Problem 24 · 2001 AMC 8
Stretch
Logic & Word Problems
careful-counting
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Answer: B — 5 white pairs.
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Hint 1 of 2
Every triangle on the top half lands on exactly one on the bottom — nothing vanishes. So just track one half and watch each color get used up.
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Hint 2 of 2
Work color by color: count how many reds and blues are "spent" by the given pairs, see what's forced to pair with white, and the leftover whites pair with each other.
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Approach: account for each color on one half (conservation)
- One half has 3 red, 5 blue, 8 white. The 2 red-red pairs spend 2 reds, leaving 1 red; the 3 blue-blue pairs spend 3 blues, leaving 2 blue.
- Now the leftovers must pair with whites. The 2 red-white pairs spend that last red and 1 white. The 2 leftover blues can't make a 4th blue-blue pair (only 3 are allowed), so each pairs with a white — using 2 more whites.
- Whites spent: 1 (with red) + 2 (with blue) = 3, leaving 8 − 3 = 5 whites per half. Those face each other as 5 white-white pairs.
- The engine here is conservation: a folded figure pairs everything one-to-one, so once you know how each "colored" piece is consumed, the remainder of any one color is forced.
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