Challenge Level

The numbers $2$, $34$ and $47$ are such that the sum of any pair of these numbers is a perfect square. Find a method for choosing three square numbers and from them finding a corresponding set of three integers with this property and give some examples.

The integers $-208$, $224$, $352$ and $737$ also have the property that the sum of any pair of these numbers is a perfect square. Find other sets of four integers with this property.