Nine L-ements
Other Possibilities

Rick Eason

First a Note on Piece Dimensions

The base dimensions chosen for this puzzle were 3 and 4. Any other two numbers can alternately be used for the two base dimensions with equal results; i.e., a puzzle with such dimensions will have the same number of solutions and each solution will correspond to one of these solutions if you substitute the new dimensions (with some exceptions, see below). The two dimensions do not have to be integers (e.g., use pi and e), but of course the two base dimensions must be different from one another.

If the dimensions chosen are A and B, then the piece sizes are:

Piece

Dimensions

A

(A+B) x (A+B) x A - A x A

B

(A+B) x (A+B) x A - B x B

C

2B x 2B x A - B x B

D

(A+B) x (A+B) x B - A x A

E

(A+B) x (A+B) x B - B x B

F

2B x 2B x B - B x B

G

(A+B) x (A+B) x B - B x A

H

(A+B) x 2B x B - B x B

I

2B x (A+B) x B - B x A

and the cube dimensions are (A + 2B) x (A + 2B) x (A + 2B).

We can use the term DUAL to represent the special case where the values of A and B are swapped from an original; i.e., use 4 and 3 rather than 3 and 4. For example, in the Nine L-ements puzzle as presented, pieces A and E are duals of one another, pieces B and D are duals of one another, and the dual of piece C would have dimensions of 6x6x4 - 3x3, etc. The dual puzzle would have a final cube dimension of 10x10x10. Again, the number of solutions would be the same as the original.

3 and 4 were chosen for the base dimensions in Nine L-ements as they produce pieces which look fairly similar to one another (and look almost like the "L" tricube), but are not so similar that they are difficult to distinguish from one another.

Exceptions

It is not completely true that just any two dimensions can be used for the base dimensions with equal result. The 169 solutions to Nine L-ements all have one A unit and two B units spanning the cube at any point. Additional solutions become possible when the cube can also be spanned by some integer combination of the chosen units other than A + 2B. For example, if A=2 and B=1 (cube dimension is 4x4x4), then 2A = A+2B, so two A's can span the cube rather than requiring one A and two B's, leading to additional solutions (1011 total unique solutions in this case).

In general, to avoid the possibility of additional solutions, we require that "m A + n B" is not equal to "A + 2B" for any non-negative integers m and n other than 1 and 2, respectively.

Other Variations

Other piece sets can also be used to make puzzles of the "Nine L-ements type." Although for a given set of base dimensions there are twenty possible pieces, we can create all possible "Nine L-ements type" puzzles by adding only three more pieces to our set. These three new pieces have the same outline as pieces G, H and I, but they have a thickness of A units. I.e., the extended piece set also includes the following three pieces.

Piece

Dimensions

J

(A+B) x (A+B) x A - B x A

K

(A+B) x 2B x A - B x B

L

2B x (A+B) x A - B x A

The complete set of all "Nine L-ements type" puzzles is then given by the following table

Pieces

Number of solutions

ABCDEFGHI

169

ABCDFGHIL

204

BCDEFGHIK

416

BCDFGHIKL

1219

ACDFGHIJK

1316

DEFGHIJKL

1508