Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 606 — #614
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Chapter 15
Cardinality Rules
8
0Z0Z0Z0s
7
Z0Z0Z0Z0
6
0Z0Z0Z0Z
5
Z0Z0Z0Z0
4
0Z0Z0Z0Z
3
Z0Z0Z0Z0
2
0Z0Z0Z0Z
1
s0Z0Z0Z0
a b c d e f g h
(a) valid
8
0Z0Z0Z0Z
7
Z0Z0Z0Z0
6
0Z0s0Z0Z
5
Z0Z0Z0Z0
4
0Z0Z0Z0Z
3
Z0Z0Z0Z0
2
0Z0Z0Z0Z
1
Z0ZrZ0Z0
a b c d e f g h
(b) invalid
Figure 15.2 Two ways to place 2 rooks (R) on a chessboard. The configuration
in (b) is invalid because the rooks are in the same column.
But now there’s a snag. Consider the sequences:
.1; a; 8; h/ and .8; h; 1; a/
The first sequence maps to a configuration with a rook in the lower-left corner and
a rook in the upper-right corner. The second sequence maps to a configuration with
a rook in the upper-right corner and a rook in the lower-left corner. The problem is
that those are two different ways of describing the same configuration! In fact, this
arrangement is shown in Figure 15.2(a).
More generally, the function f maps exactly two sequences to every board configuration;
f is a 2-to-1 function. Thus, by the quotient rule, jAj D 2 jBj. Rearranging
terms gives:
jBj D jAj
2
.8 7/2
D :
2
On the second line, we’ve computed the size of A using the General Product Rule
just as in the earlier chess problem.
15.4.2 Knights of the Round Table
In how many ways can King Arthur arrange to seat his n different knights at his
round table? A seating defines who sits where. Two seatings are considered to be
the same arrangement if each knight sits between the same two knights in both