"Measure still for measure."
_Measure for Measure_, v. 1.


Apparently the first printed puzzle involving the measuring of a given
quantity of liquid by pouring from one vessel to others of known
capacity was that propounded by Niccola Fontana, better known as
"Tartaglia" (the stammerer), 1500-1559. It consists in dividing 24 oz.
of valuable balsam into three equal parts, the only measures available
being vessels holding 5, 11, and 13 ounces respectively. There are many
different solutions to this puzzle in six manipulations, or pourings
from one vessel to another. Bachet de Méziriac reprinted this and other
of Tartaglia's puzzles in his _Problèmes plaisans et délectables_
(1612). It is the general opinion that puzzles of this class can only be
solved by trial, but I think formulæ can be constructed for the solution
generally of certain related cases. It is a practically unexplored field
for investigation.

The classic weighing problem is, of course, that proposed by Bachet. It
entails the determination of the least number of weights that would
serve to weigh any integral number of pounds from 1 lb. to 40 lbs.
inclusive, when we are allowed to put a weight in either of the two
pans. The answer is 1, 3, 9, and 27 lbs. Tartaglia had previously
propounded the same puzzle with the condition that the weights may only
be placed in one pan. The answer in that case is 1, 2, 4, 8, 16, 32 lbs.
Major MacMahon has solved the problem quite generally. A full account
will be found in Ball's _Mathematical Recreations_ (5th edition).

Packing puzzles, in which we are required to pack a maximum number of