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I assume by "square numbers" you mean perfect squares.
You didn't say how many of each were allowed:
1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1²= 23
1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 2² = 23
1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 2² + 2² = 23
1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 3² = 23
1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 2² + 2² + 2² = 23
1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 2² + 3² = 23
1² + 1² + 1² + 1² + 1² + 1² + 1² + 2² + 2² + 2² + 2² = 23
1² + 1² + 1² + 1² + 1² + 1² + 1² + 4² = 23
1² + 1² + 1² + 1² + 1² + 3² + 3² = 23
1² + 1² + 1² + 2² + 2² + 2² + 2² + 2²= 23
1² + 1² + 1² + 2² + 4² = 23
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Which two square numbers add up to 225?
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The two square numbers which add up to 13 are 9 (which is equal to 3 squared), and 4 (which is equal to 2 squared).
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16,25,36 add to 77. They are the squares of consecutive numbers 4,5,6
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