## The Sperner Game

### Text II

To sum up, we have raised the following question: is it or is it not possible to fill the large triangle with colored disks, according to the rules concerning the colors to be used on the sides of the triangle, so that no small triangle with vertices of three different colors occurs? Should the answer be positive, one could give a direct justification: it would suffice to give a concrete example of a properly colored triangle with no small triangles with three different colors on its vertices. On the other hand, should the answer be negative, a different type of justification is in order.

In case the sides of the large triangle are just made out of three disks,

one can still check "by hand" all possible colorings and observe that, in each case, a small triangle with three colors exists. Do verify this by selecting $$n=3$$ and, just after you have done so, place the mouse pointer over the image and confirm all eight possibilities.

It is impossible to follow such a strategy if no restrictions are made on the number $$n$$ of disks on the sides of the large triangle, as there would be infinitely many values of $$n$$ to check ($$n=3, 4, 5$$, etc.) and, for each one, a growing number of cases to consider.

We will now present a complete explanation, covering all values of $$n$$ and all possible ways the game could be played. Before we move forward with our reasoning, we will provide you with the means to reach such an explanation on your own. Please proceed by clicking here.