On the first day, the status of every cell is flipped. On the second day, the status of every even-numbered cell is flipped. On the third day, the status of every cell whose number is divisible by three is flipped. And so forth.
So at the end, the status of every cell will have been flipped a number of times equal to the number of divisors the cell’s number has, including 1 and the cell number itself. For example, 24 has 8 divisors (1,2,3,4,6,8,12,24), so its status was flipped 8 times. Since 8 is even, the status of the cell is the same as when it started (“-” with Sreeram’s correction). Cells whose numbers have an even number of divisors have the same symbol at the end as when they started; cells whose numbers have an odd number of divisors have the opposite symbol.
Every non-square number has an even number of divisors, because the divisors can be paired off to give products that equal the original number (1×24, 2×12, 3×8, 4×6). Square numbers have an odd number of divisors, because there’s one divisor, the number’s square root, which doesn’t get paired with a different number. E.g., 36 has 7 divisors (1,2,3,6,12,18,36) – if you try to pair them off you get 1×36, 2×18, 3×12, 6×6…but oops, you don’t count 6 twice among its divisors. So cells with non-square numbers end up with the same symbol they started with, and cells with square numbers end up with the opposite symbol.
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