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

Step 1: Understanding the Question: We have houses numbered \(1, 2, 3, \ldots, 49\). Let the "magical house number" be \(x\).
The condition is that the sum of the numbers of houses to the left of \(x\) (i.e., \(1\) to \(x-1\)) is equal to the sum of the numbers of houses to the right of \(x\) (i.e., \(x+1\) to \(49\)).

Step 2: Key Formula or Approach: We will use the formula for the sum of the first \(n\) integers: \(S_n = \frac{n(n+1)}{2}\).
The sum of numbers to the left of \(x\) is \(S_{x-1}\).
The sum of numbers to the right of \(x\) can be expressed as the total sum \((S_{49})\) minus the sum up to house \(x\) \((S_x)\). The equation to solve is: \(S_{x-1} = S_{49} - S_x\).

Step 3: Detailed Explanation: 1. Sum of numbers to the left of x:
\[ S_{\text{left}} = S_{x-1} = \frac{(x-1)x}{2} \] 2. Sum of numbers to the right of x: First, calculate the total sum for all 49 houses:
\[ S_{\text{total}} = S_{49} = \frac{49(49+1)}{2} = \frac{49 \times 50}{2} = 1225 \] Now, calculate the sum up to house \(x\):
\[ S_{\text{up to } x} = S_x = \frac{x(x+1)}{2} \] The sum to the right is the total sum minus the sum up to \(x\):
\[ S_{\text{right}} = S_{49} - S_x = 1225 - \frac{x(x+1)}{2} \] 3. Set up and solve the equation:
\[ S_{\text{left}} = S_{\text{right}} \] \[ \frac{x(x-1)}{2} = 1225 - \frac{x(x+1)}{2} \] To eliminate the fraction, multiply the entire equation by 2:
\[ x(x-1) = 2(1225) - x(x+1) \] \[ x^2 - x = 2450 - (x^2 + x) \] \[ x^2 - x = 2450 - x^2 - x \] Add \(x^2\) to both sides:
\[ 2x^2 - x = 2450 - x \] Add \(x\) to both sides:
\[ 2x^2 = 2450 \] Divide by 2:
\[ x^2 = 1225 \] Take the square root:
\[ x = \sqrt{1225} = 35 \]

Step 4: Final Answer:
The magical house number is \(35\).