New Scientist Enigma No 526 – Ewe to Move

by Keith Austin

From Issue #1678, 19th August 1989

Each of the four fields at Sunny Meadows Farm contains some sheep and some cows. On the gate of each field is hung a sign saying what fraction of the animals in that field are sheep. The signs are 1/4, 1/3, 1/2, 2/3.

Farmer Gillian explained that if she exchanged the signs on two of the fields then, by simply moving some sheep from one of the two fields to the other, she could return to a situation where each sign again correctly indicated the fraction of the animals that were sheep in that field.

As I walked round I noticed that the total number of animals on the farm was between 300 and 350.

How many sheep, and how many cows, were on the farm?

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The ratios (sheep / cows) in each field are as follows:$1/3, 1/2, 1/1, 2/1$ The fractions of sheep to move between any two fields in order to exchange their two ratios is given by $\begin{array}{|r|rrrr|} \hline \text{from} & & \text{to:} & 1/3 & 1/2 & 1/1 & 2/1\\ \hline 1/3 & & & & 1/6 & 2/3 & 5/3 \\ \hline 1/2 & & & & & 1/2 & 3/2 \\ \hline 1/1 & & & & & & 1/1 \\ \hline \end{array}$Since we can only move whole sheep we need to scale this up to give:
$\begin{array}{|l|rrrr|r|} \hline \text{field (sheep / cows)} & 1/3 & 1/2 & 1/1 & 2/1 & \text{totals} \\ \hline \text{sheep} & 2 & 3 & 6 & 12 & 23\\ \hline \text{cows} & 6 & 6 & 6 & 6 & 24 \\ \hline \text{animals} & 8 & 9 & 12 & 18 & 47 \\ \hline \end{array}$
$\begin{array}{|l|rrrr|r|} \hline \text{field (sheep / animals)} & 1/4 & 1/3 & 1/2 & 2/3 & \text{totals} \\ \hline \text{sheep} & 14 & 21 & 42 & 84 & 161\\ \hline \text{cows} & 42 & 42 & 42 & 42 & 168 \\ \hline \text{animals} & 56 & 63 & 84 & 126 & 329 \\ \hline \end{array}$