Twenty-One Years 2 - 6

with a photo from your classroom.
doug@blackdouglas.com.au

### Summary

This easy to learn game can be the basis of a significant problem solving exercise. To be successful children have to ask What number do I have to reach to be certain of being first to 21?. They also have to keep their strategy secret, because if both players know a successful set of numbers, there is no point playing. Unless of course we challenge their thinking What happens if we change the target number? or another variation. Suitable for threading.

This activity requires the same type of reasoning as Task 86, Thirty-One.

### Materials

• One calculator between two players.

### Procedure

The players take turns entering a non-zero single digit number and +. The aim is to be the first to reach a predetermined target number such as 21. A player who overshoots loses.
• Are there significant numbers prior to reaching the target number?
• Discuss the strategies children are using to arrive at the target.
If the target is 21 and you are 'further back' than 12 when it's your turn, you can't get to 21 in one press. If you are 'further forward' than 12 when it's your turn you must win (if you play like an expert). So, the special number is 12 because you can always win if the calculator is on 12 (or more) when it is your turn.

But that can only happen for sure if your opponent has to play from 11. If on 11, whatever your opponent plays means you will be facing 12 or more. Therefore you have to force your opponent to be on 11.

So now examine what happens if you go first and...

• ...press 9. Then your opponent adds 2 to force you to 11 and you lose.
• ...press 8. Then your opponent adds 3 to force you to 11 and you lose.
• ...press 7. Then your opponent adds 4 to force you to 11 and you lose.
• ...press 6. Then your opponent adds 5 to force you to 11 and you lose.
• ...press 5. Then your opponent adds 6 to force you to 11 and you lose.
• ...press 4. Then your opponent adds 7 to force you to 11 and you lose.
• ...press 3. Then your opponent adds 8 to force you to 11 and you lose.
• ...press 2. Then your opponent adds 9 to force you to 11 and you lose.
• ...press 1. Then your opponent adds 10 to ... No! they can't do that!
... and no matter what they press you can take them to 11 and they must lose. Yay!

### Content

• algebra, linear
• group (or skip) counting
• mathematical conversation
• multiplication
• pattern generalisation
• pattern interpretation
• pattern recognition
• problem solving
• recording - calculator
• recording - written
• subtraction
• times tables
But happens if your opponent goes first?

• If they press 1, they possibly know the strategy. You can give up if they force you to 11 on their next move.
• If they don't press 1, they either don't know the strategy or they are trying to find out if you do. Watch and wait. See what they do next time they go first.
• If they don't consistently press 1, win as many games as you can before they catch on.
Do you notice anything about 1, 11 and 21?

Encourage children to record an explanation of their strategy once they have become confident with it.

### Variations

What happens if we...
1. Start at the target and work back to zero?
2. The person who reaches the target first is the loser?
3. Change the target number at the start of a new round?
4. Restrict the single digit numbers to, say, 1, 2, 3?

Calculating Changes ... is a division of ... Mathematics Centre