Water Measuring Puzzle

Can you make the right volume with two wrong-sized cups?


There is a classic puzzle that goes like this : if you have a 3 gallon tub, a 4 gallon tub and a source of water, how can you measure out exactly 2 gallons? Although you can't directly measure 2 gallons with these tubs, you can get 2 gallons like so :
  • fill up the 3 gallon tub. Now, the 3 gallon tub is full, and the 4 gallon tub is empty.
  • pour the 3 gallon tub into the 4 gallon tub. Now the 4 gallon tub contains 3 gallons.
  • fill up the 3 gallon tub. Now both tubs contain 3 gallons.
  • top up the 4 gallon tub from the 3 gallon tub. Now the 3 gallon tub contains 2 gallons!
There are many ways to vary this puzzle. You can change the sizes of the tubs, or even add more tubs. You can change the amount to be measured out. Sometimes, this means that many steps will be needed, and a lot of water gets discarded.

In fact, sometimes it means that the puzzle can't be solved. Try making 7 pints with a 3 pint tub and a 9 pint tub, for example! There's a very simple rule to determine what solutions are possible, for given tub sizes :

Work out the greatest common factor of the tub sizes. You can make any multiple of this, up to the size of the biggest tub.

Not sure how that works? The form below will work out for you what amounts of water you can make up, for two given tubs.

Tub 1 : units
Tub 2 : units

Here's a playable version of this game. You can see two cups, and a tap. Move water around by clicking on the tap or the cups - click first on the place you want to take water from, then click on the place you want the water to go to.

Click Here To Start A New Game

Difficulty :

You don't need to be online to play this game. Draw some cups or buckets on a piece of paper, and use a pencil to keep track of how much water is in each bucket. As long as the goal follows the rule I mentioned earlier, the puzzle has a solution. If you aren't sure, here's some suggestions. I've rated each suggestion according to how difficult it ought to be. If these are not enough, you could always click 'start new game' above and record the goal and the volumes of the cups.

  • Make 2 litres, with a 3 litre bucket and a 4 litre bucket (easy)
  • Make 3 gallons, with a 5 gallon drum and a 7 gallon drum (easyish)
  • Make 2 ounces, with a 5 ounce cup and an 8 ounce cup (moderate)
  • Make 5 mL, with a 10 mL spoon and a 13 mL spoon (tougher)
  • Make 12 pints, with a 24 pint barrel and a 29 pint barrel (tougher still)

The game can also be played with three containers - in the puzzle below, you'll always need to use all three cups to reach the goal.

Click Here To Start A New Game

Difficulty :

In reality, this is not a practical way to get the right amount. If you make even a tiny error with each pour, these errors will add up, so that at the end, the volume you have is completely random.

(By the way, the nice photo of a tap was not taken by me. It's called Memories of summer, and was taken by flickr user tetu. Thanks, tetu, for making this pic available for use under a creative commons licence!)

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