2002 AMC 10A Problems/Problem 21

Problem

A set of tiles numbered 1 through 100 is modified repeatedly as follows: 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?

$\text{(A)}\ 10 \qquad \text{(B)}\ 11 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 19 \qquad \text{(E)}\ 20$

Solution

Given $n^2$ tiles, a step removes $n$ tiles, leaving $n^2 - n$ tiles behind. Now, $(n - 1)^2 = n^2 - n + (1 - n) < n^2 - n < n^2$ , so in the next step $n - 1$ tiles are removed. This gives $(n^2 - n) - (n - 1) = n^2 - 2n + 1 = (n - 1)^2$ , another perfect square, and the process repeats.

Thus each two steps we cycle down a perfect square, and in $(10 - 1)\times 2 = 18$ steps, we are left with $1$ tile.

2002 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 20	Followed by Problem 22
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
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2002 AMC 10A Problems/Problem 21

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