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.
Show 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).