11.5. In one of the vertices of a cube, there are $N$ butterflies, and the other seven vertices are empty. Every minute, from one of the vertices of the cube, one butterfly flies to each of the three adjacent vertices along the edges of the cube, one to the opposite vertex (relative to the center) of the cube, and one flies away and does not return. Find all values of $N$ for which, after some time, the same number of butterflies can be in each vertex of the cube.
Answer. Any $N$, divisible by 45.

Solution. 1. We will show that for any $N=45k, k \in N$ the situation where after some time the same number of butterflies can be in each vertex is possible. Let's number the vertices of the cube, first numbering the vertices of the lower base clockwise from 1 to 4, then the vertices of the upper base clockwise from 5 to 8, so that the fifth vertex is above the first, the sixth above the second, and so on, and all the butterflies initially sit on the first. Let the butterflies fly only from the first vertex for the first $9k$ minutes, after which there will be $9k$ butterflies on vertices $2,4,5,7$, and the other vertices will be empty. During the next $4k$ minutes, $k$ butterflies fly out from each of the vertices $2,4,5,7$, after which there will be $4k$ butterflies on each vertex of the cube, as required by the problem.