SET is an extremely fun thoughtful card game intended for children and adults. What many wouldn’t assume while playing SET is the fact that mathematics is hidden in the core of the game. Apart from being fun and highly addictive, the game associates with many mathematical problems from the field of affne geometry, linear algebra, set theory, combinatorics and probability. This paper is based on the geometric interpretation of the game SET with the help of finite vector and affne space \(\mathbb Z_3^4\). During the course of the game, many problems arise for which the answers are found by mathematical ”tools”. The main problem of the paper is getting an answer to one of those questions. In the mathematical interpretation of SET, solving a set problem comes down to finding the maximum value of a subset of points inside affne space \(\mathbb Z_3^4\) which doesn’t contain a line, the so-called 4-cap. Generally, the maximum value of an \(n\)-cap in \(\mathbb Z_3^n\) is known for \(n\leq6\), whereas for other dimensions it’s possible only to determine the upper limit, which is done with the help of the theorem we’re stating in the paper. For the first four dimensions with the help of the game SET we are coming across an example of a maximum \(n\)-cap, then we mathematically prove that the value of found n-cap is indeed the highest possible. Besides that, the paper deals with some basic properties of affne space \(\mathbb Z_3^n\) and especially the geometry in \(\mathbb Z_3^n\) for \(n\) = 1, 2, 3, 4. The last chapter is dedicated to the application of the game SET in mathematics classes. SET is a card game so it opens many combinatorial and probabilistic questions which can be presented to high school students and encourage them to think logically and abstractly. SET also presents a concrete model of sets so we are presenting a way in which elementary and high school students would be able to grasp basic set operations with the help of a fun game.