itself exactly; but if you look at it with one of the other sides at the bottom, you get another way that is not identical.  Then if you reflect these two ways in a mirror you get two more ways.  Now, all the other eleven solutions are non-symmetrical, and therefore each of them may be presented in eight ways by these reversals and reflections.  It will thus be seen why the twelve fundamentally different solutions produce only ninety-two arrangements, as I have said, and not ninety-six, as would happen if all twelve were non-symmetrical.  It is well to have a clear understanding on the matter of reversals and reflections when dealing with puzzles on the chessboard.

Can the reader place the eight queens on the board so that no queen shall attack another and so that no three queens shall be in a straight line in any oblique direction?  Another glance at the diagram will show that this arrangement will not answer the conditions, for in the two directions indicated by the dotted lines there are three queens in a straight line.  There is only one of the twelve fundamental ways that will solve the puzzle.  Can you find it?

301.—­THE EIGHT STARS.

[Illustration: 

+—–­+—–­+—–­+—–­+—–­+—–­+—–­+—–­+
|///|   |   |   |   |   |   |///|
+—–­+—–­+—–­+—–­+—–­+—–­+—–­+—–­+
|   |///|   |   |   |   |///| * |
+—–­+—–­+—–­+—–­+—–­+—–­+—–­+—–­+
|   |   |///|   |   |///|   |   |
+—–­+—–­+—–­+—–­+—–­+—–­+—–­+—–­+
|   |   |   |///|///|   |   |   |
+—–­+—–­+—–­+—–­+—–­+—–­+—–­+—–­+
|   |   |   |///|///|   |   |   |
+—–­+—–­+—–­+—–­+—–­+—–­+—–­+—–­+
|   |   |///|   |   |///|   |   |
+—–­+—–­+—–­+—–­+—–­+—–­+—–­+—–­+
|   |///|   |   |   |   |///|   |
+—–­+—–­+—–­+—–­+—–­+—–­+—–­+—–­+
|///|   |   |   |   |   |   |///|
+—–­+—–­+—–­+—–­+—–­+—–­+—–­+—–­+

]

The puzzle in this case is to place eight stars in the diagram so that no star shall be in line with another star horizontally, vertically, or diagonally.  One star is already placed, and that must not be moved, so there are only seven for the reader now to place.  But you must not place a star on any one of the shaded squares.  There is only one way of solving this little puzzle.

302.—­A PROBLEM IN MOSAICS.

The art of producing pictures or designs by means of joining together pieces of hard substances, either naturally or artificially coloured, is of very great antiquity.  It was certainly known in the time of the Pharaohs, and we find a reference in the Book of Esther to “a pavement of red, and blue, and white, and black marble.”  Some of this ancient work that has come down to us, especially some of the Roman mosaics, would seem to show clearly, even where design is not at first evident, that much thought was bestowed upon apparently disorderly arrangements.  Where, for example, the work has been produced with a very limited number of colours, there are evidences of great ingenuity in preventing the same tints coming in close proximity.  Lady readers who are familiar with the construction of patchwork quilts will know how desirable it is sometimes, when they are limited in the choice of material, to prevent pieces of the same stuff coming too near together.  Now, this puzzle will apply equally to patchwork quilts or tesselated pavements.