## Ken's POTW

The Blind Bartender
The following game is played between a customer and a bartender. The customer places four glasses on a revolving tray, arranged in a circle. Each glass is either right-side-up or upside down. The bartender is blindfolded and cannot see which way the glasses are placed, but the goal is to turn all the glasses the same direction.

In each round, the tray is spun, and the bartender is allowed to touch only two glasses, turning over either or both of them. After each round, the bartender is told if all glasses are oriented the same and the game is over. What is the best strategy for the bartender? Is there a maximum number of moves, after which the bartender can be certain all the glasses are identically oriented?

There are several variations to this problem worth trying as well:

1. The above problem
2. Problem 1, but there are now 6 glasses in a circle, and any 4 of them can be touched in each round.
3. Problem 1. Any number of glasses may be flipped, but the bartender wears boxing gloves and cannot determine the orientation of the glasses. An additional requirement in this version is that the game is over when all glasses are right-side up.
4. Problem 3. Only one or two glasses may be flipped at one time.

Source: 1. rec.puzzles. 2. Original. 3. Duke U. Assignment Web page 4. Penn State Math Dept. Web page

Solution
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