These criteria are referred to as the "anti-slide" requirement wherein covering the top of the box and shaking it vigorously in any direction will not dislodge a piece. If the 64-cubelet enclosing box is opaque, then the 27-cubelet Soma set has the appearance of having magically grown to fill it completely.

- A flat surface of 16 cubelets is presented at the top of the box.
- Soma pieces in the top level can be moved upwards for removal, but not in any other direction.
- Soma pieces in lower levels cannot move in any of the six directions (up, down, left, right, front, back).

*Inverted & Mirrored Puzzles*

When solving Shrinking Soma puzzles, it's often convenient to work on it upside down, where the base of the puzzle consists of the cubelets that will end up in the top level of the enclosing box. Pieces are added so that they will be locked into position when the enclosing sides are present, as well as providing support for other pieces that are temporarily below them. When complete, the arrangement of Soma pieces can be put into its enclosing box all at once simply by inverting the box over the pieces and then turning it right side up while holding the pieces in place within the open side of the box.

In some cases, it can be useful to consider the mirror image of a Shrinking Soma solution, which is created by exchanging the 3 and 9 pieces and reversing the rows and columns in the solution.

The problem has been simplified into finding 4 cubelets in Level 3 that are
adjacent to a

It was previously established that the cubelets protruding down into Layer 3
consist of a row of two cubelets as part of the 3 or 9 piece, plus two single
cubelets, one from the Y piece and the other from the remaining 3 or 9 piece.
It follows that only three sides of the square can be blocked. Therefore, and
by symmetry, the only

By excluding the T + V or Z + V combination, arrangements of the remaining 20 cubelets in the top layer and the three blocks can each be tested for feasibility. Due to symmetry, only the following possibilities need to be examined:

Discarded solutions have one or more unsupported pieces and so violate the anti-slide requirement. In the results indicated by *, there are four solutions, three of which have unsupported pieces. The one remaining possibility has the L piece supported by the 3 piece, which in turn is supported by the square. Only the

Blocks Results If T + V used for support If Z + V used for support AB D F Impossible Impossible AB D G 1 discarded solution Impossible AB E F Impossible Impossible AB E G Impossible 2 discarded solutions BC D F Impossible 1 discarded solution BC D G Impossible Impossible BC E F 1 discarded solution Impossible BC E G Impossible Impossible CD A F * Impossible CD A G Impossible 5 discarded solutions CD B F Impossible 1 discarded solution CD B G Impossible Impossible DE A F 2 discarded solutions Impossible DE A G Impossible Impossible

What Shrinking Soma cuboids can be created from the seven pieces in one Soma
set? Obviously, the cuboid volume

This requirement can be generalized to A cubelets in any level of a Shrinking Soma puzzle. The minimum number of additional cubelets f(A) that are needed for support in the level immediately below it is the minimum number of Soma pieces that comprise the A cubelets. The support cubelets can be tabulated as follows:

Each level in turn is supported by cubelets in the level below it, so the total minimum number of cubelets in the upper region of the puzzle is

Cubelets A Minimum pieces f(A) Description 1-4 1 1 four-cubelet flat piece 5-8 2 2 four-cubelet flat pieces 9-12 3 3 four-cubelet flat pieces 13-15 4 All 4 flat pieces 16-18 5 All 4 flat pieces plus 1 non-flat piece 19-20 6 All 4 flat pieces plus 2 non-flat pieces

A + f(A) + f(f(A)) + f(f(f(A))) + ...,where the additions are repeated until the number of support cubelets in the last level is reduced to 1, or until all 27 cubelets in the Soma set are exhausted. This process is best illustrated by an example, say a puzzle with

Therefore, the upper portion of this puzzle consists of three levels comprising at least 23 cubelets. At most four cubelets are available for lower levels.

Level Cubelets Cubelets remaining Minimum support pieces L (top) 16 27 - 16 = 11 f(16) = 5 L-1 5 11 - 5 = 6 f(5) = 2 L-2 2 6 - 2 = 4 f(2) = 1 L-3 1

Turning our attention to the lower portion of the puzzle, what theoretical maximum height can be achieved with a given number of cubelets? The maximum height of a single Soma piece is 3 cubelets, achieved with the L, Z or T piece. The anti-slide criterion requires that each Soma piece be blocked laterally by at least one additional cubelet that is not part of the column of three cubelets. Therefore, each increase of 3 levels in a vacant area of the bounding box consumes at least 5 cubelets: 4 in the Soma piece plus a blocking cubelet. Any individual cubelets left over when all multiples of 5 are used could contribute to 1 or 2 more levels if they were part of an L piece, for example. These results are consolidated into the following table:

and can be summarized with the mathematical expression

Cubelets c Levels L 1 1 2-4 2 5 3 6 4 7-9 5 10 6 ... ...

L = 3⌊c/5⌋ + min{(c mod 5),2},where ⌊ ⌋ represents the floor function, and modulo arithmetic is used.

The results for the upper and lower regions of Shrinking Soma puzzles can be combined to define theoretical limits on the height (number of levels) in a puzzle with a given horizontal top-level area A:

R C Area A Height (L) range Utilization range 2 2 4 7 - 15 45% - 96% 2 3 6 5 - 13 35% - 90% 2 4 8 4 - 13 26% - 84% 3 3 9 3 - 11 27% - 100% 2 5 10 3 - 10 27% - 90% 2 6 12 3 - 10 23% - 75% 3 4 2 7 14 2 - 7 28% - 96% 3 5 15 2 - 7 26% - 90% 2 8 16 2 - 5 34% - 84% 4 4 2 9 18 2 - 5 30% - 75% 3 6 2 10 20 2 68% 4 5

To address the first challenge, it is easier to orient the 4 x 4 x 4 solution so
that the wall adjacent to its

The non-flat pieces in the second Soma set can be positioned to meet these blocking requirements, as long as each has at least one of its cubelets supported by the

The Y piece is not required to block the 4 x 4 x 4 puzzle solution in its original form, but its raised cubelet will be needed as a block later.

Next, we'll look at filling out the 16 cubelets in the rest of the top level.
The remaining four pieces of the second Soma set (L, Z, T and V) are all flat
but together consist of only 15 cubelets, so we'll need to take one from the
smaller puzzle solution to make up the shortage. This is challenging to remove
without creating violations of the anti-slide requirement. After some trial and
error, it was found that the L piece in the

Here is a list of K'nex boxes with piece counts:

Lateral cubeletsGreensWhitesPiece fit2 1 0.5 Loose 3 0 1.5 Snug 4 0 2 Normal 5 0 2.5 Loose 6 0 3 Loose 7 2 2 Normal

Vertical cubeletsGreensWhites3 2 0 4 2 0.5 5 2 1 8 1 3

Puzzle K'nex connectors K'nex rods OtherTotalR C L Volume Sets Piece fitImage Link Purple White Yellow Red Green White Blue 2 5 3 30 1 Loose Image Link 24 1 15 2 29 18 4 93 2 6 3 36 1 Loose Image Link 26 2 16 3 32 22 5 106 3 3 4 36 1 Snug Image Link 24 6 8 25 20 8 91 3 4 3 36 1 Normal Image Link 22 6 7 2 23 19 2 81 2 5 4 40 1 Loose Image Link 28 7 13 2 43 22 6 2 light gray connectors 123 2 7 3 42 1 Loose Image Link 26 4 16 2 34 22 8 112 3 3 5 45 1 Snug Image Link 28 6 8 21 26 12 101 3 5 3 45 1 Normal Image Link 24 4 12 18 26 8 92 3 4 4 48 1 Snug Image Link 26 8 10 29 25 9 107 4 4 4 64 1 Normal Image Link 24 9 16 32 36 8 125 4 4 5 80 2 Normal Image Link 32 9 12 16 48 16 133 5 5 4 100 2 Loose Image Link 32 16 16 49 40 12 165 5 5 5 125 2 Normal Image Link 36 16 12 37 48 22 4 dark gray connectors, 2 hinges 177 4 4 8 128 2 Normal Image Link 48 17 4 24 64 24 181

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