The three jars problem

Here is a relatively well known puzzle (it appears, for example, in the film Fermat's Room.

"You are given three closed tins of sweets that are labelled "Lemon Sherbet", "Toffees" and "Mixture". One contains lemon sherbets, one contains toffees and the third contains a mixture of the two, and they are all wrongly labelled. What is the minimum number of sweets that you have to remove in order to ascertain which jar contains which variety?"

If you don't know the problem, work out the answer now before reading any more. So that you can't immediately see the answer I give below, I'm interposing a totally irrelevant image:

(Bottles of sherbet powder in Goodies Sweet Shop, Steep Hill, Lincoln: photograph by Andy Dingley from Wikipedia Commons.)

The book answer is one. If you take one from the tin labelled "mixture" it tells you what is in that tin, since it isn't the mixture. You can then easily identify the other two, knowing that neither is correctly labelled.

I think this answer is wrong. The true answer is zero. (And this is not because of any issue with the wording, that you might be able to feel or smell what is in the mixture tin without actually removing anything.)

You don't actually have to remove a whole sweet. You could open the mixture tin and break one of the sweets into two, remove half a sweet and work out the answer. You could take a smaller piece than a half - it could be a third, or a quarter, or a fifth, or even smaller.

In fact, for any epsilon greater than zero, you could remove a piece smaller than that size and it would give you the solution. So the number of pieces you have to examine is smaller than any positive number. The only such number is zero, so zero pieces suffice.