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Mathematics LibreTexts

2.1: Dots and Boxes

  • Page ID
    9828
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    Here are some dots; in fact there are nine of them:

    9dots.png

    We’re going to a play an “exploding dots” game. Here’s the only rule for the game:

    The 1←2 Rule

    Whenever there are two dots in single box, they “explode,” disappear, and become one dot in the box to the left.

    Example \(\PageIndex{1}\): Nine dots 1←2 in the system

    We start by placing nine dots in the rightmost box.

    Explode1a-300x87.png

    Two dots in that box explode and become one dot in the box to the left.

    Explode1b-300x83.png

    Explode1c-300x84.png

    Once again, two dots in that box explode and become one dot in the box to the left.

    Explode1d-300x81.png

    Explode1e-300x82.png

    We do it again!

    Explode1f-300x86.png

    Explode1g-300x89.png

    Hey, now we have more than two dots in the second box, so those can explode and move!

    Explode1h-300x82.png

    Explode1i-300x85.png

    And the rightmost box still has more than two dots.

    Explode1j-300x89.png

    Explode1k-300x84.png

    Keep going, until no box has two dots.

    Explode1l-300x86.png

    Explode1m-300x83.png

    Explode1n-300x86.png

    Explode1o-300x88.png

    After all this, reading from left to right we are left with one dot, followed by zero dots, zero dots, and one final dot.

    Answer:

    The 1←2 code for nine dots is: 1001.

    On Your Own. Here’s a diagram showing what happens for seven dots in a 1←2 box. Trace through the diagram, and circle the pairs of dots that “exploded” at each step.

    Explode2a-300x85.png

    Explode2b-300x81.png

    Explode2c-300x83.png

    Explode2d-300x84.png

    Explode2e-300x87.png

    Answer:

    The 1←2 code for seven dots is: 111.

    Problem 1

    Note: In solving this problem, you don’t need to draw on paper; that can get tedious! Maybe you could use buttons or pennies for dots and do this by hand.

    1. Draw 10 dots in the right-most box and perform the explosions. What is the 1←2 code for ten dots?
    2. Find the 1←2 code for eighteen dots.
    3. What number of dots has 1←2 code 101?

    Think / Pair / Share

    After you worked on the problem, compare your answer with a partner. Did you both get the same code? Did you have the same process?