Pentaminos

The pentaminos are the puzzle's pieces represented on the beside picture. They are the 12 different pieces that you can obtain joining 5 identical cubes side by side.
The rule of the game is to reconstitute a given shape. Most of the time, you have to put them into a 8 x 8 array, in which 4 cases must stay empty. Here are some examples:
but there are many others.
I decided on day to find the number of solutions, for a given array. I realised a little program to do it, first in C, then in Objective Caml later. It is a backtracking algorithm, which stops whenever an area to fill has a surface which is not a multiple of 5.

Some answers computed by the program.

You can download:

Variant

A variant consists in arranging the 12 pentaminos in a rectangle. Here are the number of solutions (up to the 4 symmetries): The Caml program
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Jean-Christophe.Filliatre[at]lri.fr (formatted with yamlpp).