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.

Description:
BACKGROUND OF THE INVENTION  
         [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&#39;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&#39;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. FIG. 1). With this knowledge, it becomes trivially easy to place the six squares and thereby complete the cube packing.  
           [0006]    It is only slightly more difficult to find a solution quickly to Conway&#39;s eighteen-block 5×5×5 cube puzzle, once the same 3-dimensional stained-glass window theory is applied. Conway&#39;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. FIG. 2) that is a perfect analog of the chain of three small cubes in the Slothouber-Graatsma Puzzle, since the chain of long bars must also extend between diagonally opposite corners of the parent cube, with the middle long bar touching the other two long bars only at their respective comers. The long axes of the three long bars are necessarily parallel, respectively, to three mutually orthogonal edges of the 5×5×5 cube.  
           [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  
         SUMMARY OF THE INVENTION  
         [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  9 x 9 x 9  cube packing in the interior of which there is a nested concentric cube packing. For CASCARA 5-in-9, the interior cube is a 5×5×5 cube, for CASCARA 7-in-9, the interior cube is a 7×7×7 cube. In both of these cases, it is required that the inner cube be composed of a specifically designated subset of the thirty-six blocks of which the entire 9×9×9 cube is composed. For CASCARA 5-in-9, the constituent blocks of the inner cube define one of the six puzzle sets of this invention-INCUBUS. For CASCARA 7-in-9, the constituent blocks of the inner cube define another of the six puzzle sets of this invention—PIPEDS. Packing solutions for the outer shell in both of these nested cube puzzles are extremely difficult to find, even though the number of distinct solutions in each case is known to be greatly in excess of 100.  
           [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&#39;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 3 ), the number of possible partitions of the blocks into flat squares exceeds one hundred. In some of these cases, discovering solutions requires almost as much ingenuity as for the cube-packing problems. Partitions into squares of the rectangular blocks of each puzzle set provide geometrical illustrations of a problem that has been studied by mathematicians since ancient times—the expression of a positive integer as a sum of squares of integers. 
       
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0017]    [0017]FIGS. 1 and 2 are each a perspective view of a prior-art cube puzzle;  
         [0018]    [0018]FIG. 3 is a perspective view of the ten blocks that comprise the inventory for the INCUBUS puzzle;  
         [0019]    [0019]FIG. 4 is a perspective view of the nine blocks that comprise the CUBELET cube puzzle;  
         [0020]    [0020]FIG. 5-A,  5 -B and  5 -C,  6 -A to  6 -Z and  7 -A to  7 -J are each a perspective view of an assembly of the nine blocks shown in FIG. 4;  
         [0021]    [0021]FIG. 8 is a plan view of the outlines of the packings shown in FIGS.  6 -A to  6 Z and FIGS.  7 -A to  7 -J.  
         [0022]    [0022]FIG. 9 is a perspective view of the inventory of ten blocks that comprise the INCUBUS cube puzzle;  
         [0023]    [0023]FIGS. 10, 11 and  12  are each a perspective view of an assembly of the ten blocks shown in FIG. 9;  
         [0024]    [0024]FIG. 13 is a perspective view of the eighteen blocks that comprise the ‘THE GREAT DIVIDE’ two-cube puzzle;  
         [0025]    [0025]FIGS. 14, 15,  16 ,  17 -A to  17 -F and  18  are each a perspective view of an assembly of blocks shown in FIG. 13; and  
         [0026]    FIGS.  19 -A to  19 -D,  20 -A to  20 -H and  21 -A to  22  are views of blocks used in the CASCARA 5-in-9 and CASCARA7-in-9 cube puzzles. 
     
    
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS  
       [0027]    The invention may best be understood from the following detailed description thereof, having reference to the accompanying drawings.  
         [0028]    [0028]FIG. 1 is a perspective view of the three small cubes of the Slothouber-Graatsma cube puzzle.  
         [0029]    [0029]FIG. 2 is a perspective view of the three long bars of Conway&#39;s cube puzzle.  
         [0030]    [0030]FIG. 3 is a perspective view of the ten blocks that comprise the inventory for the INCUBUS puzzle, contained inside a ‘corner reflector’ case designed to hold the blocks of a partially or completely assembled cube packing. All the blocks are of unit thickness; LMIN=2 and LMAX=5. Each block is distinct from all the other blocks in the set. Ignoring trivial rotations, reflections, or exchanges in position of convex sub-assemblies of blocks, the cube-packing solution shown in FIG. 3 is unique. As a result of this remarkable circumstance, finding a cube packing for INCUBUS is moderately difficult.  
         [0031]    [0031]FIG. 4 is a perspective view of the nine blocks that comprise the CUBELET cube puzzle. All the blocks are of unit thickness; LMIN=1 and LMAX=4, but the 1×1×1 unit cube is excluded from the set. The length and width of each block are defined by the pair of integer labels shown for that block. Since it is understood that the thickness of each block is equal to 1, no label for the thickness is displayed.  
         [0032]    [0032]FIG. 5-A is a perspective view of the nine blocks shown in FIG. 4, assembled to form a 4×4×4 cube packing.  
         [0033]    [0033]FIG. 5-B is a perspective view of the nine blocks shown in FIG. 4, assembled to form a packing of a square annulus.  
         [0034]    [0034]FIG. 5-C is a perspective view of the nine blocks shown in FIG. 4, assembled to form a tiling of the square 8 2 .  
         [0035]    FIGS.  6 -A to  6 -Z are perspective views of the nine blocks shown in FIG. 4, assembled to form packings of the twenty-six letters of the English alphabet.  
         [0036]    FIGS.  7 -A to  7 -J are perspective views of the nine blocks shown in FIG. 4, assembled to form packings of the ten decimal digits 0 to 9.  
         [0037]    [0037]FIG. 8 is a plan view of the outlines of the packings shown in FIGS.  6 -A to  6 -Z and FIGS. 7A to  7 -J.  
         [0038]    [0038]FIG. 9 is a perspective view of the inventory of ten blocks that comprise the INCUBUS cube puzzle, including integer labels for the lengths and widths of the blocks.  
         [0039]    [0039]FIG. 10 is a perspective view of the ten blocks shown in FIG. 9, assembled to form a cube packing, including integer labels for the lengths and widths of the blocks.  
         [0040]    [0040]FIG. 11 is a perspective view of the ten blocks shown in FIG. 9, partitioned between tilings of the two squares 5 2  and 10 2 .  
         [0041]    [0041]FIG. 12 is a perspective view of the ten blocks shown in FIG. 9, partitioned between tilings of the two squares 2 2  and 11 2 . The two partitions into squares shown in FIGS. 11 and 12 are the only possible partitions into squares for this set of blocks.  
         [0042]    [0042]FIG. 13 is a perspective view of the eighteen blocks that comprise the ‘THE GREAT DIVIDE’ two-cube puzzle. All of the blocks except for the two 3×3×3 cube blocks are of unit thickness; the pair of integer labels shown for each block of unit thickness defines its length and width. Every block of unit thickness for which the length is equal to the width is called a square block; every square block occurs once in the set, and every non-square block of unit thickness occurs twice. This set is packaged in the form of a 5×5×10 square prism; the principal puzzle challenge is to divide the pieces between two 5×5×5 cube packings.  
         [0043]    [0043]FIG. 14 is a perspective view of the eighteen blocks shown in FIG. 13, assembled to form a packing of a 5×5×10 square prism. While it is fairly easy to find packings of this square prism by the eighteen blocks of the set, it is somewhat more difficult to find examples of such packings that—like the one shown in FIG. 14—are fault-free, i.e., have no continuous planes of cleavage that intersect the prism assembly and thereby allow the packing to be separated into two distinct parts on opposite sides of a common plane boundary.  
         [0044]    [0044]FIGS. 15 and 16 are perspective views of the eighteen blocks shown in FIG. 13, divided between two packings of 5×5×5 cubes. Finding examples of such partitions into two cubes is unexpectedly difficult. Although there are exactly seventy-one different ways to divide the eighteen blocks between two subsets of equal volume (the volume of a 5×5×5 cube), only seven of these seventy-one partitions allow packings of both of the 5×5×5 cubes. Consequently, even though it is fairly easy to choose a subset of the eighteen blocks that packs one 5×5×5 cube, the odds against finding a packing of a second 5×5×5 cube with the remaining blocks are at least 10:1. Since the two 3×3×3 cubes cannot both fit inside one 5×5×5 cube packing, they must be assigned to separate 5×5×5 cubes.  
         [0045]    [0045]FIG. 17-A is a perspective view of the sixteen non-cube blocks shown in FIG. 13, partitioned among tilings of the three squares 4 2 , 6 2 , and 12 2 .  
         [0046]    [0046]FIG. 17-B is a perspective view of the sixteen non-cube blocks shown in FIG. 13, partitioned among tilings of the four squares 7 2 , 7 2 , 7 2 , and 7 2 .  
         [0047]    [0047]FIG. 17-C is a perspective view of the sixteen non-cube blocks shown in FIG. 13, partitioned among tilings of the four squares 2 2 , 8 2 , 8 2 , and 8 2 .  
         [0048]    [0048]FIG. 17-D is a perspective view of the sixteen non-cube blocks shown in FIG. 13, partitioned among tilings of the five squares 4 2 , 5 2 , 5 2 , 7 2 , and 9 2 .  
         [0049]    [0049]FIG. 17-E is a perspective view of the sixteen non-cube blocks shown in FIG. 13, partitioned among tilings of the six squares 3 2 , 4 2 , 4 2 , 5 2 , 7 2 , and 9 2 .  
         [0050]    [0050]FIG. 17-F is a perspective view of the sixteen non-cube blocks shown in FIG. 13, partitioned among tilings of the seven squares 2 2 , 3 2 , 4 2 , 5 2 , 5 2 , 6 2 , and 9 2 .  
         [0051]    In none of the partitions into squares shown in FIGS.  17 -A through  17 -F is the arrangement of the blocks unique.  
         [0052]    [0052]FIG. 18 is a perspective view of a symmetrical pyramid composed by stacking the seven squares shown in FIG. 17-F. Similar constructions can be made from any partitions of rectangular blocks into squares.  
         [0053]    [0053]FIG. 19-A is a perspective view of the shapes of the thirty-six blocks that comprise both the CASCARA 5-in-9 and CASCARA 7-in-9 cube puzzles. Every block is of unit thickness; the pair of integer labels shown for each block defines its length and width. Every square block occurs once in the set; every non-square block occurs twice.  
         [0054]    [0054]FIG. 19-B is a plan view of a 27×27 square tiled by the blocks shown in FIG. 19-A.  
         [0055]    By simple exchanges of rows and of columns, the pattern of blocks shown in plan view in FIG. 19-B can be transformed into the arrangement shown in plan view in FIG. 19-C, where the thirty-six blocks shown in FIG. 19-A are partitioned among tilings of nine squares 9 2 .  
         [0056]    [0056]FIG. 19-D is a perspective view of a stratified packing of a 9×9×9 cube each of whose nine square layers of unit thickness is one of the square tilings shown in FIG. 19-C. This packing arrangement is trivially easy to discover and does not present a serious puzzle challenge except to a young child. However, the present invention includes a requirement that transforms this puzzle into an extremely difficult cube-packing problem: the blocks in the set are identified by membership in two appropriately defined subsets that are distinguished by color, texture, or material, and it is then required that all the blocks of a particular one of the subsets be assembled to form an interior cube surrounded concentrically, without gaps, by an exterior cubic shell packed by the second subset, thereby forming a 9×9×9 cube whose outer surface is of uniform color, texture, and material. In one version of this puzzle, called CASCARA 5-in-9, the interior cube is identical to the 5×5×5 INCUBUS cube and is composed of the blocks shown in FIG. 9. In the other version, called CASCARA 7-in-9, the interior cube is identical to the 7×7×7 PIPEDS cube and is composed of the dark shaded blocks shown in FIG. 21-A. Although finding a solution for the 7×7×7 PIPEDS interior cube is somewhat more difficult than for the 5×5×5 INCUBUS interior cube, finding solutions for the exterior shells is extremely difficult in both cases.  
         [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]    [0058]FIG. 20-A is a perspective view of the thirty-six blocks that comprise the CASCARA 5-in-9 cube puzzle; ten of these blocks, which are shown shaded dark, are distinguished from the remaining twenty-six by color, texture, or material. These ten blocks are the same as the blocks of the INCUBUS puzzle.  
         [0059]    [0059]FIG. 20-B is a diagrammatic perspective view of the edges of a 9×9×9 cube that contains a concentric 5×5×5 core cube. The core cube contains the ten INCUBUS blocks shown in FIG. 9, which are shown in dark shading in FIG. 20-A, while the surrounding cubic shell contains the blocks shown in light shading in FIG. 20-A.  
         [0060]    [0060]FIG. 20-C is a perspective view of a packing of a 5×5×5 core cube by the ten INCUBUS blocks of FIG. 9, shown in dark shading in FIG. 20-A.  
         [0061]    [0061]FIG. 20-D is a perspective view of a packing, by the twenty-six blocks shown in light shading in FIG. 20-A, of the 9×9×9 hollow cubic shell shown in FIG. 20-B.  
         [0062]    FIGS.  20 -E,  20 -F,  20 -G, and  20 -H are an exploded perspective view of a packing of the 9×9×9 hollow cubic shell shown in FIG. 20-D.  
         [0063]    [0063]FIG. 21-A is a perspective view of the thirty-six blocks that comprise the CASCARA 7-in-9 cube puzzle. Nineteen of these blocks, shown in dark shading, are distinguished from the remaining seventeen, shown in light shading, by color, texture, or material. The nineteen blocks shown in dark shading are the blocks of the PIPEDS puzzle.  
         [0064]    FIGS.  21 -B and  21 -C are exploded perspective views of a particular packing of the 7×7×7 PIPEDS cube by the nineteen blocks shown in dark shading in FIG. 21-A. FIG. 21-C is an exterior view of the complete PIPEDS cube packing. Six blocks that are in the interior of this packing and are not visible in FIG. 21-C are shown in FIG. 21-B. Exclusion of the 1×6×7 and 1×7×7 blocks from the PIPEDS inventory is required in order that the combined volume of all the blocks in the inventory be equal to that of a cube whose edge length is an integer multiple-in this case seven-of the unit thickness of the blocks. More than ten distinct packings are known for PIPEDS.  
         [0065]    [0065]FIG. 21-D is a diagrammatic perspective view of the edges of a 9×9×9 cube and a concentric 7×7×7 core cube. The core cube contains the nineteen PIPEDS blocks shown in dark shading in FIG. 21-A; the surrounding cubic shell contains the seventeen blocks shown in light shading in FIG. 21-A.  
         [0066]    [0066]FIG. 21-E is a perspective view of a packing of a 9×9×9 hollow cubic shell by the seventeen blocks shown in light shading in FIG. 21-A.  
         [0067]    [0067]FIG. 21-F is an exploded perspective view of the 9×9×9 hollow cubic shell shown in FIG. 21-E.  
         [0068]    [0068]FIG. 22 is a perspective view of the thirty-six blocks shown in FIG. 19-A, partitioned among tilings of the ten squares 2 2 , 3 2 , 4 2 , 5 2 , 6 2 , 7 2 , 9 2 , 12 2 , 13 2 , and 14 2 . More than one hundred other partitions of these thirty-six blocks into sets of squares are possible.  
         [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: