167. House Number. One person said that his friend's house is located on a long street (and on the side where the house is located, the houses are numbered in order: $1,2,3$ and so on) and that the sum of the house numbers from the beginning of the street to his friend's house is the same as the sum of the house numbers from his friend's house to the end of the street. It is also known that on the side of the street where the friend's house is located, there are more than 50 but fewer than 500 houses.

What is the house number where the friend of the storyteller lives?
167. The sum of the numbers of the houses located on one side of a given house will equal the sum of the numbers on the other side in the following cases: 1) if the number of the given house is 1 and there are no other houses; 2) if the number is 6 and there are a total of 8 houses; 3) if the number is 35, and there are a total of 49 houses; 4) if the number of the house is 204, and there are a total of 288 houses; 5) if the number of the house is 1189, and there are a total of 1681 houses.

ANSWER

and so on. However, we know that the number of houses is greater than 50 and less than 500; therefore, the required number is 204.

Solving the equation \(\frac{x^{2}+x}{2}=y^{2}\) in integers, we get the answers:

\begin{tabular}{rc} 
Number & Number \\
of houses \(x\) & of the house \(y\) \\
1 & 1 \\
8 & 6 \\
49 & 35 \\
288 & 204 \\
1681 & 1189
\end{tabular}

and so on.