Problem 26 · 2010 Math Kangaroo
Stretch
Algebra & Patterns
sum-constraintwork-backward
The numbers from 1 to 10 are written 10 times each on a board. Now the children play the following game: one child deletes two numbers off the board and writes instead the sum of the two numbers minus 1. Then a second child does the same, and so forth until there is only one number left on the board. The last number is
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Answer: B — 451.
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Hint 1 of 2
Each move replaces two numbers with one, so track how the count and the total change.
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Hint 2 of 2
The total drops by exactly 1 every move, regardless of which numbers are chosen.
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Approach: track the invariant: total minus number of moves
- The starting numbers sum to 10×(1+...+10) = 550, and there are 100 numbers.
- Each move removes one number and lowers the total by 1; reaching one number takes 99 moves.
- The last number is 550 − 99 = 451.
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