Question

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?

$10$

$11$

$18$

$19$

$20$

Solution:

Starting from $n^2$ tiles, one operation removes the $n$ perfect squares, leaving $n^2-n.$ The next operation removes $n-1$ perfect squares, leaving $n^2-n-(n-1)=(n-1)^2.$

So every two operations reduce $n^2$ to $(n-1)^2.$ Going from $10^2=100$ down to $1^2=1$ takes $2(10-1)=18$ operations.

Thus, the correct answer is C.

2002 AMC 10A Problem 22

Solution:

Problem 22 in Other Years