**Math concepts:** subtraction within 100, number patterns, mental math

**Number of players:** 2 or 3

**Equipment:** printed hundred chart (also called a *hundred board*), and highlighter or translucent disks to mark numbers — or use this online hundred chart

## Set Up

Place the hundred chart and highlighter where all players can reach them.

## How to Play

- Allow the youngest player choice of moving first or second; in future games, allow the loser of the last game to choose.
- The first player chooses a number from 1 to 100 and marks that square on the hundred chart.
- The second player chooses and marks any other number.
- On each succeeding turn, the player subtracts any two marked numbers to find and mark a difference that has not yet been taken.
- Play alternates until no more numbers can be marked.

## Endgame

The player who marks the last number wins the game.

## For Advanced Students

When you play Euclid’s game on paper, circle the original pair of numbers. Collect several finished game boards. Compare the pattern of the numbers marked on each game.

- Can you explain why some games have few numbers marked and others have many?
- If you knew the first two numbers, would you be able to predict how many squares would be marked in the end?

For a hint, check out this page at Amby’s Math:

## Comments

I first discovered Euclid’s game at Alexander Bogomolny’s outstanding website, Cut the Knot. You can play a Java version online here:

## Update

Now you can play the 100-chart version online. too:

This post is an excerpt from my book *Addition & Subtraction: Math Games for Elementary Students*, available now from your favorite online book dealer.

Excellent! Don’t know why I didn’t think of this streamlining of the paper-pencil game. thank you.

I’m glad you liked it. Playing with pencil on scratch paper is quick and easy, but using the hundred chart as a game board makes it feel more like a “real” game to me — and the number patterns show up clearly on a chart.

lol lets play math lol

For me math was allways anything other then playing. Greeting, Niki Buchen

I hate math, or should I say Math hates me. But, I don’t know why I enjoy challenging myself with games that has math on it.

Love it! I am so glad that I came across your website tonight. I am starting my first homeschooling year with my 6 year old this fall and I am really excited about our math lessons. This game will be a great addition to our activities. We are doing a lot with bean bag tossing games for addition and subtraction practice.

oh wow sounds like fun ❤

I love this! I played it for a couple of days with my seven year old, and then wanted to look up Euclid and find out more about it, and that led to us playing with ratios and having quite a bit of fun. Thank you so much for your wonderful blog! And the ebook, which I will try to comment on when I have read more of it.

Hi, Christy! I’m glad to hear that you’ve enjoyed the game, and I love your extension of it in Playing with Ratios.

Could you please explain to me how this is related to Euclid? Thank you

The game is named for Euclid’s Algorithm, his method for finding the greatest common divisor (GCD) of any two numbers. The mechanics of the game are very similar to — though less efficient than — the method Euclid used. Euclid’s Game finds the GCD of the original two numbers (the smallest number marked when the game is finished) and all of its multiples that are less than the larger of the original numbers.