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Amusements in Mathematics

Measuring, Weighing, And Packing Puzzles
"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 articles of
given dimensions into a box of known dimensions, are, I believe, of quite recent
introduction. At least I cannot recall any example in the books of the old writers. One
would rather expect to find in the toy shops the idea presented as a mechanical puzzle,
but I do not think I have ever seen such a thing. The nearest approach to it would appear
to be the puzzles of the jig-saw character, where there is only one depth of the pieces to
be adjusted.
One Christmas Eve three Weary Willies came into possession of what was to them a
veritable wassail bowl, in the form of a small barrel, containing exactly six quarts of fine
ale. One of the men possessed a five-pint jug and another a three-pint jug, and the
problem for them was to divide the liquor equally amongst them without waste. Of
course, they are not to use any other vessels or measures. If you can show how it was to
be done at all, then try to find the way that requires the fewest possible manipulations,
every separate pouring from one vessel to another, or down a man's throat, counting as a