Title:

Kind
Code:

A1

Abstract:

Six puzzles for constructing cube packings and for tiling sets of squares are composed of hierarchically structured sets of rectangular blocks of length and width equal to an integer multiple of the block thickness. For five of the puzzles, it is required that the blocks be arranged to pack a single cube. For two of these five, it is further required that a smaller cube, composed of a specified subset of the pieces, be concentrically nested in the interior of this cube. Included in the inventory of blocks for the sixth puzzle are two small cubes; it is required that the entire inventory of blocks be divided between two cube packings of the same overall size. The blocks of the invention may be used as recreational puzzles, as educational tools, for esthetic purposes, and for a variety of other uses.

Inventors:

Schoen, Alan H. (Carbondale, IL, US)

Application Number:

09/810510

Publication Date:

11/01/2001

Filing Date:

03/16/2001

Export Citation:

Assignee:

SCHOEN ALAN H.

Primary Class:

International Classes:

View Patent Images:

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Primary Examiner:

WONG, STEVEN B

Attorney, Agent or Firm:

Henry C. Nields (Westboro, MA, US)

Claims:

1. A set of rectangular blocks for packing a cube, and for other purposes, said set comprising at least one specimen of every block of unit thickness whose length and width assume a continuous sequence of values within the range one and seven.

2. A set of nine rectangular blocks in accordance with claim 1 for packing a 4×4×4 cube, and for other purposes, said set comprising one specimen of every block of unit thickness whose length and width assume every integer value from one to four inclusive, with the single exception of the 1×1×1 block, which is omitted from the inventory of blocks.

3. A set of ten rectangular blocks in accordance with claim 1 for packing a 5×5×5 cube, and for other purposes, said set comprising one specimen of every block of unit thickness whose length and width assume every integer value from two to five inclusive.

4. A set of nineteen rectangular blocks in accordance with claim 1 for packing a 7×7×7 cube, and for other purposes, said set comprising one specimen of every block of unit thickness whose length and width assume every integer value from two to seven inclusive, with the exception of the 1×6×7 and 1×7×7 blocks, which are omitted from the inventory of blocks.

5. A set of eighteen blocks in accordance with claim 1 for packing two 5×5×5 cubes, and for other purposes, said set comprising two 3×3×3 cubes, one specimen of every rectangular block of unit thickness whose length and width are equal and assume every integer value from two to five inclusive, and two specimens of every block of unit thickness whose length and width are unequal and assume every integer value from two to five inclusive.

6. A set of thirty-six rectangular blocks in accordance with claim 1 for packing a 9×9×9 cube, and for other purposes, said set comprising one specimen of every block of unit thickness whose length and width are equal and assume every integer value from two to seven inclusive, and two specimens of every block of unit thickness whose length and width are unequal and assume every integer value from two to seven inclusive, a selected ten-block subset of said thirty-six blocks being distinguished from the remaining blocks by color, texture, or material and capable of being sequestered to constitute a 5×5×5 cube nested concentrically in the interior of said 9×9×9 cube.

7. A set of thirty-six rectangular blocks in accordance with claim 1 for packing a 9×9×9 cube, and for other purposes, said set comprising one specimen of every block of unit thickness whose length and width are equal and assume every integer value from two to seven inclusive, and two specimens of every block of unit thickness whose length and width are unequal and assume every integer value from two to seven inclusive, a selected nineteen-block subset of said thirty-six blocks being distinguished from the remaining blocks by color, texture, or material and capable of being sequestered to constitute a 7×7×7 cube nested concentrically in the interior of said 9×9×9 cube.

Description:

[0001] 1. Field of the invention

[0002] This invention relates to cube packings by assemblies of convex rectangular blocks of prescribed shapes. Practical applications of this field include the production of toys, games, and educational tools.

[0003] 2. Description of the Prior Art

[0004] A well-known puzzle based on the assembling of a set of blocks of prescribed shapes to pack a cube without gaps is Piet Hein's Soma puzzle, which was marketed for several years by Parker Brothers. In contrast to the puzzles described in the present invention, however, all of the blocks of the Soma cube are non-convex. Two of the advantages of convex blocks are ease of manufacture and—because of their simple shape—the creation of the attractive but mistaken illusion that the puzzle of which they are the constituent parts must therefore also be simple and easy to solve.

[0005] Two examples of cube puzzles made from convex rectangular blocks are the Slothouber-Graatsma Puzzle and Conway's Puzzle. The nine-block Slothouber-Graatsma Puzzle consists of six 1×2×2 blocks (‘squares’) and three 1×1×1 blocks (‘small cubes’), which can be assembled to pack a 3×3×3 cube. Since this puzzle is very easy to solve even without any special clues, it is of interest only because of the underlying mathematical theory (‘3-dimensional stained-glass window theory’). Application of this theory leads to the conclusion that the three small cubes must form a linear chain extending from one corner to a diagonally opposite comer of the 3×3×3 cube (‘parent cube’), the small cube in the middle touching the other two small cubes only at their respective corners (cf.

[0006] It is only slightly more difficult to find a solution quickly to Conway's eighteen-block 5×5×5 cube puzzle, once the same 3-dimensional stained-glass window theory is applied. Conway's puzzle consists of thirteen 1×2×4 blocks, one 2×2×2 block, one 1×2×2 block, and three 1×1×3 blocks (‘long bars’). From application of the theory, it follows that the three long bars must be placed inside the 5×5×5 cube in a connected chain (cf.

[0007] Publications disclosing prior art include the following:

[0008] “Mathematical Gems II”, Ross Honsberger, 1976, Mathematical Association of America, ISBN 0-88385-319-1

[0009] “Tilings and Patterns”, Branko Grünbaum and G. C. Shephard, 1987, W. H. Freeman and Co., New York, ISBN 0-7167-1193-1

[0010] “Polyominoes: Puzzles, Patterns, Problems, and Packings”, Solomon W. Golomb, 1994, Princeton University Press, Princeton, N.J. ISBN 0-691-08573-0

[0011] The six puzzle sets of the present invention, which are called CUBELET, INCUBUS, PIPEDS, THE GREAT DIVIDE, CASCARA 5-in-9, and CASCARA 7-in-9, differ from all cube-packing schemes of the prior art, in that the sizes and shapes of the pieces in each puzzle set are defined in a completely systematic way, resulting in an inventory of blocks that exhibits uniform increments in size and shape between successive blocks in the inventory.

[0012] The great range in sizes and shapes of blocks challenges the ingenuity of the user, who is forced to invent appropriate strategies for deciding on both the order in which the pieces are selected for placement in the cube and also on the positions and orientations in which they are placed. Although—in contrast to the Slothouber-Graatsma and Conway cube puzzles—there is no single special condition that must be fulfilled to make a solution possible, it is unlikely for a solution to be found at all unless a ‘greedy algorithm’ is judiciously applied. Such an algorithm is a command to solve as much of the problem as possible at every step. In practice, this means placing the largest blocks first, leaving the smaller blocks to the last. It is not claimed that this rule must be observed in an absolutely strict way; in the last analysis, flexibility and ingenuity are required. All six of these puzzle sets teach the effectiveness of the greedy algorithm, which is of such fundamental importance that it is widely employed throughout science and technology.

[0013] The basic rule that-with minor exceptions noted below—defines the inventory of blocks in each of the six sets of this invention is that all of the blocks are right rectangular parallelepipeds (‘rectangular blocks’) of unit thickness, with lengths and widths equal to every integer multiple of that unit between some minimum value LMIN and some maximum value LMAX, inclusive.

[0014] By a remarkable coincidence, in five of the six puzzle sets, the inventory of rectangular blocks has a total volume precisely equal to the volume of a single cube whose edge length is an integer multiple (four for CUBELET, five for INCUBUS, seven for PIPEDS, and nine for CASCARA 5-in-9 and CASCARA 7-in-9) of the unit thickness of the blocks. The inventory of blocks of the sixth puzzle set, THE GREAT DIVIDE, includes two 3×3×3 cubes in addition to the blocks of unit thickness, resulting in a combined volume equal to that of two 5×5×5 cubes (which, like the INCUBUS cube, have edge lengths equal to five times the unit thickness of the rectangular blocks).

[0015] The values for LMIN and LMAX that define the upper and lower limits for the lengths and widths of the rectangular blocks of each puzzle set endow that set with its own characteristic level of complexity and difficulty. Packing a cube with the nine blocks of ‘CUBELET’, which is the smallest of the six puzzle sets, is so easy that it offers a challenge only to young children. The number of distinct solutions is in excess of one hundred. At the opposite extreme, both CASCARA 5-in-9 and CASCARA 7-in-9, it is required to find a

[0016] A property of all the puzzle sets of the present invention that is not shared either by the Slothouber-Graatsma Puzzle or by Conway's Puzzle is that in spite of their three-dimensional character, they also lend themselves to a great variety of two-dimensional square tiling puzzle activities. In the case of the two CASCARA puzzles, for which the total volume is equal to 729 (=9

[0017]

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[0021]

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[0024]

[0025]

[0026] FIGS.

[0027] The invention may best be understood from the following detailed description thereof, having reference to the accompanying drawings.

[0028]

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[0031]

[0032]

[0033]

[0034]

[0035] FIGS.

[0036] FIGS.

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[0051] In none of the partitions into squares shown in FIGS.

[0052]

[0053]

[0054]

[0055] By simple exchanges of rows and of columns, the pattern of blocks shown in plan view in

[0056]

[0057] Remarkably, it is not difficult in either of the CASCARA puzzles to find a near-packing of the exterior shell that contains the smallest possible packing error: a hole in the packing of the exterior cubic shell that contains two units of volume, accompanied by the projection of one block outside of the exterior shell, such projection also containing two units of volume.

[0058]

[0059]

[0060]

[0061]

[0062] FIGS.

[0063]

[0064] FIGS.

[0065]

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[0067]

[0068]

[0069] Having thus described the principles of the invention, together with illustrative embodiments thereof, it is to be understood that although specific terms are employed, they are used in a generic and descriptive sense, and not for purposes of limitation, the scope of the invention being set forth in the following claims: