Using a 7-quart and a 3-quart jug, how can you obtain exactly 5 quarts of water from a well?

That’s a water-fetching puzzle, a familiar task in puzzle books. Most such problems can be solved fairly easily using intuition or trial and error, but in *Scripta Mathematica*, March 1948, H.D. Grossman describes an ingenious way to generate a solution geometrically.

Let *a* and *b* be the sizes of the jugs, in quarts, and *c* be the number of quarts that we’re seeking. Here, *a* = 7, *b* = 3, and *c* = 5. (*a* and *b* must be positive integers, relatively prime, where *a* is greater than *b* and their sum is greater than *c*; otherwise the problem is unsolvable, trivial, or can be reduced to smaller integers.)

Using a field of lattice points (or an actual pegboard), let O be the point (0, 0) and P be the point (*b*, *a*) (here, 3, 7). Connect these with *OP*. Then draw a zigzag line *Z* to the right of *OP*, connecting lattice points and staying as close as possible to *OP*. Now “It may be proved that the horizontal distances from *OP* to the lattice-points on *Z* (except *O* and *P*) are in some order without repetition 1, 2, 3, …, *a* + *b* – 1, if we count each horizontal lattice-unit as the distance *a*.” In this example, if we take the distance between any two neighboring lattice points as 7, then each of the points on the zigzag line *Z* will be some unique integer distance horizontally from the diagonal line *OP*. Find the one whose distance is *c* (here, 5), the number of quarts that we want to retrieve.

Now we have a map showing how to conduct our pourings. Starting from *O* and following the zigzag line to *C*:

- Each horizontal unit means “Pour the contents of the
*a*-quart jug, if any, into the*b*-quart jug; then fill the*a*-quart jug from the well.” - Each vertical unit means “Fill the
*b*-quart jug from the*a*-quart jug; then empty the*b*-quart jug.”

So, in our example, the map instructs us to:

- Fill the 7-quart jug.
- Fill the 3-quart jug twice from the 7-quart jug, each time emptying its contents into the well. This leaves 1 quart in the 7-quart jug.
- Pour this 1 quart into the 3-quart jug and fill the 7-quart jug again from the well.
- Fill the remainder of the 3-quart jug (2 quarts) from the 7-quart jug and empty the 3-quart jug. This leaves 5 quarts in the 7-quart jug, which was our goal.

You can find an alternate solution by drawing a second zigzag line to the left of *OP*. In reading this solution, we swap the roles of *a* and *b* given above, so the map tells us to fill the 3-quart jug three times successively and empty it each time into the 7-quart jug (leaving 2 quarts in the 3-quart jug the final time), then empty the 7-quart jug, transfer the remaining 2 quarts to it, and add a final 3 quarts. “There are always exactly two solutions which are in a sense complementary to each other.”

Grossman gives a rigorous algebraic solution in “A Generalization of the Water-Fetching Puzzle,” *American Mathematical Monthly* 47:6 (June-July 1940), pp. 374-375.