Sunday Times Teaser 2907, Combinatorial Cards

Sunday Times Teaser 2907 asks how many cards there would be in a set with eight images per card, each image appearing on exactly eight cards and the total number of images being the same as the number of cards.

Astute readers will recognise this as an example of a finite projective plane, and may also recognise the set of cards as being similar to the game Doppel or Spot It!

Jim Randell produced a Python module to generate a set of cards, and a similar Python program by Joel Grus can be found here

I thought I would try to produce a Python version that allows sets of cards to be produced based on a projective plane associated with a finite field of the form GF(p^n) for arbitrary prime p an positive integer n.

The tricky bit if being able to do multiplication in the finite field, which requires an irreducible polynomial of degree n over the field. My code achieves this for values of n up to 3, and for a few other combinations of p and n that have pre-calculated irreducibles.

The code can be found here. It consists of 

• a general purpose finite field module (adapted from the one I created for Sunday Times Teaser 777 )
• a module to generate sets of cards based on the projective plane of GF(p^n)