Problem 24 · 2009 Math Kangaroo
Stretch
Logic & Word Problems
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In Funny-Foot-Land men and women wear the same sort of shoes. Each man has a left foot that is two sizes bigger than his right foot. Each woman has a left foot that is one size bigger than her right foot. However, shoes are only sold in pairs of the same size. To save money some friends decide to buy shoes together. After putting on their new shoes, two shoes are left over — one of size 36 and one of size 45. What is the minimum number of people in that group?
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Answer: A — 5
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Hint 1 of 2
Each person uses two different shoe sizes; shoes come only in same-size pairs.
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Hint 2 of 2
Think of going from size 36 up to size 45 in steps of 1 (a woman) or 2 (a man) — how few steps reach 45?
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Approach: link size 36 to size 45 with the fewest people
- A man uses sizes that differ by 2; a woman uses sizes that differ by 1.
- Because exactly the sizes 36 and 45 are each left with one spare shoe, the people must form a chain of shared sizes from 36 to 45.
- From 36 to 45 is a gap of 9; using mostly steps of 2, the fewest people needed is 5.
- So the minimum number of people is 5.
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