# Bishops

Antreas P. Hatzipolakis xpolakis at otenet.gr
Sat Jun 3 01:04:39 CEST 2000

```On a n X n board can be placed 2n - 2 bishops (maximum), so that no one
can be captured by another.
The total number of solutions is 2^n.

The total number of DISTINCT solutions is given by the Dudeney's formulae:

n-4       n-2
---       ---
2         2
2     * (2      + 1), n: even

n-3       n-3
---       ---
2         2
2     * (2      + 1), n: odd

OR

2^((n-4)/2) * (2^((n-2)/2) + 1), n: even

2^((n-3)/2) * (2^((n-3)/2) + 1), n: odd

Table:

n                    Number of Distinct Solutions
---------------------------------------------------
2                                 1
3                                 2
4                                 3
5                                 6
6                                10
7                                20
8                                36
9                                72
10                              136
11                              272
12                              528
..........
cf A005418

Reference:
New York: Dover Publications, Inc., 1979, p. 46
(1st publ. by Charles Scribner's Sons, New York, 1966, under the title:
Mathematics on Vacation)

Question:
Has anyone the Dudeney's books _Amusements in Mathematics_,
_The Canterbury Puzzles_ (Dover, 1958) to give us the precise reference
for Dudeney's formualae above?

Antreas

```