Water Jug Puzzles: Solutions

How the puzzles work

A move is one of three things. Fill a jug from the tap, empty a jug into the sink, or pour one jug into another. A pour always goes until the first jug is empty or the second one is full, because without markings there is no way to stop halfway. Some puzzles have no tap and no sink: the water you start with is all there is, and you can only pour it between the jugs.

The goal is shown above the jugs. In the tap puzzles it is an amount in any one jug; in the sharing puzzles it is an exact amount in two of the jugs. The fewest moves earn three stars.

Three habits that help

• Look at what a pour leaves behind. Pour a full 5 litre jug into an empty 3 litre jug and 2 litres stay behind. Those leftovers are the only new amounts you can ever make, so most solutions are a chain of them.
• Keep one rhythm. With two jugs and a tap, pick one jug to fill, always pour it into the other, and empty the other whenever it is full. Sooner or later every amount you can measure turns up.
• Without a tap, the total never changes. Picture where every litre has to end up and ask which jug can hold the leftover on the way.

1. Four Litres

A 3 litre jug, a 5 litre jug and a tap; measure 4 litres.

1. Fill the 5 litre jug.
2. Pour it into the 3 litre jug, leaving 2 litres.
3. Empty the 3 litre jug.
4. Pour the 2 litres into the 3 litre jug.
5. Fill the 5 litre jug again.
6. Pour from it into the 3 litre jug until that is full. Exactly 4 litres are left.

Six moves. Filling the 3 litre jug first also works, but takes eight.

2. Sharing the Cider

A full 8 litre jug, an empty 5 and an empty 3, no tap; finish with 4 and 4.

1. Pour the 8 into the 5 (3, 5, 0).
2. Pour the 5 into the 3 (3, 2, 3).
3. Pour the 3 back into the 8 (6, 2, 0).
4. Pour the 5 into the 3 (6, 0, 2).
5. Pour the 8 into the 5 (1, 5, 2).
6. Pour the 5 into the 3 (1, 4, 3).
7. Pour the 3 into the 8 (4, 4, 0).

Seven moves. The amounts in brackets are the 8, the 5 and the 3 litre jugs.

3. Six Litres

A 4 litre jug, a 9 litre jug and a tap; measure 6 litres. The idea is to fill the big jug and take away fours: 9 leaves 5, then 1, which goes into the small jug so that the next 9 can lose only 3.

1. Fill the 9. 2. Pour into the 4, leaving 5. 3. Empty the 4. 4. Pour into the 4, leaving 1. 5. Empty the 4. 6. Pour the 1 litre into the 4. 7. Fill the 9. 8. Pour into the 4 until it is full: 6 litres stay in the 9.

4. Halves of Ten

A full 10, an empty 7 and an empty 3, no tap; finish with 5 and 5. The 3 litre jug works like a scoop, moving water out of the 7 and back into the 10.

1. Pour the 10 into the 7 (3, 7, 0).
2. Pour the 7 into the 3 (3, 4, 3).
3. Pour the 3 into the 10 (6, 4, 0).
4. Pour the 7 into the 3 (6, 1, 3).
5. Pour the 3 into the 10 (9, 1, 0).
6. Pour the 7 into the 3 (9, 0, 1).
7. Pour the 10 into the 7 (2, 7, 1).
8. Pour the 7 into the 3 (2, 5, 3).
9. Pour the 3 into the 10 (5, 5, 0).

5. Three Jugs at the Tap

Jugs of 5, 8 and 13 litres and a tap; measure 9 litres. The big jug is only needed at the very end: first make 1 litre with the 5 and the 8, then add a full 8 to it.

1. Fill the 8. 2. Pour into the 5, leaving 3. 3. Empty the 5. 4. Pour the 3 into the 5. 5. Fill the 8. 6. Pour into the 5 until full, leaving 6. 7. Empty the 5. 8. Pour the 8 into the 5, leaving 1. 9. Pour that 1 litre into the 13. 10. Fill the 8. 11. Pour it into the 13: 9 litres.

6. Two Litres

A 7 litre jug, an 11 litre jug and a tap; measure 2 litres. Keep the rhythm: always fill the 7, always pour it into the 11, and empty the 11 whenever it is full.

1. Fill the 7. 2. Pour into the 11 (0, 7). 3. Fill the 7. 4. Pour into the 11 until full (3, 11). 5. Empty the 11. 6. Pour the 3 into the 11. 7. Fill the 7. 8. Pour into the 11 (0, 10). 9. Fill the 7. 10. Pour into the 11 until full (6, 11). 11. Empty the 11. 12. Pour the 6 into the 11. 13. Fill the 7. 14. Pour into the 11 until it is full: 2 litres are left in the 7.

Fourteen moves. Starting with the 11 instead takes longer.

7. The Sixteen Litre Barrel

A full 16 litre barrel, an empty 11 and an empty 6, no tap; finish with 8 and 8. The 6 litre jug carries water from the barrel to the 11, and the 11 empties back into the barrel each time it fills.

1. Pour the barrel into the 6 (10, 0, 6).
2. Pour the 6 into the 11 (10, 6, 0).
3. Pour the barrel into the 6 (4, 6, 6).
4. Pour the 6 into the 11 (4, 11, 1).
5. Pour the 11 into the barrel (15, 0, 1).
6. Pour the 6 into the 11 (15, 1, 0).
7. Pour the barrel into the 6 (9, 1, 6).
8. Pour the 6 into the 11 (9, 7, 0).
9. Pour the barrel into the 6 (3, 7, 6).
10. Pour the 6 into the 11 (3, 11, 2).
11. Pour the 11 into the barrel (14, 0, 2).
12. Pour the 6 into the 11 (14, 2, 0).
13. Pour the barrel into the 6 (8, 2, 6).
14. Pour the 6 into the 11 (8, 8, 0).

When a puzzle has no answer

With two jugs and a tap, you can only ever measure multiples of the largest number that divides both sizes. Jugs of 4 and 6 litres can make 2, 4 and 6, but never 1, 3 or 5. Jugs whose sizes share no divisor, like 3 and 5 or 7 and 11, can make every whole amount up to the bigger jug, which is why every puzzle here works.