MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

Mathematical Recreations 121
Figure 4.25: Tartaglian Measuring Puzzle [62]
In the ternary diagrams of Figure 4.25, the horizontal lines correspond to
the contents of the the 8-pint vessel while the downward/upward slanting lines
correspond to that of the 5/3-pint vessel, respectively. The closed highlighted
parallelogram corresponds to the possible states of the vessels with those on
the boundary corresponding to the states with one or more of the vessels either
completely full or completely empty, i.e. the valid intermediate states in any
proposed solution. The number triplets indicate how much each vessel holds
at any stage of the solution process in the order (8-pint, 5-pint, 3-pint).
Starting at the apex marked by the initial state 800, the first move must
be to fill either the 5-pint vessel (Figure 4.25 left) or the 3-pint vessel (Figure
4.25 right). Thereafter, we follow the path of a billiard ball bouncing on the
indicated parallelogram until finally reaching the final state 440. The law of
reflection is justified by the fact that each piece of the broken lines, shown
hashed in Figure 4.25, are parallel to a side of the outer triangle of reference
and so represent the act of pouring liquid from one vessel into another while
the third remains untouched. Figure 4.25 left thereby yields the seven-step
solution:
800 → 350 → 323 → 620 → 602 → 152 → 143 → 440,
while Figure 4.25 right generates the eight-step solution:
800 → 503 → 530 → 233 → 251 → 701 → 710 → 413 → 440.
A more detailed study of this technique, especially as to its generalizations
and limitations, is available in the literature [118, 229, 296]
Recreation 22 (Barrel Sharing [284]). Barrel sharing problems have been
common recreational problems since at least the Middle Ages [284]. In their
simplest manifestation, N full, N half-full and N empty barrels are to be shared