Lazy people need to be smart

Question:

Four glasses are placed on the corners of a square Lazy Susan (a square plate which can rotate about its center). Some of the glasses are upright (up) and some upside-down (down).

A blindfolded person is seated next to the Lazy Susan and is required to re-arrange the glasses so that they are all up or all down, either arrangement being acceptable (which will be signaled by the said ringing of a bell).

The glasses may be rearranged in turns with subject to the following rules: Any two glasses may be inspected in one turn and after feeling their orientation the person may reverse the orientation of either, neither or both glasses. After each turn the Lazy Susan is rotated through a random angle.

The puzzle is to devise an algorithm which allows the blindfolded person to ensure that all glasses have the same orientation (either up or down) in a finite number of turns. (The algorithm must be deterministic, i.e. non-probabilistic )

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Solution:

This algorithm guarantees that the bell will ring in at most five turns:

  1. On the first turn, choose a diagonally opposite pair of glasses and turn both glasses up.
  2. On the second turn, choose two adjacent glasses at least one will be up as a result of the previous step. If the other is down, turn it up as well. If the bell does not ring, then there are now three glasses up and one down.
  3. On the third turn, choose a diagonally opposite pair of glasses. If one is down, turn it up and the bell will ring. If both are up, turn one down. There are now two glasses down, and they must be adjacent.
  4. On the fourth turn, choose two adjacent glasses and reverse both. If both were in the same orientation then the bell will ring. Otherwise there are now two glasses down and they must be diagonally opposite.
  5. On the fifth turn, choose a diagonally opposite pair of glasses and reverse both. The bell will ring.

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