Question

There are 49 houses in a row, numbered consecutively. A scientist finds a magical house number such that the sum of house numbers on its left is equal to the sum on its right. Find the magical house number.

Hint:

A quick shortcut for this problem type: if there are (N) houses, the magical number (x) must satisfy
(x^2 = frac{N(N+1)}{2}).
This means the sum of all house numbers must be a perfect square, and the house number is the square root of that sum. For (N=49), the sum is (S_{49}=1225), and (sqrt{1225}=35). This works because (1225) is a perfect square. This type of problem only has an integer solution if (frac{N(N+1)}{2}) is a perfect square (these are called square triangular numbers).

Answer 1