the invention may best be understood from the following detailed description thereof , having reference to the accompanying drawings . [ 0028 ] fig1 is a perspective view of the three small cubes of the slothouber - graatsma cube puzzle . [ 0029 ] fig2 is a perspective view of the three long bars of conway &# 39 ; s cube puzzle . [ 0030 ] fig3 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 fig3 is unique . as a result of this remarkable circumstance , finding a cube packing for incubus is moderately difficult . [ 0031 ] fig4 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 ] fig5 - a is a perspective view of the nine blocks shown in fig4 assembled to form a 4 × 4 × 4 cube packing . [ 0033 ] fig5 - b is a perspective view of the nine blocks shown in fig4 assembled to form a packing of a square annulus . [ 0034 ] fig5 - c is a perspective view of the nine blocks shown in fig4 assembled to form a tiling of the square 8 2 . fig6 - a to 6 - z are perspective views of the nine blocks shown in fig4 assembled to form packings of the twenty - six letters of the english alphabet . fig7 - a to 7 - j are perspective views of the nine blocks shown in fig4 assembled to form packings of the ten decimal digits 0 to 9 . [ 0037 ] fig8 is a plan view of the outlines of the packings shown in fig6 - a to 6 - z and fig7 a to 7 - j . [ 0038 ] fig9 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 ] fig1 is a perspective view of the ten blocks shown in fig9 assembled to form a cube packing , including integer labels for the lengths and widths of the blocks . [ 0040 ] fig1 is a perspective view of the ten blocks shown in fig9 partitioned between tilings of the two squares 5 2 and 10 2 . [ 0041 ] fig1 is a perspective view of the ten blocks shown in fig9 partitioned between tilings of the two squares 2 2 and 11 2 . the two partitions into squares shown in fig1 and 12 are the only possible partitions into squares for this set of blocks . [ 0042 ] fig1 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 ] fig1 is a perspective view of the eighteen blocks shown in fig1 , 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 fig1 — 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 ] fig1 and 16 are perspective views of the eighteen blocks shown in fig1 , 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 ] fig1 - a is a perspective view of the sixteen non - cube blocks shown in fig1 , partitioned among tilings of the three squares 4 2 , 6 2 , and 12 2 . [ 0046 ] fig1 - b is a perspective view of the sixteen non - cube blocks shown in fig1 , partitioned among tilings of the four squares 7 2 , 7 2 , 7 2 , and 7 2 . [ 0047 ] fig1 - c is a perspective view of the sixteen non - cube blocks shown in fig1 , partitioned among tilings of the four squares 2 2 , 8 2 , 8 2 , and 8 2 . [ 0048 ] fig1 - d is a perspective view of the sixteen non - cube blocks shown in fig1 , partitioned among tilings of the five squares 4 2 , 5 2 , 5 2 , 7 2 , and 9 2 . [ 0049 ] fig1 - e is a perspective view of the sixteen non - cube blocks shown in fig1 , partitioned among tilings of the six squares 3 2 , 4 2 , 4 2 , 5 2 , 7 2 , and 9 2 . [ 0050 ] fig1 - f is a perspective view of the sixteen non - cube blocks shown in fig1 , partitioned among tilings of the seven squares 2 2 , 3 2 , 4 2 , 5 2 , 5 2 , 6 2 , and 9 2 . in none of the partitions into squares shown in fig1 - a through 17 - f is the arrangement of the blocks unique . [ 0052 ] fig1 is a perspective view of a symmetrical pyramid composed by stacking the seven squares shown in fig1 - f . similar constructions can be made from any partitions of rectangular blocks into squares . [ 0053 ] fig1 - 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 ] fig1 - b is a plan view of a 27 × 27 square tiled by the blocks shown in fig1 - a . by simple exchanges of rows and of columns , the pattern of blocks shown in plan view in fig1 - b can be transformed into the arrangement shown in plan view in fig1 - c , where the thirty - six blocks shown in fig1 - a are partitioned among tilings of nine squares 9 2 . [ 0056 ] fig1 - 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 fig1 - 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 fig9 . 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 fig2 - 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 . 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 ] fig2 - 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 ] fig2 - 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 fig9 which are shown in dark shading in fig2 - a , while the surrounding cubic shell contains the blocks shown in light shading in fig2 - a . [ 0060 ] fig2 - c is a perspective view of a packing of a 5 × 5 × 5 core cube by the ten incubus blocks of fig9 shown in dark shading in fig2 - a . [ 0061 ] fig2 - d is a perspective view of a packing , by the twenty - six blocks shown in light shading in fig2 - a , of the 9 × 9 × 9 hollow cubic shell shown in fig2 - b . fig2 - 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 fig2 - d . [ 0063 ] fig2 - 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 . fig2 - 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 fig2 - a . fig2 - 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 fig2 - c are shown in fig2 - 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 ] fig2 - 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 fig2 - a ; the surrounding cubic shell contains the seventeen blocks shown in light shading in fig2 - a . [ 0066 ] fig2 - 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 fig2 - a . [ 0067 ] fig2 - f is an exploded perspective view of the 9 × 9 × 9 hollow cubic shell shown in fig2 - e . [ 0068 ] fig2 is a perspective view of the thirty - six blocks shown in fig1 - 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 . 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 :
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 .