This is a beautiful puzzle from the book “Mathematical Circle Diaries, Year 1” by Anna Burago (Chapter 9, last puzzle). For parents of middle school children, this is a highly recommended book.
At midnight, six guards assume their positions at the six corner towers of a hexagonal fortress (one guard per tower). Every 15 minutes, two random guards get bored. As soon as the guards get bored, they change their positions: each bored guard relocates to an adjacent tower of his choice, moving clockwise or anticlockwise. Prove that no matter where the bored guards go, all six guards will never end up at the same tower at the same time.
As always, please send your answers directly to me at email@example.com. If you like the puzzle, please share it with others. If you have interesting puzzles to share, please send them to me at my e-mail given above.