Measuring water puzzle

Question

Given two unmarked jugs, one which holds 7 liters, and another which holds 11 liters, an unlimited supply of water, and no need to conserve, how do you measure exactly 6 liters?

I would also like to know whether there is a single solution , many solution or no solution? Can it measure any amount?

See the references in my answer to this question; one of them is to a web site. — Brian M. Scott, Nov 11 '13 at 19:17

Thanos Darkadakis · Accepted Answer · 2013-11-11T19:31:05.287

6

Multiples of 7: 7,14,21,28,35...

Multiples of 11: 11,22,33.

We observe that the difference 6 appears between 28 and 22

So you keep filling the 7 liter jug and emptying it into the 11 liter jug

After 4 fillings you will be left with 6 liters in the 7-liter jug

Edit:adding 2nd solution.

Likewise you can search for a multiple of 11 that is 6 more than a multiple of 7.

55=5*11, 49=7*7

So you can fill the 11 liter jug 5 times and empty it in the 7 liter jug. In the end you will be left with 6 liters in the 11-liter jug

edited Nov 11 '13 at 19:31

answered Nov 11 '13 at 19:21

Thanos Darkadakis

1,869

Nice. So basically we need to calculate the multiples and see the difference? – user108097 Nov 11 '13 at 19:24
that's a way for sure. if you can find the difference you're looking for, you can solve it. – Thanos Darkadakis Nov 11 '13 at 19:24
so will the difference go from 1 to 10? i mean can we measure every integer till the bigger jar limit? – user108097 Nov 11 '13 at 19:25
from 1 to N-1, where N the largest number. I am adding a second solution... – Thanos Darkadakis Nov 11 '13 at 19:27

score 4 · Answer 2 · answered Nov 11 '13 at 19:36

4

This is an answered question so let me provide something that might be new to you :
You have to know that if one jug holds $a$ and the other one $b$ liters with $gcd(a,b)=d>1$ then it is possible to measure only multiples of $d$.
For example it is impossible to measure $6$ liters of water if one jug holds $8$ liters and the other one $12$ liters since $gcd(8,12)=4$ and $6$ is not a multiple of $4$
Also Bezout's identity will help you in order to find the solution in any case.

answered Nov 11 '13 at 19:36

Konstantinos Gaitanas

9,115

oh! and here 7 and 11 are co-prime that is why we can measure everything till 11 – user108097 Nov 11 '13 at 19:38
Yes exactly as you said. For example we can measure 47 liters from 2 jugs containing 101 and 23 liters respectively.I don't know the way of doing this but i know we can. – Konstantinos Gaitanas Nov 11 '13 at 19:53

Adam Stelmaszczyk · Answer 3 · 2013-11-11T19:36:23.373

1

Having $11$ and $7$ litres, we can have $11-7=4$ litres.

So, we can also have $7-4 = 3$ litres.

So, we can also have $4-3=1$ litre.

So, we can also have $7- 1 = 6$ litres.

In general, if the jugs capacities are relatively prime, we can measure any integer amount of water.

edited Nov 11 '13 at 19:36

answered Nov 11 '13 at 19:30

Adam Stelmaszczyk

826
9
28

score 0 · Answer 4 · answered Nov 11 '13 at 19:21

one solution:

fill the 7 liter jug and empty it into the 11 liter jug leaving do this again leaving 3 liters in the 7 liter jug
then empty the 11 liter jug then put the 3 liters in the 11 liter jug
then add another 7 liters to the 11 liter jug leaving 10 liters in it
then top it off from a full 7 liters leaving 6 liters in the 7 liter jug

score 0 · Answer 5 · answered Nov 11 '13 at 19:27

Fill the 7 liter jug and pour it into the 11 liter

Fill the 7 liter jug again and finish filling the 11 liter jug

There is now 3 liters in the 7 liter jug

Empty the 11 liter jug

Pour the 3 liters in the 7 liter jug into the 11 liter jug

Fill the 7 liter jug and pour it into the 11 liter jug, making 10 liters in the 11 liter jug.

Fill the 7 liter jug and finish filling the 11 liter jug. This will take 1 liter and leave 6 liters in the 7 liter jug.

score 0 · Answer 6 · answered Nov 11 '13 at 19:28

The generic answer is:
Fill the 11. Pour 11 into the 7 and you have n (=4) left.
Now empty the 7 and pour the remaining n (=4) into the 7.
Fill the 11. Pour 11 into the 7 and you have n (=8) left.
Now empty the 7 and pour the remaining n (=8) into the 7.
Now empty the 7 and pour the remaining n (=1) into the 7. This step is twice because the remainder was greater than 7.
Fill the 11. Pour 11 into the 7 and you have n (=5) left.
Now empty the 7 and pour the remaining n (=5) into the 7. Fill the 11. Pour 11 into the 7 and you have n (=9) left.
Now empty the 7 and pour the remaining n (=9) into the 7.
Now empty the 7 and pour the remaining n (=2) into the 7. This step is twice because the remainder was greater than 7.
Fill the 11. Pour 11 into the 7 and you have n (=6) left.

You can reach with this algorith any number less than 11.

score 0 · Answer 7 · answered Nov 11 '13 at 19:28

Say A has 11l and B has 7l

1) fill A with water, and pour from A to B. In A you will have 4l left. Pour the 4l into B.

2) fill A and then pour it into B until it is filled. You end up with 8l in A.

3) empty B and pour 7l from A into B. You end up with 1l in A.

4) fill A and pour 6l into B until it is full. Now you have 5l in A.

5) Pour the 5l into B.

6) Fill A and pour into B. You now have 9l in A.

7) Empty B and pour 7l from A to B. Now you have 2l in A.

8) Fill A and pour 5l into B. Now you have 6l into A.

Measuring water puzzle

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