Answer to Riddle #22: The 3 & 5 Litre Die Hard Water Puzzle

22. You have a 3 and a 5 litre water container, each container has no markings except for that which gives you it's total volume. You also have a running tap. You must use the containers and the tap in such away as to exactly measure out 4 litres of water. How is this done?
Can you generalise the form of your answer?

You might remember this puzzle from Die Hard with a vengeance where Bruce Willis and Samuel L. Jackson have to diffuse a bomb by placing 4 gallons of water on a set of scales...

illustrate die hard water puzzle Before reading the answer can I interest you in a clue?

There are two ways to solve this, maybe the question could be modified to say the 5 litre can doesn't fit under the tap...

Number 1
  • Fill the 5 litre can from the tap
  • Empty the 5 litre can into the 3 litre can - leaving 2 litres in the 5 litre can.
  • Pour away the contents of the 3 litre can.
  • Fill the 3 litre can with the 2 litres from the 5 litre can - leaving 2 litres in the 3 litre can.
  • Fill the 5 litre can from the tap.
  • Fill the remaining 1 litre space in the 3 litre can from the 5 litre can.
    Leaving 4 litres in the 5 litre can.
Number 2
  • Fill the 3 litre can from the tap.
  • Empty the contents of the 3 litre can into the 5 litre can.
  • Fill the 3 litre can from the tap.
  • Empty the contents of the 3 litre can into the 5 litre can. - Leaving the 5 litre can full and 1 litre in the 3 litre can.
  • Pour away the contents of the 5 litre can
  • Pour the 1 litre from the 3 litre can into the 5 litre can.
  • Fill the 3 litre can from the tap.
  • Empty the contents of the 3 litre can into the 5 litre can.
    Leaving 4 litres in the 5 litre can.

Generalised Form

We have two solutions above which both give the same answer of 3 litres. It turns out that if you make it algebraic the answer does not have the same form and it's sort of coincidence. Lets call the smaller container A and the lager B and work it through.

Number 1: Empty the 5 litre can into the 3 litre can - leaving 2 litres in the 5 litre can. ie B - A in B
Fill the 3 litre can with the 2 litres from the 5 litre can - leaving 2 litres in the 3 litre can. As in space in A of A - (B - A) = 2A - B
Fill the remaining 1 litre space in the 3 litre can from the 5 litre can. Leaving 4 litres in the 5 litre can. So the amount in B is B - (2A - B) or 2B - 2A

So the generalised form of solution 1 is 2B - 2A in B.

Number 2: Fill the 3 litre can from the tap. Empty the contents of the 3 litre can into the 5 litre can. Which gives us B - A space in B.
Fill the 3 litre can from the tap. Empty the contents of the 3 litre can into the 5 litre can. - Leaving the 5 litre can full and 1 litre in the 3 litre can. ie. removing (B - A) from A leaves A - (B - A) or 2A - B in A
Pour away the contents of the 5 litre can. Pour the 1 litre from the 3 litre can into the 5 litre can. ie 2A - B in B
Fill the 3 litre can from the tap. Empty the contents of the 3 litre can into the 5 litre can. Leaving 4 litres in the 5 litre can. ie A + (2A - B) or 3A - B in B.

So the generalised form of solution 2 is 3A - B in B.

There are some restrictions, such as in solution 2; 2A < B (2*A must be less than B,) but other wise we can make up other puzzles like 5 Litres and 9 Litres to get 6 Litres. or 8 Litres. All exactly the same form.





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