USA
You are given a set of tiles which are numbered from 1 to 121. You do the following operations repeatedly: you remove all those tiles that are numbered with a perfect square, and renumber the remaining tiles consecutively starting with 1. How many times must you perform the operation before you are left with 1 tile?
Answer:
20
- Let us see the number of tiles with perfect square number on them: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121.
- 11 tiles will be removed, and we're now left with 121 - 11 tiles, or 110 tiles. Now the tiles with perfect square number on them will be: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
- 10 tiles are removed again, now we're left with 110 - 10 = 100.
- Similarly this time the maximum number is 100, so there will be 10 tiles with perfect square on them.
- After 10, there'll will be 100 - 10 = 88, so there will be 9 numbers with perfect squares.
- We can see that we removed 10 tiles two times, and then 9 tiles two times, and in a similar manner we'll have to remove them two times till the number reaches 1.
- This means the number of times this operation has been re done will be 2 x 10 = 20.
