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

by : $85$

 * The House Number

Solution
The man lived at no. $204$ in a street of $288$ houses.

Proof
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:

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}$

Formally, we have:

From Pell's Equation: $x^2 - 2 y^2 = 1$, we have:

Hence we can create the solutions:


 * $\begin{array} {r|r|r|r} x & y & m = \dfrac {x - 1} 2 & n = \dfrac y 2 \\ \hline

3 & 2 & 1 & 1 \\ 17 & 12 & 8 & 6 \\ 99 & 70 & 49 & 35\\ 577 & 408 & 288 & 204 \\ 3363 & 2378 & 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.