Problem:
A set of tiles numbered through is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with . How many times must the operation be performed to reduce the number of tiles in the set to one?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The first application removes ten tiles, leaving . The second and third applications each remove nine tiles leaving and , respectively. Following this pattern, we consecutively remove tiles before we are left with only one. This requires applications.
Starting with tiles, the first application leaves tiles. The second application reduces the number to tiles. Since two applications reduce the number from to , it follows that applications reduce the number from to , and .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions