Problem 26 · 2010 Math Kangaroo
Stretch
Algebra & Patterns
work-backward
The numbers from 1 to 10 are written on a board. The children now play the following game: one child erases two of the numbers and writes in their place the sum of the two numbers minus 1. Then a second child does the same, and so on, until only one number is left on the board. The last number is …
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Answer: C — 46
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Hint 1 of 2
Watch what each move does to the running total of all numbers on the board.
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Hint 2 of 2
Every move lowers that total by exactly 1, no matter which numbers are chosen.
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Approach: track an invariant (total drops by 1 per move)
- Replacing two numbers by (their sum − 1) lowers the board's total by 1 and the count by 1.
- Starting from the numbers 1–10 (total 55, ten numbers), reaching one number takes 9 moves, dropping the total by 9.
- The final number is 55 − 9 = 46, independent of the order.
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