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Is there any pattern in the relationship between the numbers?
If you spot a pattern can you prove it always occurs?
The pattern can be generalised to four times the sum of four squares and so on (see the solution).
eg $1 = 1^2 + 0^2$
Sums of Squares
Age 16 to 18
Challenge Level 
Why do this problem?
It provides an opportunity for learners to experiment with numerical examples, to observe a pattern and make a conjecture, to try to explain their own conjecture and to formulate a proof of the conjecture which involves only simple algebra.Possible approach
Suggest that the learners make up their own examples similar to the ones given and look for a pattern.Key questions
Can you see any relationship between the numbers in the examples?Is there any pattern in the relationship between the numbers?
If you spot a pattern can you prove it always occurs?
Possible extensions
The problen page suggests an extension to three times the sum of three squares.The pattern can be generalised to four times the sum of four squares and so on (see the solution).
Possible support
Suggest that learners find out which of the numbers from 1 to 10 can be written as the sum of 2 squares.eg $1 = 1^2 + 0^2$
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