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.
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
particular, x + y + z = C. A typical state - distribution of
water - of the puzzle is described by a triple (x, y, z). For
the original problem, the initial state is thus (0, 0, C) with
C = 8.
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 (r, r,
r) which, in the homogeneous form, appear as r : r : r,
or simply as 1:1:1. On the other hand, in the homogeneous barycentric
coordinates, 1:1:1 corresponds to the centroid
(also called the center of gravity or barycenter) of ABC. For equilateral triangles, the
two system of coordinates coincide.
The description of the 3 Jugs problem as a triple of quantities
(x, y, z) fits nicely with trilinear coordinates. Draw an
equilateral triangle ABC and let the vertices have trilinear coordinates
(C, 0, 0), (0, C, 0), and (0, 0, C)
- in that order. The sides AB, BC, and CA are defined by z =
0, x = 0, and y = 0, respectively. The
other lines on which one of the coordinates is a constant integer form a
triangular grid of lines parallel to the sides of ABC. For example, lines x = const are
parallel to the side BC.
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., 0
xA) 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 z = const (mod C) will eventually be
covered.
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
big. Condition A + B = C prevents this from happening.
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.
References
- W.W. Rouse Ball and H.S.M. Coxeter, Mathematical Recreations and Essays, Dover, 1987
- H.S.M. Coxeter and S.L. Greitzer, Geometry Revisited, MAA 1967
- E. Kasner and J. Newman, Mathematics and the Imagination, Simon and Schuster, 1958
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
alexb@cut-the-knot.com
Copyright © 1996-2000 Alexander Bogomolny
