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Consider the follow magic square highlighted in yellow. The sum of its rows and columns are in green and the sum of the diagonals in red. All of its sums are a square number with the sum of the whole square also being a square number.

Magic Square of Different Square Sums

The above square has a repeated sum in one each of the diagonal and one row and one column.

question 1
What is the smallest set numbers that you can put into a 3x3 grid so ALL 8 of its sums are all different square numbers and the sum of the whole also being a square number?

question 2
Using triangle numbers instead of square numbers 1, 3, 6, 10, 15, 21, 28, 36, 45,......... and so on What is the smallest set of number to result in all of its sums being triangle numbers with the sum of the whole also being a triangle number?

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answer 1 (square sums)

enter image description here

This is the smallest because...

441 is the smallest square that is the sum of three distinct squares in two non-intersecting ways. This solution also has the lowest possible maximum value of 157.

answer 2 (triangular sums)

enter image description here

This is smallest because...

Isn't it obvious?

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  • $\begingroup$ ...is the smallest square that is the sum of three distinct squares in two non-intersecting ways. How do you know this? (I know because wikipedia knows.) $\endgroup$
    – loopy walt
    Feb 16 at 22:51
  • $\begingroup$ @loopywalt Interesting. I computed, via code, all sums of three distinct squares less than 1000, essentially generating the data shown on that page. $\endgroup$
    – Daniel Mathias
    Feb 16 at 23:27
  • $\begingroup$ The triangle solution has a cell of 34. Is there another solution that uses the smallest numbers to populte the grid $\endgroup$
    – Maff
    Feb 18 at 9:54
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    $\begingroup$ @Maff If you allow one of the numbers to be zero, then a solution with a maximum of 26 is possible. $\endgroup$
    – Daniel Mathias
    Feb 18 at 10:58
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    $\begingroup$ @Maff See updated solutions. $\endgroup$
    – Daniel Mathias
    Feb 19 at 17:36

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