6. In five 15-liter buckets, there are 1, 2, 3, 4, and 5 liters of water respectively. It is allowed to triple the amount of water in any container by pouring water from one other container (if there is not enough water to triple the amount, then it is not allowed to pour from this bucket). What is the maximum amount of water that can be collected in one bucket using such actions?
# Answer

The answer depends on the interpretation of the condition -- whether it is allowed to pour NOT all the contents of the bucket (essentially -- whether it is possible to measure OUT ONE LITER of water)

A) If it is not allowed, then the answer is 9 liters

B) If it is allowed, then the answer is 12 liters

## Solution:

Variant $A$ )

We will show how to collect 9 liters in one of the buckets:

$1,2,3,4,5=>1,6,3,0,5=>1,0,9,0,5$.

## Variant B)

We will show how to collect 12 liters in one of the buckets:

$1,2,3,4,5=>1,6,3,4,1=>1,0,9,4,1=>1,0,1,12,1$

In any case, it is necessary to prove that this is the maximum number (that it is impossible to get MORE).

Let the maximum number of liters be $n>=9$.

Consider the last operation with this bucket (at least one operation was performed

--- otherwise $n<=5$ ).

Since $n$ is the maximum number of liters, the last operation could not have involved pouring from this bucket

(otherwise, there would have been more before), i.e., it was filled, so $n$ is a multiple of 3.

There are 15 liters in all the buckets combined.

Note that after each step, there is a non-empty bucket, the number of liters in which is a multiple of three. (solution)

Assume that $n=15$.

Then, on the previous step, there were exactly two non-empty buckets, one with 5 and the other with 10 liters.

But neither of these numbers is a multiple of 3. Contradiction with (*)

## Variant A)

Assume that $n=12$.

Then, on the previous step, there should have been two non-empty buckets: 4 and 8 liters.

Then, due to condition (*), the remaining 3 liters should have been in one bucket.

But then, one step earlier, the amount of water in the buckets should have been 1, 2, 4, 8, which contradicts the condition (*).

Thus, $n=9$, an example of how to get 9 liters is provided above.

Variant B) Thus, $n=12$, an example of how to get 12 liters is provided above.