Sunday Times Teaser 3322 – Three Houses In Long Lane

by BRG on 24 May 2026

by Bernardo Recamán

Published Sunday May 24 2026 (link)

My two best friends and I all live along Long Lane, a row of 60 houses numbered consecutively from 1 to 60. I live at number 10, while my two friends have higher house numbers. One day, I was walking from one friend’s house to the other, and I kept my mind busy by adding up all the house numbers that I passed (not including their two house numbers). Thinking about this sum for a while, I realised that it is precisely the product of their two house numbers.

In which two houses do my friends live?

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# m and n are the friend's house numbers

for m in range(11, 61):

for n in range(m + 1, 61):

if 2 * m * n == (n - 1) * n - m * (m + 1):

print(f"{m} and {n}")

# solve for n: n^2 - (2.m + 1).n - m.(m + 1) == 0

# n = [(2.m + 1) +/- sqrt(8.m^2 + 8.m + 1)] / 2

for m in range(11, 60):

d2 = 8 * m * (m + 1) + 1

if (d := round(d2 ** (1 / 2))) ** 2 == d2:

n = (2 * m + 1 + d) // 2

if m < n <= 60:

print(f"{m} and {n}")

The first solution runs in 92 microseconds, the second in 19 microseconds. Or we can solve it using a quadratic diophantine equation:

from number_theory import Qde

# solve the quadratic diophantine equation:

# (2.m + 2.n + 1)^2 - 8.n^2 == 1

for a, n in Qde(8, 1).solve():

if a & 1:

m = a // 2 - n

if 10 < m < n <= 60:

print(f"{m} and {n}")

break

Sunday Times Teaser 3322 – Three Houses In Long Lane

by Bernardo Recamán

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