[Illustration]

If the reader should cut out the above diagram, fold it in the form of a cube, and stick it together by the strips left for that purpose at the edges, he would have an interesting little curiosity.  Or he can make one on a larger scale for himself.  It will be found that if we imagine the cube to have a complete chessboard on each of its sides, we may start with the knight on any one of the 384 squares, and make a complete tour of the cube, always returning to the starting-point.  The method of passing from one side of the cube to another is easily understood, but, of course, the difficulty consisted in finding the proper points of entry and exit on each board, the order in which the different boards should be taken, and in getting arrangements that would comply with the required conditions.

341.—­THE FOUR FROGS.

The fewest possible moves, counting every move separately, are sixteen.  But the puzzle may be solved in seven plays, as follows, if any number of successive moves by one frog count as a single play.  All the moves contained within a bracket are a single play; the numbers refer to the toadstools:  (1—­5), (3—­7, 7—­1), (8—­4, 4—­3, 3—­7), (6—­2, 2—­8, 8—­4, 4—­3), (5—­6, 6—­2, 2—­8), (1—­5, 5—­6), (7—­1).

This is the familiar old puzzle by Guarini, propounded in 1512, and I give it here in order to explain my “buttons and string” method of solving this class of moving-counter problem.

Diagram A shows the old way of presenting Guarini’s puzzle, the point being to make the white knights change places with the black ones.  In “The Four Frogs” presentation of the idea the possible directions of the moves are indicated by lines, to obviate the necessity of the reader’s understanding the nature of the knight’s move in chess.  But it will at once be seen that the two problems are identical.  The central square can, of course, be ignored, since no knight can ever enter it.  Now, regard the toadstools as buttons and the connecting lines as strings, as in Diagram B. Then by disentangling these strings we can clearly present the diagram in the form shown in Diagram C, where the relationship between the buttons is precisely the same as in B. Any solution on C will be applicable to B, and to A. Place your white knights on 1 and 3 and your black knights on 6 and 8 in the C diagram, and the simplicity of the solution will be very evident.  You have simply to move the knights round the circle in one direction or the other.  Play over the moves given above, and you will find that every little difficulty has disappeared.

[Illustrations:  A B C D E]