U.S. patent application number 09/810510 was filed with the patent office on 2001-11-01 for set of blocks for packing a cube.
Invention is credited to Schoen, Alan H..
Application Number | 20010035606 09/810510 |
Document ID | / |
Family ID | 26888131 |
Filed Date | 2001-11-01 |
United States Patent
Application |
20010035606 |
Kind Code |
A1 |
Schoen, Alan H. |
November 1, 2001 |
Set of blocks for packing a cube
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) |
Correspondence
Address: |
Henry C. Nields
Nields & Lemack
Suite 8
176 E. Main Street
Westboro
MA
01581
US
|
Family ID: |
26888131 |
Appl. No.: |
09/810510 |
Filed: |
March 16, 2001 |
Related U.S. Patent Documents
|
|
|
|
|
|
Application
Number |
Filing Date |
Patent Number |
|
|
60192504 |
Mar 28, 2000 |
|
|
|
Current U.S.
Class: |
273/153R |
Current CPC
Class: |
A63F 9/10 20130101; A63F
9/12 20130101 |
Class at
Publication: |
273/153.00R |
International
Class: |
A63F 009/08 |
Claims
I claim:
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.times.4.times.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.times.1.times.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.times.5.times.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.times.7.times.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.times.6.times.7 and
1.times.7.times.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.times.5.times.5 cubes, and for other purposes, said set
comprising two 3.times.3.times.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.times.9.times.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.times.5.times.5 cube nested
concentrically in the interior of said 9.times.9.times.9 cube.
7. A set of thirty-six rectangular blocks in accordance with claim
1 for packing a 9.times.9.times.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.times.7.times.7 cube
nested concentrically in the interior of said 9.times.9.times.9
cube.
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'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.times.2.times.2 blocks (`squares`) and three 1.times.1.times.1
blocks (`small cubes`), which can be assembled to pack a
3.times.3.times.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.times.3.times.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's eighteen-block 5.times.5.times.5 cube puzzle,
once the same 3-dimensional stained-glass window theory is applied.
Conway's puzzle consists of thirteen 1.times.2.times.4 blocks, one
2.times.2.times.2 block, one 1.times.2.times.2 block, and three
1.times.1.times.3 blocks (`long bars`). From application of the
theory, it follows that the three long bars must be placed inside
the 5.times.5.times.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.times.5.times.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 Grunbaum 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.times.3.times.3 cubes
in addition to the blocks of unit thickness, resulting in a
combined volume equal to that of two 5.times.5.times.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 9x9x9 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.times.5.times.5 cube, for CASCARA 7-in-9, the interior
cube is a 7.times.7.times.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.times.9.times.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'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.sup.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] FIGS. 1 and 2 are each a perspective view of a prior-art
cube puzzle;
[0018] FIG. 3 is a perspective view of the ten blocks that comprise
the inventory for the INCUBUS puzzle;
[0019] FIG. 4 is a perspective view of the nine blocks that
comprise the CUBELET cube puzzle;
[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] FIG. 8 is a plan view of the outlines of the packings shown
in FIGS. 6-A to 6Z and FIGS. 7-A to 7-J.
[0022] FIG. 9 is a perspective view of the inventory of ten blocks
that comprise the INCUBUS cube puzzle;
[0023] FIGS. 10, 11 and 12 are each a perspective view of an
assembly of the ten blocks shown in FIG. 9;
[0024] FIG. 13 is a perspective view of the eighteen blocks that
comprise the `THE GREAT DIVIDE` two-cube puzzle;
[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] FIG. 1 is a perspective view of the three small cubes of the
Slothouber-Graatsma cube puzzle.
[0029] FIG. 2 is a perspective view of the three long bars of
Conway's cube puzzle.
[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] 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.times.1.times.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] FIG. 5-A is a perspective view of the nine blocks shown in
FIG. 4, assembled to form a 4.times.4.times.4 cube packing.
[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] FIG. 5-C is a perspective view of the nine blocks shown in
FIG. 4, assembled to form a tiling of the square 8.sup.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] 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] 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] 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] FIG. 11 is a perspective view of the ten blocks shown in
FIG. 9, partitioned between tilings of the two squares 5.sup.2 and
10.sup.2.
[0041] FIG. 12 is a perspective view of the ten blocks shown in
FIG. 9, partitioned between tilings of the two squares 2.sup.2 and
11.sup.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] 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.times.3.times.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.times.5.times.10 square prism; the
principal puzzle challenge is to divide the pieces between two
5.times.5.times.5 cube packings.
[0043] FIG. 14 is a perspective view of the eighteen blocks shown
in FIG. 13, assembled to form a packing of a 5.times.5.times.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] FIGS. 15 and 16 are perspective views of the eighteen blocks
shown in FIG. 13, divided between two packings of 5.times.5.times.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.times.5.times.5 cube), only seven
of these seventy-one partitions allow packings of both of the
5.times.5.times.5 cubes. Consequently, even though it is fairly
easy to choose a subset of the eighteen blocks that packs one
5.times.5.times.5 cube, the odds against finding a packing of a
second 5.times.5.times.5 cube with the remaining blocks are at
least 10:1. Since the two 3.times.3.times.3 cubes cannot both fit
inside one 5.times.5.times.5 cube packing, they must be assigned to
separate 5.times.5.times.5 cubes.
[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.sup.2, 6.sup.2, and 12.sup.2.
[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.sup.2, 7.sup.2, 7.sup.2, and 7.sup.2.
[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.sup.2, 8.sup.2, 8.sup.2, and 8.sup.2.
[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.sup.2, 5.sup.2, 5.sup.2, 7.sup.2, and 9.sup.2.
[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.sup.2, 4.sup.2, 4.sup.2, 5.sup.2, 7.sup.2, and
9.sup.2.
[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.sup.2, 3.sup.2, 4.sup.2, 5.sup.2, 5.sup.2, 6.sup.2, and
9.sup.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] 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] 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] FIG. 19-B is a plan view of a 27.times.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.sup.2.
[0056] FIG. 19-D is a perspective view of a stratified packing of a
9.times.9.times.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.times.9.times.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.times.5.times.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.times.7.times.7 PIPEDS cube and
is composed of the dark shaded blocks shown in FIG. 21-A. Although
finding a solution for the 7.times.7.times.7 PIPEDS interior cube
is somewhat more difficult than for the 5.times.5.times.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] 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] FIG. 20-B is a diagrammatic perspective view of the edges of
a 9.times.9.times.9 cube that contains a concentric
5.times.5.times.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] FIG. 20-C is a perspective view of a packing of a
5.times.5.times.5 core cube by the ten INCUBUS blocks of FIG. 9,
shown in dark shading in FIG. 20-A.
[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.times.9.times.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.times.9.times.9 hollow cubic shell shown
in FIG. 20-D.
[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.times.7.times.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.times.6.times.7 and
1.times.7.times.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] FIG. 21-D is a diagrammatic perspective view of the edges of
a 9.times.9.times.9 cube and a concentric 7.times.7.times.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] FIG. 21-E is a perspective view of a packing of a
9.times.9.times.9 hollow cubic shell by the seventeen blocks shown
in light shading in FIG. 21-A.
[0067] FIG. 21-F is an exploded perspective view of the
9.times.9.times.9 hollow cubic shell shown in FIG. 21-E.
[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.sup.2,
3.sup.2, 4.sup.2, 5.sup.2, 6.sup.2, 7.sup.2, 9.sup.2, 12.sup.2,
13.sup.2, and 14.sup.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:
* * * * *