In this post I talked about the riddle of the water buckets. Now I challenge you to prove that in some situations it is impossible to solve it!

A grayscale image with 3 buckets

Problem statement

You have \(n\) buckets, each bucket with capacity for \(c_i\) litres of water, \(i = 1, \cdots, n\). You want to manipulate the buckets in such a way that one of them holds exactly \(t\) litres of water, knowing that the only moves you can do are:

  • completely fill up bucket \(i\) so that it holds \(c_i\) litres of water;
  • completely empty bucket \(i\) so that it now holds \(0\) litres of water;
  • move water from bucket \(i\) to bucket \(j\), until bucket \(i\) becomes empty or bucket \(j\) becomes full, whatever happens first.

Prove that, if \(t\) is not a multiple of the greatest common divisor of the \(c_i\), \(i = 1, \cdots, n\) then it is impossible for a single bucket to hold exactly \(t\) litres of water.

For example, if the buckets have capacities \(4\) and \(6\) and \(t = 3\), then you can't perform the moves above to get exactly \(3\) litres of water into one of the two buckets as the greatest common divisor of \(4\) and \(6\) is \(\gcd(4, 6) = 2\) and \(3\) is not a multiple of \(2\).

Give it some thought... and give it an actual shot! Take out a piece of paper and a pencil and get your brain working!

If you need any clarification whatsoever, feel free to ask in the comment section below.

Solution

You can read the solution here to compare with your own solution.

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