Activities for Families

It is a bad idea to pick a number, such as 46, that is not a power of 2 times some number between 1 and 20 – it will never get hit. Some numbers with lots of factors of 2, such as 32, are more likely to be hit because more starting numbers can get to them.

You can triple the number each time instead of doubling it. You can double it and add 1 each time. For younger players, select numbers above 10 and not above 60, and select a random number from 1 to 10.

Introduction

Players start the game by secretly picking 5 distinct numbers larger than 20 and not bigger than 120. After they are selected, they are written where all can see them. Using Number Cards or some other device, a random number from 1 to 20 is created. That number is repeatedly doubled until either someone’s number is hit for the first time or the number becomes bigger than 120. The first player to have all five numbers hit is the winner.

Analysis

The question is: Which are the best five numbers to pick? Here are some ideas to think about.

Rule Always pick a number that is a power of 2 times a number from 1 to 20.

If you pick a number such as 23 or 46, they can never be hit and you are guaranteed to lose.

Rule Never pick a number that is twice another number you could have picked but didn’t.

If you pick 44, why not pick 22 instead? If the other person picks 22, you will miss a round.

Further analysis The numbers from 1 to 20 are equally likely to be picked. However, because 9 leads to 18, 18 is twice as likely as a starting point than say 11 is. If you combine the ways to get different starts, the starting points have the following probabilities:

11 – 1/20 (from 11)
12 – 3/20 (from 3, 6, and 12)
13 – 1/20 (from 13)
14 – 2/20 (from 7 and 14)
15 – 1/20 (from 15)
16 – 5/20 (from 1, 2, 4, 8, and 16)
17 – 1/20 (from 17)
18 – 2/20 (from 9 and 18)
19 – 1/20 (from 19)
20 – 3/20 (from 5, 10, and 20)

Clearly the best numbers to use are multiples of 16, 12, and 20. A simple strategy is to use the five numbers: 32, 64, 24, 48, and 40. These numbers will not always win, but they should do very well for you over time.

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!