The houses are numbered consecutively from 1 to 49, which forms a sequence of natural numbers. From the context, using the method of Gauss, the sum of the first n positive integers is given by the formula:

$$S_n = \frac{n(n+1)}{2}$$

Let the house number be $x$. The sum of the numbers of the houses preceding the house numbered $x$ is the sum of the first $x-1$ natural numbers, denoted as $S_{x-1}$. The sum of the numbers of the houses following it is the total sum $S_{49}$ minus the sum up to $x$, denoted as $S_{49} - S_x$.

According to the problem, the sum of preceding numbers is equal to the sum of following numbers. Using the hint provided, we can write the equation as:

$$S_{x-1} = S_{49} - S_x$$

Substituting the formula for the sum of natural numbers into the equation:

$$\frac{(x-1)x}{2} = \frac{49 \times 50}{2} - \frac{x(x+1)}{2}$$

To solve, multiply the entire equation by 2 to eliminate the denominators:

$$(x-1)x = 49 \times 50 - x(x+1)$$ $$x^2 - x = 2450 - x^2 - x$$

Simplifying this, we get:

$$2x^2 = 2450$$ $$x^2 = 1225$$

Taking the square root of both sides, we find the value of x:

$$x = 35$$

Thus, the house number is 35.