The SET card game consists of cards containing shapes with different prop
erties. The goal of the game is to find sets of three cards that fulfill the SET
rule. A cap set is a subset of the affine space Zn
3 where no three elements
are collinear. The cap set problem explores the maximum possible size of
cap sets with regards to the dimension n. In the particular case of n = 4
the affine space Z4
3 can be represented with the 81 cards that are contained
within the SET card game. The cap set problem in this instance searches
for the largest amount of cards possible, such that no three cards fulfill the
SET rule. This thesis aims to explore and present some simple and easy
to understand approaches for attempting to solve the cap set problem using
counting arguments and combinatorics. The main result of the thesis is the
hyperplane counting method, which gives us upper bounds for the problem
in dimensions up to n = 8 as well as an application that helps visualize the
considered affine spaces of dimensions n = 2,3,4 as well as their subsets.
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