Cut The Knot!An interactive column using Java applets
by Alex Bogomolny
In [Ball & Coxeter, p. 27] the problem appears with the remark that "the solution presents no difficulty." It precedes another problem with 4 jugs of capacities 5, 11, 13, and 24 for which a solution "can be worked out only by trial." The problem is presented by the narrative:
If nothing else, such problems dress up a meaningful counting exercise that can be handed out to children in early grades. A variant with the largest vessel replaced with an unlimited source of water was offered as a Middle School Problem of the Week (October 4, 1999) at the Math Forum.
But there is also some worthwhile mathematics involved that was mostly overlooked by teachers and students alike. (This is judging by the number of questions I received concerning existence of a solution.)
Label the jugs A, B, C in the increasing order of their
capacities. Let's agree to use the same letters for the capacities
themselves. Let x, y, z denote the quantities of water in the jugs. In
Here's the statement of existence:
Start with (0, 0, C), and pour from C to A and from A to B to obtain
The problem and the proof have a surprising geometric interpretation [Coxeter & Greitzer, p. 89] in terms of triangular coordinates. Cartesian and polar coordinates are great tools in the analytic geometry of the plane. Spherical coordinates are suitable for the geometry of the sphere, as are cylindrical coordinates on a cylinder. There are several systems of coordinates in which vertices and sides of a triangle are treated in an equitable manner. The most important are the barycentric and trilinear coordinates.
For a point P in the plane of ABC, the triple of its (signed) distances to the sides BC, CA, and AB is called the trilinear coordinates of P (with respect to ABC.) The distances are signed such that, for example, the distance to AB is positive or negative depending on whether P is located on the same or different side of AB as vertex C. The barycentric coordinates are defined as a triple of (signed) areas of triangles APB, BPC, and CPA. In both systems, location of P is fully defined by any two numbers; the third number is redundant. For this reason, the coordinates are considered as homogeneous.
For example, if r is the inradius
-- the radius of the inscribed circle -- of ABC, then trilinear coordinates of its incenter are
The description of the 3 Jugs problem as a triple of quantities
The applet below demonstrates how the puzzle moves are reflected on such a grid. (Play with it with the button Trace checked.)
The (integer) points that satisfy the built-in constraints of the 3 Jugs problem (e.g., 0xA) fill a parallelogram. Only the points on the boundary of that parallelogram could be attained as a result of valid puzzle moves. The basic move corresponds to an inverted "V" line with one side parallel to AC (pour from C to A), the other to AB (pour from A to B.)
The jug B is full at the points of the "western" side of the parallelogram. Close to that side, the left leg of the inverted V may not reach the bottom line of the parallelogram. In which case, a secondary move must be made: first parallel to the line BC to the "eastern" side of the paralellogram (pour B to C), and then to the bottom side (pour from A to B.)
Since we are only interested in the modular arithmetic, we may
overlook the need for secondary moves on the western side of the
parallelogram and keep applying only the basic move. Since A and C are
mutually prime, all lines
The condition A + B = C serves a double purpose. First, together with
the relative primality of A and B, it insures that all three capacities
share no common factor, save 1. Otherwise, the quantities that could be
measured with three vessels of the specified capacities would share that
common factor. For the problem to be generally solvable the mutual
primality of all three capacities is a necessary condition. However, it is
not sufficient. Anomalies also arise when the three jugs are rather
This completes the analysis of existence of a solution to the 3 Jugs problem. That of the 2 and 4 Jugs problems is left as an enticement for the future Poissons.
Alex Bogomolny has started and still maintains a popular Web site Interactive Mathematics Miscellany and Puzzles to which he brought more than 10 years of college instruction and, at least as much, programming experience. He holds M.S. degree in Mathematics from the Moscow State University and Ph.D. in Applied Mathematics from the Hebrew University of Jerusalem. He can be reached at firstname.lastname@example.org
Copyright © 1996-2000 Alexander Bogomolny