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    Why sqrt? Because the result matches the pattern.

           1  2  3  4  5  6  7  8  9 10 11        days 
           1  1  3  2  4  4  6  4  7  7  9          student count
           1  1  1  0  0  0  0  0  1  1  1          result 0 (even) / 1 (odd)
           1  1  1  2  2  2  2  2  3  3  3          square root days