Besides some arithmetic, there is the problem to build an octahedron
from two pyramids with isoceles sides of length 40 and compute the height
(which is of course 40*sqrt(2), then there is a classical SOMA 3 x 3 cube problem
with 6 quadrominos and one triomino (6*4+3=27).
For the Soma cube which appears in the movie, there are
The Soma cube is a three-dimensional version of the 12 pentomino problem in
which a 6 x 10 grid has to be covered, the later is a classic Canterbury puzzle.
Around 1982 I once wrote a Pascal computer program (with turtle graphics)
which computed by recursive backtracking all solutions and even animated them on an
apple school computer and remember having been asked to leave
the lab because a supervisor thought I play games. I learned about the combinatorial
search problem to find the number of solutions from Ralph Ehrismann who had like
me participated in the
Schweizer Jugend Forscht project and had written there a program to compute
the 2339 solutions.
I had in high school also been fascinated by the problem to find the number of n-polyominos
and tried hard to find a law. I learned only later that the problem is hard as it is a
self-avoiding random walk problem. They are also called lattice animals. It is a difficult
combinatorial problem with applications in statistical mechanics.
There are 12 pentominos.
(Tetris bricks). The 35 hexominos
can not be used to tile a rectangle however. Then there are
The 107 simply connected heptominoes can tile a 7 * 107 rectangle.
Then there are 369 octominoes
and 1285 nonominoes and
and 4655 decominos.