Problem 16 · 2023 Math Kangaroo
Hard
Logic & Word Problems
careful-counting
A tower consists of blocks that are labelled from bottom to top with the numbers from 1 to 90. Bob uses these blocks to build a new tower. For each step he takes the top three blocks from the old tower and places them on the new tower without changing their order (see diagram). How many blocks are there in the new tower between the blocks with the numbers 39 and 40?
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Answer: E — 4
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Hint 1 of 2
Blocks move in groups of three (the top three each step), and higher old numbers end up lower in the new tower.
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Hint 2 of 2
Find the group containing 39 and the group containing 40, then count what lies between them.
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Approach: track the blocks in groups of three
- Each step moves a block of three with their order kept: ..., (37,38,39), (40,41,42), ...
- Higher-numbered groups are placed first, so going up the new tower we see 40,41,42 then 37,38,39.
- Between block 40 and block 39 lie 41, 42, 37, 38.
- That is 4 blocks.
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