Mixed Singles

Mark two points on a line and label them 0 and 1, in that order. Now: In one move you can add or remove two neighboring points marked 0 0 or 1 1. Through a series of such moves, is it possible to arrive at a single pair of points labeled 1 0?

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No. Below each 1, write the number of 0s to the right of it, then add up these numbers. At the start, this sum is 0, an even number, and after each move it remains even (either it remains unchanged or it increases or decreases by 2). The goal state would produce an odd sum, 1. So the task is impossible. From Arthur Engel, Problem-Solving Strategies, 2008.

August 22, 2025August 21, 2025 | Puzzles