Five-Sided Story

pentagon

In a regular pentagon, all diagonals are drawn, as shown. Label each vertex of the pentagon and each intersection of the diagonals with the number 1. Now: In one step you can change the signs of all the numbers on a side or on a diagonal. Is it possible, by a sequence of such steps, to convert all the labels in the diagram to -1?

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No. At the start, the number of -1s on the pentagon’s perimeter is 0, an even number. Each step will change two of these numbers, so their quantity will always be even. Our goal would make it odd. So the task is impossible. From Arthur Engel’s Problem-Solving Strategies, 2003.

October 3, 2025 | Puzzles