Rotation erases the identity of each cup, but it cannot erase the geometry: pairs remain adjacent or opposite. The strategy works by reducing the possible configurations until only one class remains.
The Rotating Tray
Riddle statement
Four cups are placed at the corners of a square tray. Each cup is either upright or upside down.
You are blindfolded.
Before the first turn, and after every action you take, a bell rings and you win if all four cups have the same orientation.
On each turn, you may:
- choose two adjacent cups or two opposite cups;
- touch them to learn their orientations;
- turn over neither cup, either one, or both.
If the bell does not ring, another person rotates the tray by 0, 1, 2, or 3 quarter-turns without changing the orientation of any cup. Before your next turn, you no longer know which cup occupies each corner.
Design a strategy that guarantees the bell will ring within at most five turns.
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Solution
If the bell rings at any point, the process ends.
Turn 1 — opposite pair
Touch two opposite cups and leave both upright. If you do not win, only two classes of configuration can remain:
- three upright cups and one upside down;
- two opposite upright cups and two opposite upside-down cups.
Turn 2 — adjacent pair
Touch two adjacent cups and leave both upright.
In the second class, every adjacent pair contains one cup of each orientation, so the result is three upright and one upside down. If the configuration already had three upright and one upside down, touching the odd cup rings the bell; if you do not touch it, the same class remains.
Therefore, if you still have not won, exactly three cups are upright and one is upside down.
Turn 3 — opposite pair
Touch two opposite cups.
- If their orientations differ, leave both upright. The sole upside-down cup was among the two touched, so all four become upright.
- If both are upright, turn over exactly one. The cup that was already upside down lies on the other diagonal; the result is two adjacent upright cups and two adjacent upside-down cups.
Turn 4 — adjacent pair
Touch two adjacent cups and turn over both.
- If they had the same orientation, the other two also matched each other and had the opposite orientation. Turning over the touched pair makes all four equal.
- If their orientations differed, the configuration becomes two opposite upright cups and two opposite upside-down cups.
Turn 5 — opposite pair
Touch two opposite cups. In the only remaining class, every opposite pair contains two cups with the same orientation. Turn over both and all four cups become equal.
The intermediate rotations may change which corner contains each cup, but they preserve every configuration class used in the argument. The strategy therefore guarantees success in no more than five turns.