# Two Jugs - One's Volume Unknown

Here's a modification on more traditional two and three jugs puzzles:

Tanya is standing on the bank of a river. She has two clay jugs. one of them is a 5 liter jug, but as for the other, Tanya only remembers that it either holds 3 or 4 liters. Help Tanya to determine the capacity of the second jug. It's impossible to tell how much water there is in a jug just by looking inside it; you can only see whether it is full.

### References

1. I. Yashchenko, Invitation to a Mathematical Festival, MSRI/AMS, 2013, p 15

Solution

Tanya is standing on the bank of a river. She has two clay jugs. one of them is a 5 liter jug, but as for the other, Tanya only remembers that it either holds 3 or 4 liters. Help Tanya to determine the capacity of the second jug. It's impossible to tell how much water there is in a jug just by looking inside it; you can only see whether it is full.

### Solution

The puzzle sounds at least somewhat different from the classical ones, but there is too little leeway as to what one (or Tanya) can do with the jugs: the only allowed operations are 1) filling in, and 2) pouring out.

There are two ways to approach the problem, depending of which jug is filled in from the river.

Assume you keep filling the big, 5 liter jug and pouring from it into the smaller one. If the small one is full but the big one is not, empty the small one. How long will you be able to go on in this manner. Here's the sequence of amounts of water in the tow jugs.

For the 3 liter jug:

1) 0-0, 2) 5-0, 3) 2-3, 4) 2-0, 5) 0-2, 6) 5-2, 7) 4-3, 8) 4-0.

For the 4 liter jug:

1) 0-0, 2) 5-0, 3) 1-4, 4) 1-0, 5) 0-1, 6) 5-1, 7) 2-4, 8) 2-0.

On the next operation the difference between two possibilities will become transparent. Pouring the whole of the remaining water from the big jug into the small will overflow the 3 liter jug but not the four liter jug.