Math Detours · Grade 5 · Around the corner · 20

The race to 21

Take turns adding 1, 2 or 3. Whoever says 21 wins. Work backwards from the end and, if you go first, you can always win.

Name ____________________Date ____________

Tonight's idea

Two players start at 0. They take turns adding 1, 2 or 3 and saying the new total. Whoever says 21 wins.

It looks like luck. It is not. Start at the end and work backwards.

Suppose you say 17. The other player must say 18, 19 or 20. From each of those you can reach 21. So saying 17 is as good as saying 21.

Now 17 is your new target. The same thinking says: get to 13. Then 9, then 5, then 1. Call these the safe numbers.

+4 +4 +4 +4 +4 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
The safe numbers. From one safe number you can always reach the next, whatever the other player adds.

Why do they go down in fours? The other player adds 1, 2 or 3. You then add 3, 2 or 1. The two turns together always make 4.

Go first and say 1. After that, whatever they add, you add the number that makes 4.

Watch one: a whole game

You go first. The other player tries different things, and it never helps.

You say 1. They add 2 and say 3.
You add 2: 5. They add 3 and say 8.
You add 1: 9. They add 1 and say 10.
You add 3: 13. They add 3 and say 16.
You add 1: 17. They add 2 and say 19.
You add 2: 21. You win.

Play against the computer, then change the target: mathdetours.com/demos/race21.html.

Your turn

  1. You have just said 13. The other player adds 2. What do you say next? They then add 3. How do you finish?
  2. (a) The other player goes first and says 3. What do you say? (b) The other player goes first, says 1, and knows the secret. Can you win? Explain.
  3. Change the game: now whoever says 20 wins. You still add 1, 2 or 3. Work backwards to find the safe numbers. Do you want to go first or second?
  4. A new game: whoever says 30 wins, and you may add 1, 2, 3 or 4. (a) What do the safe numbers go down in now, and why? (b) List them. (c) First or second?

Stretch: 21 is poison

Back to adding 1, 2 or 3, but now whoever says 21 loses. Which number do you want to say near the end? Find all the safe numbers. Should you go first or second?

Answers

  1. They say 15, so you add 2 and say 17. They add 3 and say 20. You add 1 and say 21.
  2. (a) Add 2 and say 5. You are now on the safe numbers and they are not. (b) No. They can reach 5, 9, 13, 17 and 21 whatever you add. You can only win if they make a mistake.
  3. Count back in fours from 20: 16, 12, 8, 4, 0. The total starts at 0, which is already a safe number. So go second, and always add what makes 4.
  4. (a) Fives. They add 1, 2, 3 or 4, and you add 4, 3, 2 or 1. Each pair of turns makes 5. (b) 25, 20, 15, 10, 5, 0. (c) Second, because the game starts on the safe number 0.

Stretch. You want to say 20. Then the other player has to say 21. So it is the race to 20 from exercise 3: the safe numbers are 20, 16, 12, 8, 4 and 0, and you should go second.

Next year's game has several piles instead of one total: grade 6, Nim.

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