168. Another house number puzzle. Brown lives on a street with more than 20 but fewer than 500 houses (all houses are numbered in sequence: $1,2,3$ and so on). Brown discovered that the sum of all numbers from the first to his own, inclusive, is half the sum of all numbers from the first to the last, inclusive.

What is the number of his house
168. The house number of Brown is 84, and there are a total of 119 houses on the street. The sum of the numbers from 1 to 84 is 3570, and the sum of the numbers from 1 to 119 is 7140, which, as required, is exactly twice as much.

Let's write down the consecutive solutions (in integers) of the equation \(2 x-1=y^{2}\):

\[
\begin{array}{rr}
\frac{x}{1} & \frac{y}{1} \\
5 & 7 \\
29 & 41 \\
169 & 239 \\
985 & 1393
\end{array}
\]

and so on. Then the integer part \({ }^{*} \frac{x}{2}\) will give us the house number, and the integer part \(\frac{y}{2}\) will give us the total number of houses. Thus (omitting the trivial case \(0-0\)), we get \(2-3,14-20,84-119,492-696\) and so on.