Changing Places

  • Changing Places

    PoW ID: 353

    Discrete Math, High School, Discrete Math, Discrete Math

    Problem Print

    I love to play games, and the following is a neat "changing places" puzzle. Place three red chips on positions 1, 2, and 3, and three blue chips on positions 10, 11, and 12. In just 22 moves, you must make the chips change places.

    Move one chip at a time, alternating colors, along the lines from one circle to another. At no time may a chip of one color be on a circle that a chip of the opposite color could reach on the next move. Only one chip may be on a circle at any one time.



    Hint: Remember that this is the Discrete Math Problem of the Week, so try to use some graph theory and change the puzzle layout into a graph. Your answers must include your moves as well as your strategies - just being lucky will not be counted correct. You must analyze what you did and describe it thoroughly or upload a graph.

    Bonus: What does it mean for two graphs to be isomorphic?

    Teaching Tip: Have students work in pairs and discuss their moves, with one student moving the chips and the other keeping track of the moves on paper.

    Answer Check

    Show Answer

    There may be several correct answers depending upon which color you chose to start with. Here is one possible solution in 22 moves:

    1 to 8, 10 to 5, 2 to 9, 11 to 6, 3 to 4, 12 to 7, 8 to 3, 5 to 12, 9 to 10, 6 to 1, 4 to 9, 7 to 6, 3 to 4, 12 to 7, 10 to 5, 1 to 8, 9 to 10, 7 to 2, 5 to 12, 6 to 1, 4 to 11, 8 to 3.
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