2002 AMC 10A Problems/Problem 22
Problem
A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one?
Solution
Solution 1
The pattern is quite simple to see after listing a couple of terms.
Thus, the answer is .
Solution 2
Given tiles, a step removes tiles, leaving tiles behind. Now, , so in the next step tiles are removed. This gives , another perfect square.
Thus each two steps we cycle down a perfect square, and in steps, we are left with tile, hence our answer is .
See also
2002 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
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