Set - How to make a magic square of Set

What you see here is a magic square, much like the addition and subtraction squares you may have used as a child.

These magic squares are even more talented, as they all follow the rules of the card game SET®. To learn how to make one with ease, read on.

SET® cards contain four properties: color, shape, number of objects, and shading. The rules state for each property, they must all be equal, or all different. For example, if we look at the top row of the square, we see three different colors, three different shapes, three different numbers, and three different types of shading within the objects. Need more examples? Any line on the magic square yields a set.
Constructing a magic square may seem complex at first glance, but in reality anyone can make one by following this simple process:

Choose any three cards that are not a set. (It will work with a set but the square becomes redundant) For example, we will choose these:

Now place these three cards in the #1, #3, and #5 positions in the magic square.

Using our powers of deduction, we can conclude that in order to create a set in the first row, the #2 card needs to have a different color, different shape, same number, and same shading as the #1 and #3 cards. That leaves us with a solid purple oval. The rest of the square can be completed in the same way, giving us the following magic square:

A few examples will convince you that this method works. Not only does the magic square work but it can be theoretically proven through a mathematical model. This model makes an easy proof of the magic square, as well as answer any questions about how SET® works.

Proof of the Magic Square | More Math Tricks