# Jugs of Wine

You have four jugs of 9, 7, 4 and 2 litres capacity. The 9 litre
jug is full of wine, the others are empty. Can you divide the wine
into three equal quantities?

## Problem

You have four jugs of 9, 7, 4 and 2 litres capacity. The 9 litre jug is full of wine, the others are empty.

Can you divide the wine into three equal quantities?

Can you do it in different ways?

You have three jugs one of which is full and holds 8 litres. The capacity of other jugs is not known. But, it is known that when using them every whole number quantity from 1 litre to 8 litres can be accurately measured out.

What could be the capacities of the 2 other jugs?

How would you measure all the whole number quantities from 1 to 8 litres?

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## Student Solutions

This is a very well-explained solution submitted by Julia (Wymondham High School):

For the jugs holding 9, 7, 4, and 2 litres, this flow diagram shows how the solution can be achieved in three distinct ways, using eight decantings of the wine. In each case, the 9 litres of wine are being poured back and forth to achieve the required result. We order the jugs by size and use a four digit number to represent the volume of wine in each jug. For example 9000 means there are 9
litres in the 9 litre jug, and the 7, 4 and 2 litre jugs are empty. The solution is found when we have 3330, where 3 litres are in each of the 9, 7 and 4 litre jugs; and the 2 litre jug is empty.

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To measure out all the integer amounts from 1 to 8 litres using three jugs, one of which is full and holds 8 litres, there are several possible solutions. For example, for jugs with capacities 8, 3 and 2 litres, the following triples give the numbers of litres in each of the jugs at successive steps and all the amounts from 1 litre to 8 litres occur at some stage of the process: (8,0,0) (5,3,0) (5,1,2) (7,1,0) (7,0,1) (6,0,2) (6,2,0) (4,2,2). There are other solutions for capacities of 8, 4 and 3; for 8, 5 and 4; for 8, 5 and 1 etc.

Two other different students from Wymondham High School, David and Rachel, also submitted good solutions. In their answers, they included a very useful table of results which showed at a glance the state of the jugs after any particular pouring.

Much later two students from Flegg High, Luke and Ian, also submitted a successful solution to this problem. They had found their answer after "hours of trouble, and help from Mrs Fenn".