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:
Joseph S. Madachy: Madachy's Mathematical Recreations.
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
More information about the SeqFan
mailing list