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Mathematics LibreTexts

2.2: Other Rules

  • Page ID
    9829
  • [ "article:topic", "authorname:mmanes", "license:ccbysa" ]

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    Let’s play the dots and boxes game, but change the rule.

    The 1←3 Rule

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

    Example \(\PageIndex{1}\): Fifteen dots in the 1←3 system

    Here’s what happens with fifteen dots:

    Explode3a-300x85.png

    Explode3b-300x86.png

    Explode3c-300x85.png

    Explode3d-300x82.png

    Explode3e-300x83.png

    Explode3f-300x80.png

    Explode3g-300x85.png

    Answer:

    The 1←3 code for fifteen dots is: 120.

    Problem 2

    1. Show that the 1←3 code for twenty dots is 202.
    2. What is the 1←3 code for thirteen dots?
    3. What is the 1←3 code for twenty-five dots?
    4. What number of dots has 1←3 code 1022?
    5. Is it possible for a collection of dots to have 1←3 code 2031? Explain your answer.

    Problem 3

    1. Describe how the 1←4 rule would work.
    2. What is the 1←4 code for thirteen dots?

    Problem 4

    1. What is the 1←5 code for the thirteen dots?
    2. What is the 1←5 code for five dots?

    Problem 5

    1. What is the 1←9 code for thirteen dots?
    2. What is the 1←9 code for thirty dots?

    Problem 6

    1. What is the 1←10 code for thirteen dots?
    2. What is the 1←10 code for thirty-seven dots?
    3. What is the 1←10 code for two hundred thirty-eight dots?
    4. What is the 1←10 code for five thousand eight hundred and thirty-three dots?

    Think / Pair / Share

    After you have worked on the problems on your own, compare your ideas with a partner.  Can you describe what’s going on in Problem 6 and why?