The number that cannot change
The numbers 1 to 12 are written on a board. Rub out any two of them and write their sum minus 2 in their place. Keep going until one is left.
What is the last number?
Something does not change, whatever you pick
- The 12 numbers 1 to 12, and one move that may be made on any two of them: rubs out both and put their sum minus 2 back. Any two can be chosen.
- Play two moves and see. Take 1 and 2: that leaves 1. Now take 1 and 3: that leaves 2. A different choice gives different numbers, so watch something else.
- Watch how many there are. Every move takes two numbers away and puts one back, so from 12 down to 1 is 12 − 1 = 11 moves, however you play.
- Now watch the total. Two numbers leave and their sum minus 2 arrives, so the total falls by 2 every move, whichever two were chosen.
- 11 moves, each costing 2, is 11 × 2 = 22. The numbers add to 78, so the last number is 78 − 22 = 56, however it is played.
How it works.
- 01
The 12 numbers 1 to 12, and one move that may be made on any two of them: rubs out both and put their sum minus 2 back. Any two can be chosen.
- 02
Play two moves and see. Take 1 and 2: that leaves 1. Now take 1 and 3: that leaves 2. A different choice gives different numbers, so watch something else.
- 03
Watch how many there are. Every move takes two numbers away and puts one back, so from 12 down to 1 is 12 − 1 = 11 moves, however you play.
- 04
Now watch the total. Two numbers leave and their sum minus 2 arrives, so the total falls by 2 every move, whichever two were chosen.
- 05
11 moves, each costing 2, is 11 × 2 = 22. The numbers add to 78, so the last number is 78 − 22 = 56, however it is played.