New Year's Countdown Day 15: Anti-magical Diagonals

Algebra Level 5

Define a heterosquare to be a square grid containing consecutive positive integers starting from 1 such that the sums of the integers in each row, column, and long diagonal are all different. Furthermore, define an anti-magic square to be a heterosquare whose sums form a consecutive sequence of integers.

If there exists a 15×1515 \times 15 anti-magic square, then there are nn possible values d1,d2,,dnd_1, d_2, \dots, d_n for the sum of its two diagonal sums. Find the value of ni=1ndi.n \displaystyle \sum_{i = 1}^n d_i.


Note: A diagonal sum is the sum of the numbers along a long diagonal of the square. For example, the two diagonal sums of the following square are 1+5+9=151 + 5 + 9 = 15 and 3+5+7=15:3 + 5 + 7 = 15: 123456789 \begin{array}{|c|c|c|} \hline 1 & 2 & 3 \\ \hline 4 & 5 & 6 \\ \hline 7 & 8 & 9 \\ \hline \end{array}


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