CEMC Galois, 2003

Combinatorics Difficulty 3.0 AMC 10/12 Prove it Canada

In the game "Switch", the goal is to make the dimes (D) and quarters (Q) switch spots. The starting position of the game with 1 quarter and 1 dime is shown below. Allowable moves are: (i) If there is a vacant spot beside a coin then you may shift to that space. (ii) You may jump a quarter with a dime or a dime with a quarter if the space on the other side is free. The game shown in the diagram takes three moves. Complete the diagram to demonstrate how the game of "Switch" that starts with 2 quarters and 2 dimes can be played in 8 moves.

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