340.—­THE CUBIC KNIGHT’S TOUR.

Some few years ago I happened to read somewhere that Abnit Vandermonde, a clever mathematician, who was born in 1736 and died in 1793, had devoted a good deal of study to the question of knight’s tours.  Beyond what may be gathered from a few fragmentary references, I am not aware of the exact nature or results of his investigations, but one thing attracted my attention, and that was the statement that he had proposed the question of a tour of the knight over the six surfaces of a cube, each surface being a chessboard.  Whether he obtained a solution or not I do not know, but I have never seen one published.  So I at once set to work to master this interesting problem.  Perhaps the reader may like to attempt it.

341.—­THE FOUR FROGS.

[Illustration]

In the illustration we have eight toadstools, with white frogs on 1 and 3 and black frogs on 6 and 8.  The puzzle is to move one frog at a time, in any order, along one of the straight lines from toadstool to toadstool, until they have exchanged places, the white frogs being left on 6 and 8 and the black ones on 1 and 3.  If you use four counters on a simple diagram, you will find this quite easy, but it is a little more puzzling to do it in only seven plays, any number of successive moves by one frog counting as one play.  Of course, more than one frog cannot be on a toadstool at the same time.

342.—­THE MANDARIN’S PUZZLE.

The following puzzle has an added interest from the circumstance that a correct solution of it secured for a certain young Chinaman the hand of his charming bride.  The wealthiest mandarin within a radius of a hundred miles of Peking was Hi-Chum-Chop, and his beautiful daughter, Peeky-Bo, had innumerable admirers.  One of her most ardent lovers was Winky-Hi, and when he asked the old mandarin for his consent to their marriage, Hi-Chum-Chop presented him with the following puzzle and promised his consent if the youth brought him the correct answer within a week.  Winky-Hi, following a habit which obtains among certain solvers to this day, gave it to all his friends, and when he had compared their solutions he handed in the best one as his own.  Luckily it was quite right.  The mandarin thereupon fulfilled his promise.  The fatted pup was killed for the wedding feast, and when Hi-Chum-Chop passed Winky-Hi the liver wing all present knew that it was a token of eternal goodwill, in accordance with Chinese custom from time immemorial.

The mandarin had a table divided into twenty-five squares, as shown in the diagram.  On each of twenty-four of these squares was placed a numbered counter, just as I have indicated.  The puzzle is to get the counters in numerical order by moving them one at a time in what we call “knight’s moves.”  Counter 1 should be where 16 is, 2 where 11 is, 4 where 13 now is, and so on.  It will be seen that all the counters on shaded squares are in their proper positions.  Of course, two counters may never be on a square at the same time.  Can you perform the feat in the fewest possible moves?