Math PicnicEvery sheet is a promise

The number that cannot change, step by step

The numbers 1 to 5 are on a board.

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The question

What is the last number?

Something does not change, whatever you pick

123456789101112
13456789101112
2456789101112
12−1=11
each move:−2
78−22=56
  1. 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. Nobody says which two, and that is the difficulty — there are hundreds of ways to play.
  2. Play two moves and see. Take 1 and 2: that leaves 1. Now take 1 and 3: that leaves 2. A different pair would have left a different board, so the numbers themselves are no help.
  3. So stop watching the numbers and watch the COUNT. Every move takes two numbers away and puts one back, so the count drops by exactly 1 — from 12 down to 1 is 12 − 1 = 11 moves, however anyone plays.
  4. Now watch the TOTAL. Two numbers leave and their sum minus 2 arrives, so the total falls by 2 every single move — no matter which two were chosen. That is the thing that does not depend on the choosing.
  5. 11 moves, each costing 2, is 11 × 2 = 22. The numbers add to 78, so the last number is 78 − 22 = 56 — and it would be 56 whichever pairs anyone chose, which is why nobody has to play it out.

How it works.

  1. 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. Nobody says which two, and that is the difficulty — there are hundreds of ways to play.

  2. 02

    Play two moves and see. Take 1 and 2: that leaves 1. Now take 1 and 3: that leaves 2. A different pair would have left a different board, so the numbers themselves are no help.

  3. 03

    So stop watching the numbers and watch the COUNT. Every move takes two numbers away and puts one back, so the count drops by exactly 1 — from 12 down to 1 is 12 − 1 = 11 moves, however anyone plays.

  4. 04

    Now watch the TOTAL. Two numbers leave and their sum minus 2 arrives, so the total falls by 2 every single move — no matter which two were chosen. That is the thing that does not depend on the choosing.

  5. 05

    11 moves, each costing 2, is 11 × 2 = 22. The numbers add to 78, so the last number is 78 − 22 = 56 — and it would be 56 whichever pairs anyone chose, which is why nobody has to play it out.

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