313.—­QUEENS AND BISHOP PUZZLE.

[Illustration:  FIG. 1.]

[Illustration:  FIG. 2.]

The bishop is on the square originally occupied by the rook, and the four queens are so placed that every square is either occupied or attacked by a piece. (Fig. 1.)

I pointed out in 1899 that if four queens are placed as shown in the diagram (Fig. 2), then the fifth queen may be placed on any one of the twelve squares marked a, b, c, d, and e; or a rook on the two squares, c; or a bishop on the eight squares, a, b, and e; or a pawn on the square b; or a king on the four squares, b, c, and e.  The only known arrangement for four queens and a knight is that given by Mr. J. Wallis in The Strand Magazine for August 1908, here reproduced. (Fig. 3.)

[Illustration:  FIG. 3.]

I have recorded a large number of solutions with four queens and a rook, or bishop, but the only arrangement, I believe, with three queens and two rooks in which all the pieces are guarded is that of which I give an illustration (Fig. 4), first published by Dr. C. Planck.  But I have since found the accompanying solution with three queens, a rook, and a bishop, though the pieces do not protect one another. (Fig. 5.)

[Illustration:  FIG. 4.]

[Illustration:  FIG. 5.]

314.—­THE SOUTHERN CROSS.

My readers have been so familiarized with the fact that it requires at least five planets to attack every one of a square arrangement of sixty-four stars that many of them have, perhaps, got to believe that a larger square arrangement of stars must need an increase of planets.  It was to correct this possible error of reasoning, and so warn readers against another of those numerous little pitfalls in the world of puzzledom, that I devised this new stellar problem.  Let me then state at once that, in the case of a square arrangement of eighty one stars, there are several ways of placing five planets so that every star shall be in line with at least one planet vertically, horizontally, or diagonally.  Here is the solution to the “Southern Cross”:  —­

It will be remembered that I said that the five planets in their new positions “will, of course, obscure five other stars in place of those at present covered.”  This was to exclude an easier solution in which only four planets need be moved.

315.—­THE HAT-PEG PUZZLE.

The moves will be made quite clear by a reference to the diagrams, which show the position on the board after each of the four moves.  The darts indicate the successive removals that have been made.  It will be seen that at every stage all the squares are either attacked or occupied, and that after the fourth move no queen attacks any other.  In the case of the last move the queen in the top row might also have been moved one square farther to the left.  This is, I believe, the only solution to the puzzle.

[Illustration:  1]