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­Upstart Puzzles: Chair Games

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Chair Games, Figure 1

The goal is to rearrange the people around the table to be in clockwise alphabetical order, using as few moves as possible, where each move involves moving a person three seats away (in either direction) from the empty chair.

A group of people is sitting around your dinner table with one empty chair. Each person has a name that begins with a different letter: A, B, C ... Because you love puzzles, you ask them to rearrange themselves to end up in alphabetical order in a clockwise fashion, with one empty chair just to the left of the person whose name begins with A. Each move involves one person from one chair to the empty chair k seats away in either direction. The goal is to minimize the number of such moves.

Warm-up. Suppose you start with eight people around the table, with nine chairs. The last name of each person begins with the letter shown, and you are allowed to move a person from one chair to an empty chair three chairs away in either direction (see Figure 1). Can you do it in four moves?



Maybe I'm missing something, but why can't the first example be solved in two moves?

F from 3 to 6
C from 9 to 3

I noticed that both example solutions have the people moving counter-clockwise only, but the problem says they can move in either direction. Is this just a coincidence? If counter-clockwise movement only is a requirement, then the printed solution to the first problem would be correct.

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