This problem originated in Russia, according to various sources, but no one’s sure precisely where:
Before you is a square table that can rotate freely. In each corner is a deep well, at the bottom of which is a tumbler that’s either upright or inverted. You can’t see the tumblers, but you can reach into the wells to feel their positions.
Periodically the table rotates and stops at random. After each stop, you can feel two of the tumblers and turn over either, both, or neither. If all four of the tumblers are in the same state — all upright or all inverted — then a bell sounds. Otherwise the table rotates again and you make another “move.”
Can you guarantee to ring the bell in a finite number of moves? If so, how?
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At first it sounds impossible, but the task can be completed within five moves:
- After the first spin, reach into wells in diagonally opposite corners of the table and set both tumblers upright.
- Let the table spin again and reach into two adjacent wells. Because of move 1, you’ll find that at least one of them is upright. If the other is inverted, turn it upright. If the bell doesn’t sound, then the position is now UUUD.
- After the next spin, reach into diagonally opposite corners. If you find an inverted tumbler, turn it up and the bell will ring. If both tumblers are upright, invert one, leaving the position UUDD (two inverted tumblers in adjacent wells).
- Let the table spin, then choose two adjacent wells and reverse both tumblers. If they were in the same position, then the bell will sound. Otherwise we know the position is now UDUD.
- After the fifth spin, reach into diagonally opposite wells and reverse both tumblers, and the bell must sound.
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