Sunday Times Teaser 2755 – Female Domination

by Michael Fletcher

In the village of Alphaville, the number of females divided by the number of males was a certain whole number. Then one man and his wife moved into the village and the result was that the number of females divided by the number of males was one less than before. Now today two more such married couples have moved into the village, but the number of females divided by the number of males is still a whole number.

What is the population of the village now?

This doesn’t really justify a programmed solution since a little algebra quickly gives a solution. But here is one:

The mathematical approach is as follows. With $$n$$ and $$k$$ integer and $$m$$ and $$f$$ as the initial number of males and females we have: $\frac{m}{n}=n$ $\frac{f+1}{m+1} = n-1$ $\frac{f+3}{m+3} = k$ Eliminating $$n$$ from the first two equations gives: $f=m(m + 2)$ and substituting this into the third then gives: $(k-m+1)(m+3)=6$ from which we quickly obtain $$m=k=3$$ and $$f=15$$, leading to a final village population of 24.

This is a truly brilliant problem.

3. I wrote a simple program to check initial populations up to 1000. (Makes slight use of routines from enigma.py).

I did some analysis to write a more sophisticated program, but by the time I got to “m + 3 is a divisor of 6 greater than 3” there was nothing left for a program to do really.

Unusually I actually sent in an entry for this puzzle. And that might explain the mysterious cheque for £20 I received in the post yesterday.