# Sunday Times Teaser 2755 – Female Domination

*by Michael Fletcher*

#### Published: 12 July 2015 (link)

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?

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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: \[\frac{f}{m}=n\] \[\frac{f+1}{m+1} = n-1\] \[\frac{f+3}{m+3} = k\] Eliminating \(n\) from the first two equations gives: \[f=m(m + 2)\] and substituting this into the third then gives: \[(k-m+1)(m+3)=6\]Since both factors on the left are integers and \(m + 3 > 3\), the right hand factors must be 1 and 6 giving \(m=k=3\) and \(f=15\), leading to a final village population of 24.

This is a truly brilliant problem.

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.

I am worried that these Sunday Times teasers will go the way of the New Scientist ‘Enigma’ puzzles when Victor Bryant retires (as I suspect he will do quite soon).

So it is a good idea to send in answers regularly in order to ensure that the Sunday Times sees these teasers as a popular feature.

It is great to hear that you have won one!