutation’ is to be applied to any tile, it is selected by the use of the 1-7 and A-G dice, as described above, to select a grid position, or by use of one or more dice to specify a number or symbol that is imprinted on the tile. Alternatively, mutations are limited to one or a few tiles which have a special symbol (e.g., a red dot) and an optional sequence designation (e.g., the numbers 1, 2, 3 or 4) to distinguish them. Then, at an appropriate time, the appropriate player rolls a special die. For example, between generations one and two, each player rolls the special die in alternating fashion four times each, for the four red-dotted and numbered tiles each player placed. The die, for example, is a twelve-sided kind, with: six sides indicating nothing happens, and the tile is replaced by a standard blank tile of the same color; three sides indicating that the tile ‘mutates’ and is replaced by a plain tile of the other player's (the player to the right, if more than two players) color; two sides indicating that the tile ‘dies’ and the grid space is vacated; and, the final side indicating that the tile becomes a ‘bio-hazard’ and is replaced by a dead zone tile.

Generation Two then Proceeds.

Such ‘mutations’ are optionally applied at later generations as well; and, occur between generations, as described, or at any time.

PROGRESSIVE PLAY WITH UNIFORM TOKENS: Although not ideal, special markings on the board permit 2vo to be played with readily-available or inexpensive tokens of two colors (although shades of those colors, or additional colors will help prevent confusion) but with uniform or near uniform (again to help avoid confusion) size. These may consist of tiddlywinks, checkers, coins (pennies and dimes), ‘go’ stones, or any other available tokens.

FIG. 11 shows that each grid unit square has a small square inscribed at its center, one of which is designated (101). In generation one, tokens are, placed here. For generation two, tokens on the central squares are removed and, for any two empty squares that are adjacent horizontally (1102) or vertically (1103), a token is placed on the small 2×1 rectangle crossing their common edge, indicating that both those squares are occupied and may not be occupied via any of their remaining three edges. For generation three, the tokens over the rectangles are removed and, for any 2×2 cell of squares that are unoccupied, a token is placed on the larger square (1104) that is at the grid intersection common to the four, and indicates that all four are occupied. Thus, the size, shape, and placement of the symbols (1101-1104) they occupy indicates the use of tokens that are uniform in size, rather than embodying the use information into the shape and size of the tokens themselves.

Since the tokens are of uniform size, to prevent confusion, as an option, for example, black and white are used in generations one and three, and red and blue are used in generation two. Or, dark, medium and light shades of each color are used progressively. Alternatively, if available, small, medium and large (but not in the correct shape and size relationship, such as small medium and large ‘go’ stones) tokens of the same two colors are used for the same purpose.

Alternatively, the markings of FIG. 11 are constructed as indentations or holes in the board and each tile has an appropriate complementary peg in its bottom (with an optional similar peg on its top serving as a handle). This guarantees that the tiles are placed properly during play, no matter what their size.

GENERALIZATION TO THREE DIMENSIONS AND BEYOND (AND BEHIND): The three-dimensional and four-dimensional embodiments that result from the following theoretical discussion are likely to only be practicable to implement in a computer-mediated version, where internal elements of the playing volume are displayed translucently, transparently, or removed transiently, to see internal structure; and where it is easy to vary the orientation of the playing volume.

The previous section disclosed, and depicted in FIG. 11, that the basic game as described:

    • is played in two dimensions, on a grid of squares;
    • alternates two types of tiles (squares and flat ‘bricks’ or 2×1 rectangles) and then returns to larger squares (2×2) as the tiles grow;
    • the intermediate rectangular tile configuration can be oriented in two orthogonal directions, while the square tile configuration that starts and ends the sequence is orientable in only one way;
    • comprises three generations as the tiles return to the first configuration (but larger);
    • the tiles comprise size 1, 2 and 4 units of area as generations progress; and,
    • the three generations are played on the: one center, four edges, and four corners of the grid squares.

FIG. 12 depicts a generalization where the game:

    • is played in three dimensions, in a grid of cubes;
    • alternates three types of tiles (cubes (1×1×1), bricks (1×2×1), and flattened cubes (2×1×2)) and then returns to larger cubes (2×2×2) as the tiles grow;
    • the two tile configurations intermediate to the cubes can each be oriented in three orthogonal directions, while the cubic tile configuration that starts and ends the sequence is orientable in only one way;
    • comprises four generations as the tiles return to the first configuration (but larger);
    • the tiles comprise size 1, 2, 4 and 8 units of volume as generations progress; and,
    • the four generations are played on the: one center, six faces, twelve edges, and eight corners of the grid cubes.

In FIG. 12, element (1201) shows how, in the first generation, an occupied cube would be depicted in a computer display by placing a solid or luminous small cube within the partially or entirely transparent grid cube. Note that each cube is occupied, or not, independent of any adjacent cube, and the unit is orientable in only one way.

Element (1202) shows how, in the second generation, an adjacent pair of cubes is marked as occupied by placing a small similar shape spanning the face the two cubes share in common. Note that: (playing volume edge effects aside) any given cube can participate in any of six such pairings, one via each face, but only one at a time. Further note that the 2×1×2 rectangular solid so made can be oriented in any of three orthogonal orientations.

Element (1203) shows how, in the third generation, four adjacent cubes in a 2×2 matrix form a 2×1×2 rectangular solid, which is marked as occupied by placing a small similar shape spanning the edge that the four cubes share in common. Note that: (playing volume edge effects aside) any given cube can participate in any of twelve such pairings, one via each edge, but only one at a time. Further note that the 2×1×2 rectangular solid so made can be oriented in any of three orthogonal orientations.

Finally, element (1204) shows how, in the fourth generation, eight adjacent cubes in a 2×2×2 matrix form a 2×2×2 cube, which is marked as occupied by placing a smaller (but larger than the first internal) cube spanning the corner that the eight cubes share in common. Note that: (playing volume edge effects aside) any given cube can participate in any of eight such pairings, one via each corner, but only one at a time. Further note that the 2×2×2 cube so made is orientable in only one way.

This pattern and play algorithm can be generalized to four dimensions (with the fourth dimension being conceptualized as temporal or hyper-spatial) or more. It may also be applied to other grids; for example, the equilateral triangles in two dimensions of FIG. 18, can be generalized to tetrahedrons in three dimensions, etc.

The scheme of 2D squares and 3D cubes is generalizable to N dimensions as follows:

Let D equal the number of dimensions. C(M,N) is the combination function, being ‘the number of ways M objects can be taken N at a time’; or, (M!)/(N!(M−N)!), where ! is the factorial function. The results of C(M,N) are conveniently arranged in Pascal's triangle where the row, counting from the top, starting at 0, is M; and, the entries in a row, counting from the left, are 0 through N. For the object that, in two dimensions is a square and, in three dimensions is a cube, the following tables are developed.

D = 1ITEMN = 0N = 1
T11C(D, N)11

D = 2ITEMN = 0N = 1N = 2
T21C(D, N)121

D = 3ITEMN = 0N = 1N = 2N = 3
T31C(D, N)1331

D = 4ITEMN = 0N = 1N = 2N = 3N = 4
T41C(D, N)14641

Each of the four tables has the same structure. The upper-left corner states the number of dimensions, D. Then, labels aside, there are columns for N=0 through N=D. The entries in rows TD1, TD2 and TD3 are the combinatorial function, C(D,N), 2N and the product of those two terms, respectively.

For the one-dimensional case, D=1, TABLE I (for example, see FIG. 12, diagram (1206)):

    • the entries in row T13 are 1 and 2; and, these correspond to the 1 center and 2 ends of the unit line segment.
    • the entries in row T11 are 1 and 1; and, these correspond to the possible orientations for the unit line segment, and the doubled line segment as the segments progress.

For the two-dimensional case, D=2, TABLE II (for example, see FIG. 11):

    • the entries in row T23 are 1, 4 and 4; and, these correspond to the 1 center, 4 sides and 4 corners of the unit square.
    • the entries in row T21 are 1, 2 and 1; and, these correspond to the possible orientations for the square, ‘brick’, and large square as the tile types cycle.

For the three-dimensional case, D=3, TABLE III (for example, see FIG. 12, diagrams (1201-1204)):

    • the entries in row T33 are 1, 6, 12 and 8; and, these correspond to the 1 center, 6 faces, 12 edges and 8 corners of the unit cube.
    • the entries in row T31 are 1, 3, 3 and 1; and, these correspond to the possible orientations for the cube, ‘brick’, ‘flat tile’ and large cube as the solid tile types cycle.

For the four-dimensional case, D=4, TABLE IV:

    • the entries in row T43 are 1, 8, 24, 32 and 16; and, these correspond to the 1 center, 8 faces, 24 edges, 32 corners and 16 hyper-corners of the unit hyper-cube.
    • the entries in row T41 are 1, 4, 6, 4 and 1; and, these correspond to the possible orientations for the five hyper-tiles as they double and cycle back to hyper-cube.

As far as practicable embodiments: a 7×7×7×7 hyper-cube is displayed as seven separate, but ordered, 7×7×7 cubes in a computer-mediated navigable display. See FIG. 12, diagram (1205). Further, a one-dimensional version of the game can be played, simply, as expected: on a board of 1×N array of squares (where values for N could comprise any number, but 13, 15, 17, 19, 21 and 23 are reasonable.) Two (or more) players alternate placing 1×1 units (1212 &1214) in generation one. In each subsequent generation: order of play is reversed; the length of the 1×N tiles are doubled; the previous tiles are removed and the larger tiles (1211 &1213) deposited in the UDU method, growing an occupied area (e.g., from 1214), by a factor of two, to the left (L) or right (R), or placed anywhere there is room. The player with the most tiles at the end wins. See FIG. 12, diagram (1210) which is not fully populated with tiles, only showing one single and one double tile for each of two colors.

In 2vo, the unit 1×1 square tile is doubled in one dimension leading to a 2×1 ‘brick’, and then doubled in the other dimension returning to a 2×2 square shape. This characteristic trait, that the alternate generation tiles (at least) are self-similar at increasing scale, is shared by several two-dimensional embodiments, and is also a characteristic associated with “fractals”. Some specific tile pairs (which are extensible onward) that exhibit this characteristic are: (201) & (203); (202) & (204); (501) & (508); (502) & (509); (1701) & (1710); (1901) & (1903); (1902) & (1904); etc. In three dimensions, the self-similarity is, generally, exhibited every third generation; and, in four dimensions, every fourth generation.

ADDITIONAL EXAMPLE EMBODIMENTS NOT BASED SOLELY ON THE UNIT SQUARE: The game is generalizable in other ways as well. For example, in FIG. 17 are shown a set of tiles that are used to play an additional embodiment exhibiting binary geometric growth, or doubling. Although this embodiment is playable on the same type of square unit grid, the base tile unit here (1701 or 302) is an “L”, shown in a double line, made of three unit squares shown in single lines that would not necessarily be drawn on the tiles. Like the squares in (201) and (203) appearing at four times the area in odd alternating generations, the “L” also appears in odd alternating generations at four times the area. Tile (1710) shows the third generation “L” tile with, again, the single lines shown for illustration only. In the intervening even generations, tiles of intermediate area (comparable to (202) for example) are used, the most straightforward embodiment of which is (1704). However, tiles (1702) through (1709) show a number (not necessarily exhaustive) of tiles of the correct area ratio, and any combination of these in any number are supplied as desired.

FIG. 18 shows an alternative board (1800) that uses equilateral triangles, e.g., (1801), in lieu of squares. Again, any number of players (within reason) can play; but, this board configuration is particularly inviting of three players. The triangular board can also be of any reasonable size; and, the number of grid units is the square of the number of triangles on the edge. Board (1800), for example, has 7 triangles on each edge and has 49 units total, like the 7×7 square board. A board with an edge length of 8 would have 64 units; with an edge length of 9, 81 units; with an edge length of 10, 100 units; etc.

FIG. 19 shows tiles suitable for playing on a board such as is shown in FIG. 18. Tiles are shown in solid lines, and the grid units shown in dashed lines for illustration only. Tile (1901) corresponds to the grid unit to be used in generation one, comparable to the square (201) tile. Tiles (1902), (1903), (1904) and (1905) correspond to (202), (203), (204) and (205) respectively; and, the same alternating shape, doubling in area, relationship in the triangles and rhombuses can go on indefinitely, just as with the squares and ‘bricks’.

With the board of length 7 and area 49, only about 9 (1003) tiles will fit at generation three, or one (1905) tile will fit at generation five. With the three generation version, this will easily result in a three-way tie, but is unlikely to for two players. The board of length 8 and area 64 will hold, at a maximum, 16 (1903) tiles or 4 (1905) tiles, better numbers for three players.

An alternative set of tiles, roughly comparable to the linear growth embodiment, comprises tiles (1901), (1902), (1906), (1903 and/or 1907) and (1908) of areas 1, 2, 3, 4 and 6 grid units, respectively, on successive generations. With the 49 triangle board, six (1907) tiles would fit at generation five; with the 81 triangle board, 10 would fit. For triangular (as well as other) embodiments of the instant invention, all of the myriad variations of board size and shape, tile size and shape, tile supply, markings on the board and tiles, adjacency and timing rules, etc., that have been described with square embodiments, are practiced as options.

An additional way to extend the biological or evolutionary metaphor is for the tiles—in addition to, or in lieu of, growing and/or becoming more complex in shape—to exhibit some form of ‘behavior’ as the ‘organism’ evolves. An example follows, as shown in FIGS. 20 and 21.

FIG. 20 shows a pattern, with two types of unit squares, that is instructive for playing another embodiment of the instant invention; however, it is also an option to play this embodiment without the darkened type of squares and no restrictions on the placement of the third generation tiles. The pattern has two mirror-image configurations (2001) and (2002); and, a board is, optionally, inscribed with areas of each The pattern of (2001) is shown, enlarged, in (2003). Note that, edges aside, each 2×2 cell of white squares is surrounded by four black squares; and, each black square is surrounded by four 2×2 cells of white squares.

This embodiment is, optionally, played with tiles (201) and (202) in generations one and two as usual. However, in generation three, the tiles shown in FIG. 21 are used instead of (203) tiles. The tiles are five unit squares in area and comprise a 2×2 white cell and an attached black square. Shown at (2101), (2102), (2103) and (2104) are such a tile, polarized suitably for the pattern shown in (2001), in four orientations at 90° rotations. Mirror image tiles are optionally supplied, but are not shown. Tiles are supplied only in one orientation, or in both; both players are supplied with the same orientation, each with only one, or each with both. If the board is unmarked the (210x) tiles are placed without restriction; if the board is marked with (2001) and/or (2002) patterns, tile placement is restricted to correspond with the pattern; or, alternatively, if the board is marked, tile placement is unrestricted, but only the tiles that correspond to the pattern will “spin”. The embodiment where tiles “spin” is more practical with computer-mediated display.

With the “spinning” tile embodiments, a 2×2 cell of white squares and the four surrounding black squares must be empty in order to place the body of a (210x) tile because, at each move (a move comprising one player's turn, or both players' turns) the tile will rotate 90°. However, optionally, so long as the tile can be placed on a move, the additional surrounding black squares are vacated only as they are needed during the next few moves. Direction of tile spin is: uniform for all tiles; dependant on the tile (e.g., by color, ‘gender’, or polarity); player's choice; assigned by the computer; or, determined by some random mechanism such as the roll of a die; etc. Spin speed is, optionally, also variable to values other than 90° per move. Particularly if computer-mediated, these elements are, optionally, varied dynamically.

In (2105) two pairs of tiles are shown. On the left (2106), if both (2107) on top and (2108) on bottom are turning counterclockwise their black appendages will collide at (2109) on the next 90° rotation (or the third rotation in the clockwise direction). However, if (2108) spins at half the speed of (2107) they will not collide. On the other hand, on the right (2110) the top tile (2111) has its black tooth in the overlapping square (2113) and, so, it will be vacating that spot as the tooth of the bottom tile (2112) moves into the conflicting square, and collision is avoided.

If a tile hits an obstacle (any type of tile present) there are several alternatives: the tile may knock off what it hits and/or itself be knocked off the board; the ‘junior’ (or, alternatively, ‘senior’ in size, or time on the board) party may be eliminated; the tile may simply stop spinning permanently, or until the blockage is removed; etc.

Spinning tiles, optionally, have some scoring or other advantage. It may be as simple as spinning tiles are worth extra points; or, tiles that ‘mesh’ with other tiles of their own player's color gain in value, while ‘meshing’ with other player's tiles reduce value. Further, the advantage is, optionally, that at the end of the game a pattern of spinning gears is established. If a player is able to place a marker on one of the teeth of one of the (or only one of his own) gears, it is transported and deposited, to be picked up by the tooth of some other gear, and so on. A goal (for example, being able to transport the marker from one edge of the board to the opposite edge), if established, creates a winning strategy (for the first to achieve it), or scores additional points.

FIG. 22 depicts two repeating grids composed of more than one shape. These are illustrative, non-limiting examples only; and, embodiments of the invention are implemented by use of, or combined with, any other pattern or algorithm (decorative or mathematical, uniform or non-uniform, periodic or non-periodic, etc.) that tiles the plane, or fills space of three or more dimensions. Grid (2201) is composed of semi-regular octagons and squares; grid (2202) is composed of the same semi-regular octagons and rhombuses twice the area of the squares. Dotted lines show how a grid of unit squares is, optionally, overlaid on either grid. Also shown is a set of tiles, any combination of which (optionally, in combination with other tiles described herein such as (202), etc.) is used to play embodiments of the instant invention on grids (2201) and/or (2202), comprising: a right isosceles triangle (2203) one half unit in area; the unit square (2204), the same as (201); a rhombus (2205) two units in area, the same as (506); a trapezoid (2206) two units in area, the same as (505); a 3×1 bar (2207), three units in area, the same as (301); a cross (2208), five units in area; and, the semi-regular octagon (2209), seven units in area.

FIG. 23 shows a straightforward physical (or computer-mediated) embodiment of the instant invention played in three dimensions. This is a non-limiting example only. Board (2300) is a ‘wedding cake’ structure comprising: a 7×7 grid of unit squares; onto which has been centered a 5×5 grid of unit cubes; onto which has been centered a 3×3 grid of unit cubes; onto which has been centered a single unit cube. This comprises a mound, of 35 unit cubes in volume, onto which an additional 112 units of volume are to be stacked to complete a 7×7×3 unit rectangular solid. No parts of a tile may exceed this boundary; and, each generation is over when no more tiles in play can be added. The solid ‘tiles’ used are: a unit cube (2301) at generation one, a 2×1 ‘brick’ at generation two; and, an “L” of three units in generation three.

In the first generation, unit tiles (2301) are added according to the following rule: a tile may be placed atop an empty square face of the board (except the central tier), or atop a face of another tile of the same color.

In the second generation (2301) tiles are removed and (2302) tiles are placed in the UDU (or other) method already described; however, it is not required to place every (or any) tiles on a particular turn This is important because, in generations two and three, the structure already made is in the process of being dismantled from smaller tiles and re-built from larger tiles. On any given turn, it may be desired (or not) to do more dismantling and ‘hit bottom’ before building up. If this is not done, smaller tiles are trapped below larger tiles; and, this may or may not be desired strategically. The rule for placing the larger tile is that, in whatever orientation it is placed: it may not exceed the edges of the 7×7×3 space; it, optionally, must be entirely supported, with no overhangs; and, optionally, each of its bottom squares must rest on squares of board, or top squares of tiles of its own color.

The third generation follows the same basic rules except that if a single unit tile was trapped and now exposed it (or two of them) are, optionally, removed in lieu of a brick on either ‘up’ phase of a move.

The bricks (2302) are laid flat, or stood on end. The L's (2303) are laid flat and ‘wrapped around’ a corner; are laid on their back with a foot in the air; or, are even stood on a foot, with an optional requirement that the overhang is supported by the board or another tile of the same color.

The winner is, optionally: the player with the most volume; or, the player with the most squares among the 48 units of the top surface (excluding the central tier) of the 7×7×3 rectangular solid.

FIG. 24 shows non-limiting examples of more general shapes of a three-dimensional board and tiles, that are representative of those used in alternative embodiments of the instant invention.

Additionally, any three-dimensional board is, optionally, played with a single layer of solid tiles, or flat tiles. Additional rules optionally determine how different levels or ‘geographic features’ of the terrain are to be played (e.g., lower levels before higher levels) and scored (e.g., extra points for higher altitudes).

BOARD WRAP-AROUND: Lastly, particularly when embodied as a computer-mediated game, the instant invention optionally employs exotic rules for board wrap-around. Normally, at the edge of a board, pieces are not permitted to hang over the edges, and certain strategic factors come into play at edges and corners. However, through the use of computer-mediated displays, situations that are physically impossible are implemented.

The first, most straightforward, is board wrap-around in either of the two directions. In that case, a piece that partially ‘hangs off’ the right side of the board (not normally permitted) shows up (to that same partial extent) on the left side of the board, and vice versa. The technique is optionally applied horizontally (left and right), vertically (top and bottom) or both. This mathematically creates a cylinder in one direction or a torus in both.

The cylinder is, optionally, projected on an annular board with a circular hole in the middle around which is the top of the cylinder; the bottom is at the outer edge of the annulus. This requires different sized tiles, as the circumference of the rows increases; however, if the game is implemented on a computer, the size of the tiles is easily adjusted accordingly. Spirals and sunflower-like patterns are also, optionally, applied to such boards.

In a somewhat more exotic ‘twist’ as the board is wrapped-around in either direction a half-twist (as in a Moebius Strip) is applied and, thus, a game piece that hangs off the top (for example) not only re-appears at the bottom of the board, but on the back as well. Further, because of the twist, the order of rows (or columns) is reversed. Because the computer is not restricted to the physical geometry of the twist, the move to the back and the reversal of rows can be implemented separately or together. A different mode is optionally applied to each direction, or each edge.

Further, other ‘impossible’ options for continuity can be mathematically implemented, and displayed, by the computer. Just two more examples are disclosed, but the inclusion of any such exotic variation is intended to be within the scope of the instant invention. First, 90° continuity is implemented by connecting (for example) the right edge to the top of the board. Finally, instead of the reversal of rows (columns) in the Moebius option, a ‘spiral’ option connects, for example, the right of each row to the left of the next lower row, and the right of the bottom row to the left of the top.

A last embodiment that is impractical, if not quite impossible, to implement without the use of a computer-mediated display involves continuity of tiles, rather than the board. In this embodiment, tiles at a particular phase are placed, and groups of adjacent tiles ‘merge’ into a continuous area Then, during a subsequent phase, sub-sets of, such an area, within boundaries of permitted configurations, are removed from the area, without regard for the boundaries of the tiles that were placed to create the area.


The graphics and layouts of boards, graphics and configuration of pieces, algorithms and rules of play, steps described and/or depicted in any flow diagram, and other elements disclosed herein, are exemplary. A number of alternatives for each element have been disclosed, as have specific choices of alternatives comprising some specific preferred embodiments. To whatever degree these alternatives are not in conflict, any and all of the alternatives for any element are practiced, in any combination, with any and all of the alternatives for other elements, in order to create alternative preferred embodiments of the instant invention. Furthermore, certain steps or other elements may be arranged differently, combined, separated, modified or eliminated entirely, without deviating from the intended scope of the invention.

Further, these elements can be combined with elements of other games, now in existence or later developed, without deviating from the intended scope of the invention. Additionally, any method of manufacture, publishing or distribution of physical game boards and pieces used to play such games, now known or later developed, is intended to be within the scope of the instant invention. Similarly, any method of designing, displaying, distributing or programming computer-mediated versions of the instant invention (including but not limited to, artificial intelligence to produce a version where a computer plays a human, stereographic display, or versions where several players communicate over a network) now known or later developed, is intended to be within the scope of the instant invention.

The contents of the disclosure of this patent document, and the accompanying figures, is copyright to the inventor. The copyright owner has no objection to the facsimile reproduction of the patent document or the patent disclosure, as it appears as issued by the Patent and Trademark Office, to the extent permitted by law. Written permission of the copyright holder must be obtained for any other use. Copyright holder otherwise reserves all copyright rights whatsoever, including the right to excerpt, compile or otherwise alter or adapt, or make any other use of, this information.

Further, the names 2vo, 1vo, Lvo, LINEAR 2vo, 45vo, Avo, ANGLED 2vo, other names used herein, the numeral and subscripted text naming convention, and any other trademarkable elements are trademarked to the inventor.

In any event, any publication of or about any of the information contained herein must contain appropriate patent, trademark and copyright notices.

It will thus be seen that the objects set forth above, among those made apparent from the preceding description, are efficiently attained and certain changes may be made in carrying out the above method and in the construction set forth. Accordingly, it is intended that all matter contained in the above description or shown in the accompanying figures shall be interpreted as illustrative and not in a limiting sense.