Baltic Way 1991 · Problem 13

Solution:

Define the distance between two small triangles to be the minimal number of steps one needs to move from one of the triangles to the other (a step here means transition from one triangle to another having a common side with it). The maximum distance between two small triangles is $8$ and this maximum is achieved if and only if one of these lies at a corner of the big triangle and the other lies anywhere at the opposite side of it. Assume now that we have assigned the numbers $1, \ldots, 25$ to the small triangles so that the difference of the numbers assigned to any two adjacent triangles does not exceed $3$. Then the distance between the triangles numbered $1$ and $25$; $1$ and $24$; $2$ and $25$; $2$ and $24$ must be equal to $8$. However, this is not possible since it implies that either the numbers $1$ and $2$ or $24$ and $25$ are assigned to the same "corner" triangle.