Round Numbers

A circle is divided into six sectors, into which are written (say, counterclockwise) these numbers:

1, 0, 1, 0, 0, 0

You can increase any two neighboring numbers by 1. By doing this repeatedly, is there a way to make all six numbers equal?

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No. Call the current six numbers a₁ … a₆. Now I = a₁ – a₂ + a₃ – a₄ + a₅ – a₆ is initially 2, and I will remain unchanged no matter what we do. So there’s no way to reach I = 0. Via Arthur Engel, Problem-Solving Strategies, 1999.