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# The water jug puzzle

Here is another well-known, medieval problem, also known as the ‘water pouring puzzle’.

You have an 8 litre jug full of water and two smaller jugs, one that contains 5 litres and the other 3 litres. None of the jugs have markings on them, nor do you have any additional measuring device.You have to divide the 8 litres of water equally between your two best friends, so that each gets 4 litres of water. How can you do this?

## The water jug puzzle - solution

• First, water is poured from the 8 litre jug into the 5 litre jug, leaving 3 litres of water in the original 8 litre jug.
• Next, water is poured from the 5 litre jug into the 3 litre jug, so we now have 3 litres of water in the 8 litre jug, 2 litres of water in the 5 litre jug and 3 litres of water in the 3 litre jug.
• The 3 litre jug is emptied into the 8 litre jug, so the 8 litre jug now contains 6 litres of water.
• The 2 litres of water in the 5 litre jug are now poured into the empty 3 litre jug.
• Water is poured from the 8 litre jug (which at this stage contains 6 litres) into the empty 5 litre jug.
We now have 5 litres of water in the 5 litre jug, 2 litres of water in the 3 litre jug and 1 litre of water in the 8 litre jug.
• Water is poured from the 5 litre jug to fill the 3 litre jug which already contains at this stage 2 litres of water.
• We are left with 4 litres of water in the 5 litre jug which is given to one friend, and 3 litres of water in the 3 litre jug that is poured back into the 8 litre jug that already contains 1 litre of water. This gives 4 litres of water which are given to the second friend.

The whole scenario can be summarised using numbers in brackets to denote the litres of water at each stage in each of the 8 litre, 5 litre and 3 litre jugs, respectively:

$[8,0,0] \rightarrow [3,5,0] \rightarrow [3,2,3] \rightarrow [6,2,0] \rightarrow [6,0,2] \rightarrow [1,5,2] \rightarrow [1,4,3] \rightarrow [4,4,0]$

This diagram is known as a state diagram

## Discussion

The water jug problem can be solved with just two jugs - one that can hold 5 litres of water and the other that can hold 3 litres of water, if there is also an unlimited supply of water from a tap and a sink. Show the series of state diagrams that solve this problem. Post your solutions in the comments!