Find formula for coordinates : Solution

The problem statement is here. Observe that the $ (x, y) $ coordinate appears in the $ (x + y) ^{th} $ solid line. The $ n^{th} $ diagonal has $ 1 + 2 + 3 + ... + n $ elements before it, i.e. $ (x + y) ^{th} $ diagonal has $ \frac{(x+y)(x+y+1)}{2} $ elements before it. Add the $ x $ elements to it to get the point at coordinate $ (x+y) $. So the answer is $ \frac{(x+y)(x+y+1)}{2} + x $.