How do you measure 4 liters of water from 5 and 3 liter bottles?

Logic Level 2

You have an infinite water supply. You want to measure 4 liters of water but you only have a 5 liter jug and a 3 liter jug, which have no liter markings on them. As a move, you can either fill a jug up completely, pour a jug's contents to another until the former is empty or the latter is full, or throw away a jug's contents. (Without any marking, these are all you can do.)

Is it really possible to measure 4 liters of water in one of the jugs? If so, what is the least amount of water (in liters) that you will have to waste (by pouring it away)?


Clarification: The image above is just for illustration. In actuality, there is no empty container being used. When you have one jug containing 4 liters, the other jug may contain anything; its contents aren't counted as wasted.

Yes, no water is wasted Yes, 3 liters Yes, 5 liters No

Hill Taeudomkul by Hill Taeudomkul , 19, Thailand

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2 solutions

Hisham Parvez
Jan 12, 2017
  1. Fill the 5L jug
  2. Pour the 5L jug into the 3L jug as much as possible (that leaves the 3L jug full and the 5L jug with 2L water)
  3. Pour what's left of the 5L into the container
  4. 5L jug is now empty, 3L is still full
  5. Pour what's in the 3L jug into the 5L jug and fill the 5L so it's completely full
  6. 5L jug is now full, 3L jug is empty
  7. Pour contents of the 5L jug into the 3L jug until the 3L jug is full
  8. That leaves the 5L jug with 2L of water which can be put into the container making 4L
  9. 3L is still full, so that's 3 litres remaining
1 Helpful 0 Interesting 0 Brilliant 1 Confused
Hill Taeudomkul
Jan 5, 2017

1.Fill the 5 liter jug completely.
2.Pour the water from the 5 jug into the 3 jug.
3.Pour out the 3 jug.
4.Pour the 5 jug into the 3 jug.
5.Fill the 5 jug.
6.Pour the 5 jug into the 3 jug.


Only step 3 requires you to pour out water(3 liters)

1 Helpful 0 Interesting 0 Brilliant 1 Confused

Playing devil's advocate, how do you know that there is no way which only wastes 2, 1 or even 0 liters of water?

Calvin Lin Staff - 4 years, 5 months ago

What if you do the following:

  1. Pour water from 3 l jug to 5 l jug.
  2. Pour water from 3 l jug to 5 l jug until full.
  3. Pour 1 l left in 3l jug to a third container (provided you have such).
  4. Pour water from 5 l to 3 l jug.
  5. Pour 3 l jug to the separate container. It will have now 1+3=4 l of water. The 5 l container will have 2 l of water left in it.

You did not waste any water (dumped it).

Maria Kozlowska - 4 years, 5 months ago

The trick is to realize that: 5 – 3 = 2 and 5 – (3 – 2) = 4 Incidentally, the reason we can find a solution is because the two numbers 5 and 3 are relatively prime–that is, they have no common divisors. The more general problem is finding integer solutions for the equation ax + by = c. Solutions (x,y) exist when the greatest common divisor of a and b is a factor of c. Additional information:http://mathforum.org/library/drmath/view/51595.html

Hill Taeudomkul - 4 years, 5 months ago

+Maria Kozlowska You do not have another container.

P.S. You mouth is not considered as another container;)

Hill Taeudomkul - 4 years, 5 months ago

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I think that needs to be clarified in the question.

Maria Kozlowska - 4 years, 5 months ago

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Thanks. I've updated the problem statement to reflect this. Those who previously answered "Yes, no water is wasted" has been marked correct.

In future, if you spot any errors with a problem, you can “report” it by selecting "report problem" in the menu. This will notify the problem creator (and eventually staff) who can fix the issues.

Brilliant Mathematics Staff - 4 years, 5 months ago

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