Unique Sums of Squares: Lower Secondary Mathematics Competition Question

This is an extension to yesterday’s question. We are still looking at how positive whole numbers can be expressed as the sums of square numbers.

This time we are looking at how some numbers can be expressed as different sums of squares. Again, we are only allowed a maximum of four squares and a minimum of, obviously, just one.

For example, 9 can be simply expressed as 3², but it can also be made from 2² + 2² + 1². The number 11 can only equal 3² + 1² + 1²; there are no other sums of squares that can equal 11 unless we use more than four of them.

So, our question this time is to find those numbers between 20 and 30 inclusive that can be expressed as the sums of squares in only one unique way and using a maximum of four squares.




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