Henry Ernest Dudeney/Modern Puzzles/85 - The House Number/Solution

by : $85$

 * The House Number
 * A man said the house of his friend was in a long street,
 * numbered on his side one, two, three, and so on,
 * and that all the numbers on one side of him added up exactly the same as all the numbers on the other side of him.
 * He said he knew there were more than fifty houses on that side of the street,
 * but not as many as five hundred.


 * Can you discover the number of that house?

Solution
Let there be $m$ houses in the street, where we are told $50 < m < 500$.

Let the man live at no. $n$.

We have that:

We can attempt to solve this using the Quadratic Formula but this won't help us much.

However, we can plug in values of $m$ to see which values give integral $n^2$.

Trying out a few values of $m$, we see:


 * $\begin{array} {r|r} m & n \\ \hline

1 & 1 \\ 8 & 6 \\ 49 & 35 \\ 288 & 204 \\ 1681 & 1189 \\ \end{array}$

Because there are between $50$ and $500$ houses in the street, we know the man lives at no. $204$ in a $288$-house street.