Wythoff's game strategy for kids: how to play it at home

Wythoff's game is a two player game with two piles of objects. On your turn you take any number from one pile, or the same number from both piles. Whoever takes the last object wins. The strategy comes down to finding a few safe pairs, and children can discover them with a parent at the table.

How the game works

Set out two piles. Say one pile has 5 and the other has 8. On each turn a player picks one of two moves. You can take as many as you like from one pile. Or you can take the same number from both piles, say 2 from each. The player who clears the table wins.

You can play it with real objects or open the board at Wythoff's game, which keeps the piles tidy and lets your child try a position again.

How I play it with a child

I spent more than twenty years as a paramedic and a fire captain. A lot of that work was staying calm and thinking one step ahead. I see the same habit in this game, and I try to model it more than explain it.

Start small. Two piles of 2 and 3 are plenty for a first game. Let your child move first and play honestly. When they lose, ask one question: "Where did that game turn?" Then set it up again and let them try a different first move.

After a few rounds, give them a puzzle instead of a game. "It is your turn and the piles are 1 and 2. What do you do?" Let them try every move. They will find that every choice leaves you a way to win. That is the first cold position, and they found it themselves.

The safe pairs

A cold position is one where the player about to move will lose if the other player makes no mistakes. The cold pairs start like this:

(0, 0), (1, 2), (3, 5), (4, 7), (6, 10), (8, 13)

The whole strategy is to hand your opponent one of these pairs. Say the piles are 5 and 9. Taking 1 from both leaves 4 and 8, which is not on the list. Taking 6 from the 9 leaves 5 and 3, and now your opponent is stuck. Working through that with your child is the real lesson.

Older kids enjoy spotting the patterns, and this is where the golden ratio comes in. Starting with (1, 2), the gap between the two numbers in each pair grows by one: 1, 2, 3, 4, 5. The golden ratio, often written phi, is about 1.618, and phi squared is about 2.618. The nth safe pair is the whole number part of n times phi, then the whole number part of n times phi squared. Try n = 3. Three times 1.618 is about 4.85, and three times 2.618 is about 7.85, which gives (4, 7). Divide the larger number by the smaller in later pairs and the answer creeps toward 1.618. You do not need to teach any of that. Leave the list on the fridge and wait for the question.

The math thinking it builds

Wythoff's game asks a child to hold two numbers in mind and compare them. It asks them to look at a position and consider every move before choosing one. It rewards working backward from the end, which is how a lot of real problem solving goes.

I built Classical Math Games at the end of 2025 after my son cried over basic math facts. I wrote about that in why I built Classical Math Games. Games like this one were part of the answer for us, because they put the thinking first and let the facts come along with it. If your child likes this one, Tower of Hanoi and Euclid's game have the same patient feel.

Facts still matter, since a child who knows 8 minus 3 without counting has more room to think about strategy. When you want steady practice there, Flash Cards and number bonds are the tools we use at our house.

Try Flash Cards for free.