TITLE: Measure different volumes based on specified capacities
QUESTION [5 upvotes]: Given three containers of specified volume, how many different volumes can you measure ?
For example, suppose we have cans with capacities 2, 3, 4 litres.
We can measure :

1 litre by filling the 3 litre can and pouring it in the 2 litre can. 
 2, 3, 4 litres are trivial.
 5 litres by filling both the 2 and 3 litre cans. 
6 litres by filling the 4 litre can with 3 litres and filling the 3 litre can fully or filling the 2 and 4 litre cans. 
 7 litres by filling the 3 and 4 litre cans.
8 litres by getting 6 litres and filling the 2 litre can.
9 litres by filling all the cans. 
The answer for this case is 9.
Is there a general way to answer this for three capacities a, b, c?

REPLY [2 votes]: For the case where we have cans with $2,3,4$ liters, consider the polynomial
\begin{equation}
(1+x^2)^3 (1+x^3)^2(1+x^4)^2\left(1+\frac{1}{x^2}\right)^2\left(1+\frac{1}{x^3}\right) \left(1+\frac{1}{x^4}\right)
\end{equation}
Look at the terms $x^k$ where $k\geq 1$. I believe all those powers of $x$ are the capacities you can measure using the three cans.
Reasoning:
Any term of the form $1+x^a$ allows for two choices - You either fill the can of volume $a$ or you don't fill it. If we multiplied three terms and wrote $(1+x^2)(1+x^3)(1+x^4)$, this will allow you to find out how many different volumes you can measure if you are allowed to fill each can at most once and if a can once filled is never emptied.
Also, in the case of cans of size $2,3$ and $4$, it is possible to fill the 2 liter can once, empty it into the 4 liter can, fill the 2 liter can again, pour it into the 4 liter can again and fill the 2 liter can a third time. This is why we have $(1+x^2)^3$. Similarly for the 3 litre can. The 4 litre can could be emptied into the 2 and 3 liter cans combined and the refilled, so we have the term $(1+x^4)^2$ as well.
We should also allow for subtracting of volumes - This implies that terms of the form $(1+\frac{1}{x^a})$ should be present. If we look at the 2 liter can, you could fill it (and empty it) from the 4 liter can twice. Therefore, we have a term $(1+1/x^2)^2$. Similarly we get the term $(1+1/x^3)$. The 4 liter can could also be emptied once after filling it from the other two cans. Combining all of this, we get
\begin{equation}
(1+x^2)^3 (1+x^3)^2(1+x^4)^2\left(1+\frac{1}{x^2}\right)^2\left(1+\frac{1}{x^3}\right) \left(1+\frac{1}{x^4}\right)
\end{equation}
This idea can be extended to other values of $a,b$ and $c$. I tried it for a few other values (using Wolfram Alpha to expand the expressions) and it seems to work.