# 6. CONDITION

On an 8x8 chessboard, 64 checkers numbered from 1 to 64 are placed. 64 students take turns approaching the board and flipping only those checkers whose numbers are divisible by the ordinal number of the current student. A "Queen" is a checker that has been flipped an odd number of times. How many "Queens" will be on the board after the last student has stepped away from it?
Solution. Obviously, each checker is flipped as many times as its number has divisors. Therefore, the number of "queens" will be the number of numbers from 1 to 64 that have an odd number of divisors, and this property is only possessed by perfect squares. Thus, the numbers of the "queens" remaining on the board will be $1, 4, 9, 16, 25, 36, 49$, and 64, which totals 8.

Answer: 8.

## Grading Criteria for Problem 6.

The answer is correct, but there are remarks on the clarity of its presentation and justification. 5 points.