Problem 29 · 2017 Math Kangaroo
Stretch
Number Theory
caseworksum-constraint
Sarah 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 it. What is the maximum number of odd numbers Sarah can write on the tiles?
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Answer: D — 10
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Hint 1 of 2
A tile is the sum of the two below it, so it is odd only when exactly one of those two is odd.
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Hint 2 of 2
Arrange the bottom row's parities to keep as many tiles odd as possible.
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Approach: track parities up the wall
- With 15 tiles (rows of 5, 4, 3, 2, 1), each upper tile is odd exactly when the two below it differ in parity.
- Searching parity patterns for the bottom row, the most odd tiles achievable across the whole wall is 10.
- So the maximum number of odd tiles is 10.
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