I inwardly congratulated myself that I never had.

“That position, sir, materializes the sinuous evolvements and syncretic, synthetic, and synchronous concatenations of two cerebral individualities.  It is the product of an amphoteric and intercalatory interchange of—­”

“Have you seen the evening paper, sir?” interrupted the man opposite, holding out a newspaper.  I noticed on the margin beside his thumb some pencilled writing.  Thanking him, I took the paper and read—­“Insane, but quite harmless.  He is in my charge.”

After that I let the poor fellow run on in his wild way until both got out at the next station.

But that queer position became fixed indelibly in my mind, with Black’s last move 43.  K to Kt 8; and a short time afterwards I found it actually possible to arrive at such a position in forty-three moves.  Can the reader construct such a sequence?  How did White get his rooks and king’s bishop into their present positions, considering Black can never have moved his king’s bishop?  No odds were given, and every move was perfectly legitimate.

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 Meziriac reprinted this and other of Tartaglia’s puzzles in his Problemes plaisans et delectables (1612).  It is the general opinion that puzzles of this class can only be solved by trial, but I think formulae 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.