Is there any way to generalise this problem?

Question

My friend posed this question to me:

I have a glass with 8 litres of water along with one empty 3 litre and an empty 5 litre glass. I have to pour 4 litres of water into his glass in one go. His glass is not calibrated. We do not know how much volume of water it can hold(but it is more than 4). I solved the problem in this way:

$\begin{matrix} 8 & 0 &0 \\ 3&5&0\\ 3&2&3\\ 6&2&0\\ 6&0&2\\ 1&5&2\\ 1&4&3\\ \end{matrix}$

Now I can pour the 4 litres from the 5 litre glass.

My question is that can I generalise this problem?

If I have $V_1$ volume of water in a glass which has a capacity of $V_1$ along with 2 other glasses of volume $V_2$ and $V_3$ such that ($V_2$ and $V_3$ $\lt$ $V_1$).My friend needs $V$ volume of water in one go.

And under what conditions will this generalisation fail?

As I remember, there is a an interpretation of a solution of such problems with a billiard ball which clears the situation. — Alex Ravsky, Nov 10 '13 at 09:46
Unfortunately, I don't have the link. But I think that this should be somewhere in Internet. — Alex Ravsky, Nov 10 '13 at 12:55
@Alex: It is, though it took a bit of searching; see my answer. — Brian M. Scott, Nov 11 '13 at 13:41

score 1 · Accepted Answer · answered Nov 11 '13 at 13:12

There is an extended discussion of such problems and a graphical method of solution in Chapter $4$, ‘Bouncing Balls in Polygons and Polyhedrons’, of Martin Gardner’s $6$th Book of Mathematical Diversions from ‘Scientific American’. He says that the graphical method, which uses rhomboidal graphs on a grid of equilateral triangles, first appears in M.C.K. Tweedie, ‘A Graphical Method of Solving Tartaglian Measuring Puzzles’, The Mathematical Gazette, Vol. $23$, No. $255$; July, $1939$, pp. $278$-$282$.

More accessible is the extensive discussion of the problem starting here at Alex Bogomolny’s Cut The Knot! site.

Is there any way to generalise this problem?

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