see if you can figure out the number?

Question

whats the smallest number that can be represented by the sum of 4 distinct squares in 3 different ways.:

hint -its a number under 100 -
and there are 3 different combinations that work.

have fun

Answer 1

The answer is 78:

1² + 2² + 3² + 8² = 1 + 4 + 9 + 64 = 78
2² + 3² + 4² + 7² = 4 + 9 + 16 + 49 = 78
1² + 4² + 5² + 6² = 1 + 16 + 25 + 36 = 78

I did this creating a spreadsheet with the first 9 squares. There were 126 ways (9 choose 4) ways to pick the four squares which then resulted in a sum. I sorted the sums and found the one that repeated itself 3 times (78).

The answer is 78.

Answer 2

I don't know whether this is minimal, but 1850=5^2+9^2+12^2+40^2=3^2+4^2+15^2+40^2=3^2+4^2+12^2+41^2.

Answer 3

78