## Monday, January 14, 2008

### A Fun Arithmetic Game That Sparks Exploration >> Fun Math Blog

Unplugged Project: Puzzles: I missed Puzzle week for the project so I offer my work now.

A fun arithmetic game that sparks exploration >> Fun Math Blog
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:
• 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.
• Decide who is going to go first, by tossing a coin or in some other mutually agreeable way.
• Players take turns writing down the positive difference between any two numbers on the sheet of paper.
• Numbers cannot appear more than once on the paper.
• The player who cannot write down a unique positive difference loses.
I have used this game in two 5th/6th grade mathematics classes so far. As simple as this game appears on the surface, I suggest that you read the rest the original post to learn its importance to your child's mathematics education. It is a rich and valuable game for them to puzzle over.

I have posted, as a comment at the original post, what my classes have learned in forty minutes of playing. I won't repeat the observations here in hope that you will play and reach your own conclusions first. But we have not even scratched the surface of the extensions of this game. I need to do some independent study about Euclid's Algorithm before I use it with my older students. I am even considering using it with my college classes.

If, after learning this game with your child, you find that there is a time when you cannot play with her, you will be interested in this quote from Wild About Math!:
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.
I use the Cut The Knot site often as a problem solving tool. There are many Java applets there for many classes of problems. We especially use it with the Josephus Flavius problem (which also has extentions and applications to many other problems) but with a far less gruesome problem.

Here are three screen grabs of Euclid's Game. In the first image, I have chosen to go first because the computer generated numbers are 29 and 18:

In the next image, I purposefully made a subtraction error to demonstrate the feedback that the applet provides:

And finally, I won!

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