Problem 21 · 2023 Math Kangaroo
Stretch
Logic & Word Problems
careful-counting
A tower is built from bricks labelled 1 to 50, from bottom to top. Bob builds a new tower: each time he takes the top two bricks off the old tower (keeping their order) and places them on top of the new tower (see picture). When he is finished, which two bricks lie directly on top of each other?
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Answer: E — 27 and 30
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Hint 1 of 2
Each move peels the top two bricks (keeping their order) and drops them on the new tower, so the new tower forms in pairs.
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Hint 2 of 2
Track the new stack pair by pair and look for which two of the listed bricks end up directly one above the other.
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Approach: simulate the pair-by-pair transfer onto the new tower
- The old tower has 50 on top of 49 on top of 48 … down to 1; each move lifts the top two as a pair and stacks them on the new tower.
- So the new tower is built bottom-up as 49,50, then 47,48, then 45,46, and so on, in steps of two.
- Following this all the way down, the pair 30 then 27 lands one directly above the other in the finished new tower.
- So the two bricks on top of each other are 27 and 30, answer E.
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