Activities for Families

After your child becomes familiar with the game, encourage them to look for the remarkably simple strategy for always winning – once they discover it, see if they can explain why it works.

Introduction

Start with any number, say 20. Let the child decide whether to go first or second. During their turn, a player may subtract any divisor of the current number from the number. The player forced to 0 loses.

Analysis

As usual, a good strategy for learning about this game is to look at a simpler version of the game, which in this case means starting with very small numbers. If it is your turn and you are faced with each of these numbers, here is what will happen: 1 – lose, 2 – win, 3 – lose, 4 – win, 5 – lose, 6 – win, 7 lose, and 8 win. By now the pattern is clear – if it is your move and you have an odd number, then you will lose; if you have an even number, then you will win.

Finding the winning strategy is a big step, but let’s go deeper. Why does this work? What are the properties of odd and even numbers that create this situation? Set this question before your child and give them a lot of time to think about it and experiment with it – there is no hurry, and this process of wrestling with a question is invaluable and should not be short circuited.

Some experimentation with small numbers quickly reveals what is going on. If you have an odd number, all of the divisors are odd, so when you subtract any divisor the result is an even number. Consequently, odd numbers on one turn always lead to an even number on the next turn. Even numbers always have both odd and even numbers for divisors. So, the situation is not quite the same. However, if you have an even number, your goal is to give your opponent an odd number, and there is an easy way to do that – select the divisor 1 and subtract it!

First and foremost, playing math games should be fun, like any other game your family plays together!

Please let your child make poor plays (mistakes) without correction, and resist the urge to tell them the best ways to play. Bit by bit, your child will get better at the game, and they will learn so much more if you let them figure things out. There is no hurry.

If you see your child make a mistake, ask them to describe why they decided to do what they did. If your child is stuck and doesn’t know which play to make, ask them to describe the pros and cons of their choices, or ask them about how they solved a similar situation in the past. If your child doesn’t remember how to do a calculation, discuss with them the methods they know for figuring it out. These conversations are important for helping your child to develop mathematically.

Through math game play and math conversations, you are helping your child learn to enjoy math and develop important problem solving skills!