Problem 26 · 2017 Math Kangaroo
Stretch
Logic & Word Problems
Number Theory
caseworkcareful-counting
Paul wants to write a positive whole number onto every tile in the number wall shown, so that every number is equal to the sum of the two numbers on the tiles that are directly below. What is the maximum number of odd numbers he can write on the tiles?
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Answer: B — 14
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Hint 1 of 2
Work in parity (odd/even): a tile is odd exactly when the two below it differ in parity.
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Hint 2 of 2
Search the bottom row patterns of the six-row wall to maximise odd tiles.
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Approach: reduce to parity and optimise the bottom row of the wall
- Only odd/even matters: a tile is odd exactly when the two tiles below it have different parities, so the whole wall is fixed once the bottom row's pattern of odds and evens is chosen.
- For the six-row (21-tile) wall, testing the bottom-row patterns, the best choice (such as even, odd, odd, even, odd, odd) makes 14 of the 21 tiles odd.
- So the maximum number of odd tiles is 14, choice B.
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