# How do we know that Icosoku always has solutions?

This is a continuation of a question I asked here. The puzzle Icosoku is now described by Wikipedia as: "The puzzle frame is a blue plastic icosahedron, and the pieces are 20 white equilateral-triangular snap-in tiles with black dots and 12 yellow pins for the corners. The pins are printed with the numbers from 1 to 12, and the triangular tiles have up to three dots in each of their corners. The object of the game is, for any arrangement of the pins, to choose the positions and orientations of the triangles so that the total number of dots on the five joining corners of each pin equals the number of the pin. The manufacturer asserts that there is a solution for each pin arrangement. The puzzle is shipped in a solved state. The pins are not suitable for children under three, but this toy is otherwise suitable for all ages."

My question is how can one prove that there is a solution for each pin arrangement, as the manufacturer asserts? There are 12 factorial ways of arranging the pins (less by a factor of 1/120 if one considers the symmetry of the icosahedron), so using brute force to prove this might be a bit challenging.