Euclid's game: the Euclidean algorithm for kids at the kitchen table

Euclid's game starts with two piles of counters. On each move you subtract the smaller pile, or a multiple of it, from the larger pile, and you keep going until one pile is empty. Along the way your child is running the Euclidean algorithm, the old method for finding the greatest common factor of two numbers.

How the game works

Pick two numbers. Say 20 and 6. The smaller pile is 6, so you may take 6, 12, or 18 from the 20. You cannot go below zero. If you take 18, the piles become 2 and 6. Now the smaller pile is 2, so you can take 2, 4, or 6 from the 6. Take 6 and that pile is empty. The puzzle is done.

Notice what was left at the end: a pile of 2. That is the greatest common factor of 20 and 6. It works every time, and that is the quiet surprise inside the game.

The version at Euclid's game is a calm solo puzzle. Your child works both piles, there is no opponent, and they can start over with new numbers whenever they like.

How I play it with a child

I have three kids, and I learned the hard way that a lesson goes better when it starts with a question instead of a rule. So I do not explain the algorithm. I sit beside them and play.

Start with friendly numbers like 12 and 8. Let your child choose each move and say out loud what they are taking. When a pile empties, ask, "What number was left?" Then try 15 and 10, and 18 and 12. After three or four puzzles, ask whether the leftover numbers have anything in common with the starting pair.

Some children see it right away. Some need a few more evenings. Both are fine. If they get stuck on a subtraction, slow down and let them use real counters. The game is patient, so you can be too.

Later, try a pair like 13 and 8. The leftover is 1, and that tells you the two numbers share no factor except 1. That is a nice thing for a child to discover on their own.

The math thinking it builds

Euclid's game gives a child steady practice with subtraction and multiples without it feeling like a drill. They have to look at the smaller number and ask how many of it fit inside the larger one. That is division, arriving from the side.

It also builds the habit of following a process all the way through and then asking what it means. Most worksheets ask for an answer. This asks a child to notice something.

When my son cried over basic math facts, I realized the facts were standing in the way of the interesting parts of math. I wrote about that in why I built Classical Math Games. A child who still counts on fingers to find 18 minus 12 will spend all their attention on the arithmetic and miss the pattern. When the facts are solid, games like this one open up.

That is why we pair the puzzles with fact practice at our house. Flash Cards cover addition, subtraction, multiplication and division, and number bonds help younger kids see how numbers split apart. If your child enjoys this kind of puzzle, Wythoff's game and Tower of Hanoi are good next steps.

When the facts feel easy, puzzles like Euclid's game feel like play.

Try Flash Cards for free.