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No Square Sums

Stage: 3 and 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

Shakil wants to remove numbers from the set $\{1, 2, 3,..., 16\}$ so that no two remaining numbers add to make a perfect square. What is the smallest number of numbers that he needs to remove?

 
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.

  

This problem is taken from the UKMT Mathematical Challenges.
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