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A fun arithmetic game that sparks exploration

Here's a game that's easy and leads to a nice exploration of number theory for those so inclined. Two people play. All you need is a sheet of paper and a pencil or pen. Here's how to play:

  1. Each person thinks of a number between 1 and 50 without telling the other person what the number is. Then, each person writes their number on the sheet of paper.
  2. Decide who is going to go first, by tossing a coin or in some other mutually agreeable way.
  3. Players take turns writing down the positive difference between any two numbers on the sheet of paper.
  4. Numbers cannot appear more than once on the paper.
  5. The player who cannot write down a unique positive difference loses.

Here's an example of how a game might go between Sol and his friend Michele.

  1. Sol thinks of the number 5. Michele thinks 3.
  2. They write 5 and 3 on the paper.
  3. Sol goes first.
  4. 5 minus 3 is 2 so Sol adds 2 to the paper.
  5. The paper now has these numbers: 5 3 2
  6. Michele notices that 5 minus 2 is 3 but 3 is already on the paper.
  7. Michele also notices that 3 minus 2 is 1 so she writes 1 on the paper.
  8. The paper now has these numbers: 5 3 2 1
  9. Sol notices that 5 minus 1 is 4. He writes 4 on the paper.
  10. The paper now has these numbers: 5 3 2 1 4
  11. Sol wins as no more unique differences can be calculated.

Here's another sample game:

  1. Sol thinks 8. Michele thinks 6.
  2. The paper has: 8 6
  3. Michele goes first.
  4. Michele notices that 8-6=2. The paper now has: 8 6 2
  5. Sol notices that 6-2=4. The paper now has 8 6 2 4.
  6. The game is over and Sol wins as no more unique differences can be calculated.

Here are some interesting exploration questions:

  1. Once both numbers are written down is there a way to determine who will win?
  2. Once both numbers are written down does strategy matter, other than who goes first?
  3. For starting numbers of 5 and 3 all numbers between 1 and 5 got written down but when 6 and 8 were the starting numbers only 2, 4, 6 and 8 were possible differences. What determines whether all numbers get used and if not which ones are used and which aren't?

This game is related to Euclid's algorithm and to the greatest common divisor of two integers. At Cut the Knot there's a Java version of this game, Euclid's Game, that you can play alone against the computer. In the computer game the computer picks the two starting number but you can practice determining who should go first.

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  1. very good game, I’m going to use it with the fifth graders in intervention! thank you.

  2. Andree,

    Do report back and let us all know how the games goes with the fifth graders.

  3. Wow, it’s been a long time since this was posted. I have finally done this with two fifth/sixth grade classes. One class was successful. The other class? Not. There are so many reasons why they weren’t and I won’t go into that here. But in that one class, I did manage to lead one student towards some observations, which are included here.

    The only information I gave them were the rules of the game, a demonstration of the game with one student, and that they should make as many observations as possible about the numbers used to see if they can determine who will win in every game. I also introduced the idea of “1st player / 2nd player” so that we had some common vocabulary to discuss strategy. Here are the student observations:

    If both numbers chosen are even, the 2nd player wins.
    If both numbers are odd, the 1st player wins.
    If one number is odd and one is even, then the 1st player wins. (The students have not confirmed this but they feel strongly that it is true.)

    Some student teams quickly learned the value of getting a difference of 1. Other teams used their time to find numbers that did not have many differences at all ( for example: 10 and 15; 3 and 6). Their observations are screaming out for a discussion of why these statements are true.

    I really like the Cut The Knot site and appreciate the link to Euclid’s Game. A confession: would you believe that never ran across Euclid’s Algorithm until I was teaching, and then only through independent reading and never in a class? So this is pretty new to me. I am still groping about in the dark on the relationship with this game and Euclid’s Algorithm and the GCD. I need to spend extended time with pencil and paper as I go through the information. I can see that this exploration, when I am prepared, can be used with the older kids at school.

    I am preparing a post with links to this post. Thank you for this post. It is far richer than it seems at first read.

  4. Andree,

    It’s great to hear back and to hear of the interest your students have with this little game.

    The more Math I learn, the more I write on this blog, the more I pay attention to what readers want, the more I realize that the very basics are what’s needed. And I don’t mean the basics as they’re taught in school. I’m talking about the most fundamental concepts that need to be explored to be internalized and appreciated.

    I attended Stanford, majoring in Math. I didn’t enjoy my mathematical experience there. I much more enjoyed the play and exploration I was doing in High School and Junior High School.

    When I attended the prestigious Ross program in the late 70’s the head of the program, Professor Arnold Ross, a renowned professor and brilliant man always reminded us to think deeply of simple things. That advice has stuck and I’m always looking for fun little explorations that spark curiosity. With curiosity students will learn themselves what they need to be successful.

    You and your students might enjoy my most recent post about 8 simple paper and pencil Math games. They can each be explored to great depth and enjoyment.

    Thanks again, Andree, for checking back in with us.

  5. Neat game! I used this earlier this week with a class of elementary teachers, and they had fun with it. With little prodding from me (which amounted to suggesting pairs of starting numbers to try once they felt satisfied they had figured it all out), they eventually honed in on the complete story. Way cool.

    While they were working on it, a variation sprang to mind: instead of starting with two integers, you can start with any two (positive) fractions. This version of the game is most interesting if we insist that only reduced fractions are allowed to be written down.

    We played that variation a couple times, but my students weren’t as excited about it. *I* thought it had potential, though. :-)

  6. great article, thanks for sharing.

  7. It was really very nice. I enjoyed it with my fridends

  8. I liked the patterns very much.It was very interesting to try with my friends.

  9. Gave this a go in my own time, really enjoyed finding a method that works. Thanks for this little game ( and I know this comment is quite late :) )

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