Chapter: Measuring, Weighing, and Packing Problems

"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 Niccolò Fontana, better known as Tartaglia (the stammerer), 1500-1559. It consists of 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. Claude Bachet 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 a 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.

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References

Project Gutenberg

1. Dudeney, H. E.: "Amusements in Mathematics", The Authors' Club, 1917

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