298.—­BISHOPS—­GUARDED.

Now, how many bishops are necessary in order that every square shall be either occupied or attacked, and every bishop guarded by another bishop?  And how may they be placed?

299.—­BISHOPS IN CONVOCATION.

[Illustration: 

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| B | B | B | B | B | B | B | B |
+—–­+—–­+—–­+—–­+—–­+—–­+—–­+—–­+
|   |   |   |   |   |   |   |   |
+—–­+—–­+—–­+—–­+—–­+—–­+—–­+—–­+
|   |   |   |   |   |   |   |   |
+—–­+—–­+—–­+—–­+—–­+—–­+—–­+—–­+
|   |   |   |   |   |   |   |   |
+—–­+—–­+—–­+—–­+—–­+—–­+—–­+—–­+
|   |   |   |   |   |   |   |   |
+—–­+—–­+—–­+—–­+—–­+—–­+—–­+—–­+
|   |   |   |   |   |   |   |   |
+—–­+—–­+—–­+—–­+—–­+—–­+—–­+—–­+
|   |   |   |   |   |   |   |   |
+—–­+—–­+—–­+—–­+—–­+—–­+—–­+—–­+
|   | B | B | B | B | B | B |   |
+—–­+—–­+—–­+—–­+—–­+—–­+—–­+—–­+

]

The greatest number of bishops that can be placed at the same time on the chessboard, without any bishop attacking another, is fourteen.  I show, in diagram, the simplest way of doing this.  In fact, on a square chequered board of any number of squares the greatest number of bishops that can be placed without attack is always two less than twice the number of squares on the side.  It is an interesting puzzle to discover in just how many different ways the fourteen bishops may be so placed without mutual attack.  I shall give an exceedingly simple rule for determining the number of ways for a square chequered board of any number of squares.

300.—­THE EIGHT QUEENS.

[Illustration: 

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|   |   |   |   ..Q |   |   |   |
+—–­+—–­+—–­+...+—–­+—–­+—–­+—–­+
|   |   ..Q..   |   |   |   |   |
+—–­+...+—–­+—–­+—–­+—–­+—–­+—–­+
| Q..   |   |   |   |   |   |   |
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|   |   |   |   |   |   | Q |   |
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|   | Q |   |   |   |   |   |   |
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|   |   |   |   |   |   |   ..Q |
+—–­+—–­+—–­+—–­+—–­+—–­+...+—–­+
|   |   |   |   |   ..Q..   |   |
+—–­+—–­+—–­+—–­+...+—–­+—–­+—–­+
|   |   |   | Q..   |   |   |   |
+—–­+—–­+—–­+—–­+—–­+—–­+—–­+—–­+

]

The queen is by far the strongest piece on the chessboard.  If you place her on one of the four squares in the centre of the board, she attacks no fewer than twenty-seven other squares; and if you try to hide her in a corner, she still attacks twenty-one squares.  Eight queens may be placed on the board so that no queen attacks another, and it is an old puzzle (first proposed by Nauck in 1850, and it has quite a little literature of its own) to discover in just how many different ways this may be done.  I show one way in the diagram, and there are in all twelve of these fundamentally different ways.  These twelve produce ninety-two ways if we regard reversals and reflections as different.  The diagram is in a way a symmetrical arrangement.  If you turn the page upside down, it will reproduce