Contents 1 Introduction 2 Invariants " " # % & ' " "
( " ) %' *
3 Crossing a River + ,
-. / %' 0 - 1 " 2 3 " ( 3 ) 4 ( . 5 )
&
1 5 ! ! $ ) $ 21
) * ! ! $ ( ) $
4 Games ( # , ( - ( ( 6 ( + 5 ( / , (( . , (( &
& (( 7 8 &
9 (( # ((( " #%: + (() ; #%: + ()
& (*
5 Knights and Knaves ) 6 << ) 6 ) ) 7 7 ) %5& ) ( = " ) ) %5 . %5 ) %5 %5 ) %' . & . %5 ) > ? 6 )( ? )) )* = ) 85 )!
& 6 Induction * %' * " * " *( 6 + *) " ? + .
41 ( ( ( (( () (! ) ) )) )* ) )$ * *
63 * *( *( *) ** *! *$ ( * $ ! !
85 !) !$ $ $ $)
** *
*! *$
+ @A +/ 3 * + *
& ?
7 The Towers of Hanoi A " % . -9 8 -=B2 8 " 8 (
& ) 8 The ! ! ! !( !) !* ! !!
Torch Problem 64 ; > & 5 5 +4 " ? "
9 Knight’s Circuit $ /# $ 5 $ $( 3 $) . > < $*
$) $$ ) )
107 ! $ $ ) *
117 ! * $
133 ( (* (! (!
Solutions to Exercises
149
Bibliography
170
Chapter 1 Introduction 8 & & 4 " . & 4 C C 4 . & &
& . B & D E / . F . . G4
4 . H .
/ & A " 4& " . /
1.1
Algorithms
. / / 4/A .
. ' ' & . .4 4 & . . ' 4 B & 4 . . + 4 8 & 4 & & 4 " 4 & 4 & " . F . I 4 4 & . 4 " A 1 2 5 . 10 " . .
4& 4 4 . 4 17 " C4 4 41 D &9C & / 5 . 4 . & / 4 J K x y L K z L " 5 . . ' G H 17 & & ? & 4 I . 4 & / 8 . / .
. . + ' .
1 3 19 20 30 .
1 4 5 6 17 " . + / & / & . & & & . "
G+ /
& 4
. H " 4 . " . . . . " & F / & . . ' C . & '& " & . '& & . < . J & 4 . 4 . . 4 & & & K L .
/ " 4 8 . ' & & I
/ & & & & 4 & & 4 " I
. 5 Æ &
1.2
Bibliographic Remarks
8 A . M6 N M N / & " 4 KE L K; L I & D .4 & 4 . 4 &
(
Chapter 2 Invariants K8L K L . & . 1 > . KL K L K4L KL " . / & " .
. ' . - 4 - A
. . & & & A & & & A Æ 8. & 1 ' " . A 4 & & A . . - . & " A 4 4 4 4& 4
C . 4 " 4 4 4 5 4 .4 - & 5 4 4 5 . 4 & " 5 & 4 4 4 " & + . & ' . " I 4 4 & B A Chocolate Bars 5 & < 4& 8 5 & . G" 4 H
)
*
+ 4 4×3 A " &
+ J
/
=4 & & 2 Empty Boxes % & ' 4
. ' ' 4
. ' ' . 102 & ' =4 & ' 2 Tumblers
4 4& G A H 8 5
4& =4
& &I 4
&
&
+ J "
+ 4 .
4& 2 ( Black and White Balls A 4
. . 4 4 " & .4 - 4
. & .4 . & " & 4 . 8. 4 4&
I . & F 4 4 4& % ' .
. & I 4 & . - . & . A
.
4 2 ) Dominoes / /. 5 62 5 G A H & .
+ J #
I 4 ' & 4 5 . 8 62 5 . 4 4 & & .
2 * Tetrominoes A . 4 5 . < " A F >/ O/ 6/ "/ 8/ G A (H " .4 ' 4 . 5 . < > . . 4 GH 4 4 .
. 5
!
+ (J >/ O/ 6/ "/ 8/ G H 4 "/ 4
. 5 . 8 G H 4 6/ 4
. 5 . 8 GH 8×8 4 >/ A. 6/ - & 2
2.1
Chocolate Bars
J 5 & < 4& 8 5 & . G" 4 H =4 & & 2
2.1.1
The Solution
" / .4 -
. &
. & "
.
. - 4 4
.
. " KL K L . . ?4 4 4 < F 4
.
. 8 4 4& 4 & "
. 4 4&
. %5&
. 4 4&
.
$
-
.
.
2.1.2
The Mathematical Solution
> . .& & + 4 % . + ' & 6
4 4 Abstraction " & 4 - p
. 4 c
. " . . " A
- K L . . G K LH & . 8 > . & 4
9 " & & "
5& 4 . . " K L .
K/4L /4 & . I
& & & > 5 .
< . " p c & . 5 . 74 & .
A 4 < . " " . 8 & & . . & / & 4 . . & C / . 9C " / & G"' / . 4 A " & .& C. ' 4 A .& 4
& C 9 " 4 " A ' . Æ A " 4
" A . & ' A
& A > A 4 I . . . F H Assignments " ' 1 . - & . p , c := p+1 , c+1 .
4 " 4 & K := L K L " .
/ . G p , c H ? & . "
/ . ' G p+1 , c+1 H " 5
. . F . ' & ' " & . . & . ' 8 ' C
.
. C & p+1 c+1 . p c & & 8 4 p K L p+1 c K L c+1 " 4 . . . . . 4 ' . + ' p−c . p , c := p+1 , c+1 E ' . G+ ' p−c ' p c H - E & & . 5& 4 . E . E . & + ' 5& A word of warning
!
" # " $ = %
& " " ' # "
Æ (() ! $
becomes
%
$ " %
p−c = (p+1) − (c+1) ,
4 . p c " p−c . p , c := p+1 , c+1 " . . 5& ' E ' E . & ' 4 4 m n 4 m , n := m+3 , n−1
- m + 3×n & m + 3×n = (m+3) + 3×(n−1) .
4 m & 3 & n & 1 . m + 3×n , ' E ls := rs E[ls := rs]
' & . E ls & ' . ' rs = ' J (p−c)[p , c := p+1 , c+1] = (p+1) − (c+1)
(m + 3×n)[m , n := m+3 , n−1] = (m+3) + 3×(n−1)
(m+n+p)[m , n , p := 3×n , m+3 , n−1] = (3×n) + (m+3) + (n−1)
" . .4 E . ls := rs . . . E
E[ls := rs] = E .
Induction " A . ' . p−c 8& p = 1 c = 0 & p−c = 1 p−c p−c = 1 4 & - 5 p = s 4 s
. 5
. c A s−c = 1 " c = s−1 "
.
. 5
" & 8 . . ' / & 4 4
& " . < . ' G & & < H 8. ' & . ' & & 4 & . ' . . ? . 8 & . < 8 . / ' & 5 G 4 < 5H Summary " . /
. / C - 4 Exercise 2.1 . "4 & I G 1 & & H 4 " 4 . & . . & 1234 & =4 & & / . 4 2 G=J H 2
2.2
Empty Boxes
"& &/ ' % & ' 4
. ' '
4
. '
' . 102 & ' =4 & ' 2 " .4 8 e f .
. &
. .
' & 8.& . e f 8.& A . e # . ' ' e f ( 8.& . )
A . f = A . e+f
? &
. '
. ' 4 . 4 & . 5 & / " & F /J 9
2.3
The Tumbler Problem
6 4 4
.
4 4& 8 5
4& =4
& &I 4 &
& + 4 .
4& 2 " 4 D
.
4 6 u " F . 4 .
"4
4& 4 " &
(
u := u+2 .
" 4
4 F C u
& 4 " & u := u−2 .
+& 4
4& G 4 4& H F u 8
& / KL K L K F L 8 ' 5 u := u ,
. & - skip & ' skip .
" . 4 . ' . A . . . . %& . skip 4 skip - . . 4 u := u+2 u := u−2 - . 4 4 . u 2 " 4 J / . u " & . u J true false 8 true . u G< 4 . H false . u G A H 6 4 even.u . 5& " (even.u)[u := u+2] = even.(u+2) = even.u .
" even.u . u := u+2 (even.u)[u := u−2] = even.(u−2) = even.u .
" even.u . u := u−2 - 4 & 4 4
& .
. /4
4 8.
4 4&
I .
4 4&
" < /4
O
4 5
. /4
)
B 4 . 4 G ( & H & . . " ' . P &
. 8.& 4 " 4& G=J . 5 H *GH 81 & *G H 4 4 C 4 *GHC ' = . & 4
2.4
Tetrominoes
8 4 . *G H " & & . & 4 "/ 4
. 5 . 8 . & . c
. 5 " & c := c+4 .
" c mod 4 G c mod 4 . c & 4 + ' 7 mod 4 3 16 mod 4 0 H 8& c 0 c mod 4 0 mod 4 4 0 c mod 4 4& 0 8 4 4 & K c . 4 &L # . 4 K &L 4 & K c . 4 L ?4 m×n G"
. 5 m
n H " c = m×n m×n . 4 + m×n . 4
m n
. 4 m n G H . 2 ? . & "/ - 4 D 4 . *GHJ . & . . " & . *GH 4& / " . Æ & & . 8 . &
*
KÆ L 6 4& KÆ L 0 & & 9 4 4 & . ' . ' 4 ' " / & 4 D = 4 &J m×n 4 ⇒
{
J c . 4 c = m×n
}
m×n . 4 ⇒
{
& .
}
m . 2 ∨ n . 2 .
" 4/ " A / K L
& K ⇒ L &
"
m×n 4 m×n . 4 G& K m×n 4 m×n . 4 L K m×n 4 m×n . 4 LH " ' 4 & .4 K ⇒ L &
4 & /
=
. .
. 5 4& . 4 G4 . H 4 . . m×n
. 5 m×n " J 8. m×n . 4 m . 2 n . 2 K ⇒ L &
A " &
K ∨ L KL ? & KL 4 / K L C & m n
. 2 / K' L 4 m . 2 n . 2 4 ' & " A " & 4 '
.
B & & . / 4 & Æ 4 . &
& 4 " . K.L 8 J
8. m×n 4 m . 2 n . 2 " & . &J / ' 4 ' 4& 8/ 4 & 4 4 ' &
4 I & & 8 &
K ⇒ L 4 ' # & 4 6 4 4 K ⇐ L &
4 4 . 4 4 . F 4& 6 4 *G H & . . "/ G8 1 . 8/ 4×1 4 1 8/ 4 . 8 H - "/ 5 D 5 5 F& . 5 . 4& 4 4 4 5 /
" "/ 4& " 4 & 4 5 4 5 4 4 5 5 - "/ G A )H 4 & . 5 .
+ )J 3 "/ - . . " b
. 5 4 w
. 4 5 8 d
. "/ l
. & d , b , w := d+1 , b+3 , w+1 .
& l , b , w := l+1 , b+1 , w+3 .
.
!
b − 3×d − l ,
(b − 3×d − l)[d , b , w := d+1 , b+3 , w+1] =
{
A .
}
(b+3) − 3×(d+1) − l =
{
}
b − 3×d − l
(b − 3×d − l)[l , b , w := l+1 , b+1 , w+3] =
{
A .
}
(b+1) − 3×d − (l+1) =
{
}
b − 3×d − l .
& . w − 3×l − d .
?4 . b − 3×d − l < 4 4& < 4
& "/ & . w − 3×l − d 4 4& < - 4 & "/ ⇒
{
. *GH 4 4 .
. 5 4
. 5 5
. 4 5
}
b=w ⇒
{
b − 3×d − l = 0 w − 3×l − d = 0
}
(b = w) ∧ (3×d + l = 3×l + d)
!" ⇒
{
$ }
(b = w) ∧ (l = d) ⇒
{
b − 3×d − l = 0 w − 3×l − d = 0
}
b = w = 4×d = 4×l ⇒
{
}
b+w = 8×d ⇒
{
b+w
. 5
}
. 5 . 8 . - 8. & "/
. 5 & 8 B 4 *G H " & *G H 4 4& 8 6 4& . 5 4 = . 4& & . G=4 & & . F . & 8 1 Æ *G H *G H & .& 4 & . "/ & & A 4 . ' & 4 & Æ 4 H *GH & & *G H , 9
2.5
Additional Exercises
Exercise 2.2 , . . D "
. D & & 4 D . F & D . 8.& ' 4 D ' 2
2.6
Bibliographic Remarks
" &/ ' 4 & - +D " . 4 . M,! N " 8 . $$$ G JPP4444P<<PH @ @ - C.5 . C . / 3 " & ' . << 4 8 . . ' " 0 & - . 4 . . 4 8 . "
8 4 %' 4 & 3 8 4 G & F . H # >& $ )
& @ KO @&<&& # >& L # 4 $!! " . &
Chapter 3 Crossing a River " '
. - . K /. L & & & 81 5 1 5 & 5 . . 4 ; . I . . 4 / . 1 . " . 4 ' <
. & ? . 4 4
. . / 4 / ' . 4
> 4 4& . 8 . & . & 4 1 4 =4 F . /. . / & & &
&
4 4 . " ' . &
& & / 5 . & 4 . B & & 4 .
/. C4 4 . C & ' & 9 =4 . . &
&
4
& & . 3 4 4 - 4 & '& .
4 . & 4.
3.1
Problems
Goat, Cabbage and Wolf . 4 .&
4. =4 & . & . 4
G 4 4
H 4. . 4 G 4 4. 4 H =4 . 2 The Jealous Couples " G 4.H 4 " & & 4 & " D . 4 4 4. 4 . & =4 2 Adults and Children . . " & &
4 =4 2 # & / & ( Overweight 3 . " & 4 & 46 49 52 3 100 1 4 =4 & 2
3.2 3.2.1
Brute Force Goat, Cabbage and Wolf
" /
/ 4./ . /. >
4 /. . .
#
/ & 4 & . 4 Æ & . . 4 .&
4. =4 & . & . 4
G 4 4
H 4. . 4 G 4 4. 4 H =4 . 2 " . . 4 " 4 & . . 4 . 4 - f G. . H g G. H c G.
H w G. 4.H 4 L G. .H R G. H . R
K L . L . ? 4& 4 . 4 4& . . C 8 /
/ 4./ 4 . . . . 4 & A + ' 4 4 4 J . . " . 4
4. . + & 4 / " 4 4 " 4 . I .
8. 4 &
- 4 / /. 8. . . 4 24 G 'H F
. =4 .
'/ " 5 . 4
'
&
(
f = g = c ∨ g = c .
" .
G f = g = c H
F G g = c H " ' 4 g c 5 F . f & 5
. 4 4. ' & & f = g = w ∨ g = w .
8. 4 4 " 4 4 F
G? 4 f g 5
. c w 4I 4 f g F c w 5 5H f 6 6 6 6 6
g 6 6 6 6 6
c 6 6 6 6 6
w 6 6 6 6 6
?4 4 4 " A D " . C 'C . C 'C " 4 & . ' K6666L 4 . . " & 4 . 4 .
4. . " & K6666L ' K66L ' + 4 % & . K6666L ' KL ' " .4 J " . " . 6666 666
#
)
RRRL LLLL
RRLL
LLRL
LRLL
RLRR RRLR
+ J ,/
LLRR
RRRR
LLLR
/ -./
" .
4 " . 666 666 " . 4. " . 666 66 ( " . " . 66 " & 4 K
L K4.L
3.2.2
State-Space Explosion
" . & & . 4 4 . 4 4 =4 & 4 . & . . & + 5 &
8 D & . K L " /
/ 4./ I C 9C
.
. - 4 5 & 4 & & 4 . 8 K4 L . . 16 I /
/ 4./ < . . . F I .
4
. . / " . KD/ L 4 = ' . 4 .
*
8. 4
. 26 64 9 " 1
. & 4 4 1
. 8 D/ A "
. 210 1024 F &
. " J K&L " K// L . . . & & A "
. A B . . " . . /. . . 4 K L 4 KL ' " / K/ ' L 4& " / K 'L & 4 8. n 2n F . 5 n 2n &
- . K'L 4
. G n ' 2n H - . . 4 '& . & 5 34 / 5& F &
. 4/ & . / 8 5 & C . < . 4 2
3.2.3
Abstraction
" / ' . & . & & I & .5 & " /
/ /4. ' 8 /
//4. K. L KL K
L K4.L 4 & 4 .2 8 . 4 K &L 4 4.
A & . 4 4.
" K &L J 4 4 &
. K4.L K
L - & K4.L K
L & F 2 6 4 / & 4 4.
8 4
$
K L
4. K L . 4 .& 4 =4 & . & . 4 =4 . 2 ?4 8 & J " 4 " .4 & & & & / . " / 4 . ' & Avoid unnecessary or inappropriate naming.
- . . . < . " . &
. K L 5&
& - . &
3.3
Jealous Couples
@& . .
" . &
Æ & & & . " . " D/ ' ' 8 & .
& F & . / J " G 4.H 4 " &
& 4 & " D . 4 4 4. 4 . & =4 2
!
3.3.1
What’s The Problem?
. 4 & 4 . & 4 K L / " & &
. K L J . 4 . . .4 & . . 4 . ?4 & K L & K L 4 . & 4 .& &
. G. D . 9 H 4
& . Æ .& &
. " & F 4 & 4
& .& 2 # & 4
& .& n 2 > &
& 4 G .& & & H 4 D 4 . & 4 .& & . 4 '
. . 4 & 8. & G H '
. < . & .
' 5 4 & . 4 . & 4 4 & . 4 . & " 4 Æ 8 4 4 C . 4 & 4 C 4 5 . . 4 4 C 4 & A .
3.3.2
Problem Structure
" . 4& 4 + " " 4 J A 4 ' K
. L 4 & A 4 .4 &
$
$
& K . &L 4 & A .4 & 4 , 4 4 . & " . & Æ & . 4 . , A 4 4 & . > D 4 & 4 D 6 . . &I 1 4 " 4 & & . 4 & 8 4 . " . . . . 4& I . . . . &
9 8. 4 & . 4 D " 4 4 G" / . /
/ 4./ ' ./ &
& & 4 " &
& 4 /
4. &
4 ' . I & /. 4 & . FH
3.3.3
Denoting States and Transitions
- & & " . . 4 4 A = 4 H W C 4. & " &
I . ' 2H 4 3C 1C,2H 4 - ' 4 I . ' 1H,1W 4. 4 . 4 1C 4. 4 .
? 4 . ' 8 & . 4 1 ?
' & - 4
4 G 4.H .
& 4 5 & ' 3H || 3W 4 4 . 4 ' . 1C,2H || 2W 4
4 4 . 4 4 " 3C || 5 A || 3C
4 ' 3H |2W| 1W I . 4 4 . 4. ? . .& . . '
.
4 .
& & " 4 P & A + '/ 1C,1W || 1C,1H G 4. 4 . 4 . H 3H |3W| & & 4 8 5 4 3C || 4 || 3C 4 . G8 & . / / F 4 H ' . 8. p q S 5 . {
p }
S {
q
}
& . 5 . S . p 4 q . ' {
2C,1H || 1W
}
3H |2W| 1W {
3H || 3W
}
& 4 4 . 4 4 4 4 . 4 4 >. 4 4& & . 8 & .4
$
3.3.4
Problem Decomposition
; 4 ' & . " 5 . S0 .& { 3C || }
S0 { || 3C } .
> &
' 4 .
" ./ &
& " . & 4 . 4
" & & S0 5 S1 S2 S3 { 3C || } S1 { 3H || 3W } , { 3H || 3W } S2 { 3W || 3H } ,
{ 3W || 3H } S3 { || 3C } .
" 5 S1 . 4 4 . " 5 S2 . S1 4 . 4 +& 5 S3 . S2 4 & S1 .4 & S2 .4 & S3 4 4 & S1 ; S2 ; S3 4 D . . G& . H A G& H " 4 4 ' &
&
&
. 4 & &
& &
. . S3 & &
. . S1 8. 4 . S3 . 4 . . . 4 S1 S3 & . - 4 . S1 S2 4 . & G8 & . . &H = 4
{
3C ||
}
1C,2H |2W| ;
{
1C,2H || 2W
}
1C,2H |1W| 1W ;
{
2C,1H || 1W
}
3H |2W| 1W {
3H || 3W
} .
" { 3C || } 1C,2H |2W| ; 1C,2H |1W| 1W ; 3H |2W| 1W
{ 3H || 3W } .
5 S3 . S1 J {
3W || 3H
}
1W |2W| 3H ;
{
1W || 2C,1H }
1W |1W| 1C,2H ;
{
2W || 1C,2H }
|2W| 1C,2H {
|| 3C
}
.
- 4 . 4 . S2 - &
? . S2 . A . " 5 . S2 .4 S1 .4 & S3 " . S2 4 . 5 8. &
& .4 . J {
3H || 3W
}
T1 ;
1C |1C| 1C
;
T2 {
3W || 3H
} .
$
? & C 1C |1C| 1C C " & .// //. I 4
& " 4 &
5 . T1 T2 8. . . & 1C || 2C 2C || 1C @ / . . . & 2C || 1C 1C || 2C " /. & 4 +& T1 8 . D 4 J {
3H || 3W
}
3H |1W| 2W ;
{
1C,2H || 2W
}
1C |2H| 2W {
1C || 2C }
.
&
& . T2 4 J {
2C || 1C }
2W |2H| 1C ;
{
2W || 1C,2H }
2W |1W| 3H {
3W || 3H
} .
+& & 4 D/ J {
3C ||
}
1C,2H |2W| ; 1C,2H |1W| 1W ; 3H |2W| 1W ;
{
3H || 3W
}
3H |1W| 2W ; 1C |2H| 2W ;
{
1C || 2C }
1C |1C| 1C ;
{
2C || 1C }
2W |2H| 1C ; 2W |1W| 3H
(
;
{
3W || 3H
}
1W |2W| 3H ; 1W |1W| 1C,2H ; |2W| 1C,2H {
|| 3C
}
.
G8 4 " & & "
H
3.3.5
A Review
4 4 D/ & .& 4 F /. " ' &
& 4 .
. 4 & / 5 4 A K L ? 4 5 . & .& . { p } S { q } .& 4 " A " 1 & . &
& 4 &
. & # ' + . . .4 J Exercise 3.1 (Five-couple Problem) " A D & ' . 3 4 2 Exercise 3.2 (Four-couple Problem) ;.& &
& 4 . & / &
" . D & '/ . &
3 " .4 & . + A & 4 & 4 . A G' H &
.& . A . . B A 4 4& . & .
% &'
)
G8 &
. &
" . . . &
. . .H 2 Exercise 3.3 4 . ' . 4 . 4 . ' . ' =J " J
. . " & 2
3.4
Rule of Sequential Composition
" { p } S { q } 4 . D/ . .& 8 !
G "& = 4 5 . . & .& . I 4 . A H A & 4 " 4 A & / p A & q . 8. S p q .
pSq
. .& & p . ' . S ' . S .4 4 .& & q + ' r d .
M &
N 4 N = 0 ,
*
G & 0 4H M = N×d + r ∧ 0 ≤ r < N .
8. S A . { N = 0 } S { M = N×d + r ∧ 0 ≤ r < N } .
. & 5 I ' . & 5 " . S1 S2 S3 S1 ; S2 ; S3 ' & A ' S1 ' S2 ' S3 " " . S1 S2 S3 5 4 8 . 4 p q r & " . S .& A
pSq & S S1 ; S2 S1 S2 .& A
pS r 1
rS q 2
.
" r . S1 . S2 8. 4 " 4 4 D/ "
3C || || 3C " 3H || 3W 3W || 3H 4 A " F 4& . . 5 " . & & & S1 I .& A S2 & & A S2 I . .& r S1 J
% &'
+ 4 8 & 4 & & 4 " 4 & 4 & " . F . I 4 4 & . 4 " A 1 2 5 . 10 " . . 4& 4 4 . 4 17 8 & A . " A 4 4 4 4 4 4 & 4 . 4 & 4 4 D& " 5 4 & & " ' 4 4 4 G8 1 4& & 4 4 ' (H ; A . 4 . . p q . .
p |5,10| q
' &
& . 5 . S1 S2 p q { 1,2,5,10 || } S1 { p,5,10 || q } { p,5,10 || q } p |5,10| q { p || q,5,10 }
{ p || q,5,10 } S2 { || 1,2,5,10 } .
- & Exercise 3.4 (The Torch Problem) . . . A t1 t2 t3 . t4 t1 ≤ t2 ≤ t3 ≤ t4 + . .
B & & .4 4 & & & .4 4 J
!
(a) " 1 1 3 3 (b) " 1 4 4 5
=J 8 A G4 t1 t2 t3 t4 1 2 5 10 &H & 4 4 =4 1 4& & & . & . B 4 . 4 . 8 & 4 .4 G? GH G H 4 & H (a) =4 & . 2 G8 H =4 & . 2 (b)
. D& 4
. 4 & &
. 4 2 - . . 4 & 2 G, & H
(c) ?4 . 4 4
- F 2 % . = . . . (d) , ; 4 &
? ' A " D & & 4 & " 4 .& 4 4 5 2 Exercise 3.5 /. G' ( H " 4 F 4& . . =4 & 4& 2 2
& (
3.5
$
Summary
8 4 /. 4 /. & # &
. / / ' . 4 4. '& & 4 & 4 & . I 4
. & 4 < . & < ' . 8 / &
& 4 .
" & A &
4& ;.& & . . 4& &
. &
& # & & 4 & 4& . 4 G 4 H / A '& . "
C . & . & /4 . 4 .4 & A 1
(
Chapter 4 Games " 4 4 4/ , & ' . & 4/ " G K LH . 4 4 " & 4 . . " & .& ' 8 .
" '
. 4 E . 4 +4 4 . &/ & .& 4 G
. .& . H 4 4 K L K# L 5 = 4 4& . 4
4.1
Matchstick Games
& 4 . "4 & # . . " 4 " & 4 & ' . 4 K8 L . & 5& & G . ' 4 & 4
& H K . L &
(
(
)
4 . 8 & 4 & &I & . . . 4 . & 4& . 4 . 4 & 4 4 & . & . . 4 4 ' . 4 1 2 " 4
. . 3 G
. 0 3 6 9 H " 4 8. m
. G m . 3 H & m mod 3 " 1 2 " 4
. . 3 " 0 . 4 & & 4
. . 3 8 4 & G & . ' H 4& 4 " .4 ' & A " . % & 4 1 - 4 2 " . % & 4 0 - 4 2
& 4 ' G 4 & 4 1 2 H2 8 4 & 4 4 4 4
N 4 N
A' 2
( " . % & 4 1 3 4 - 4 4 4 &2 ) " . % & 4 1 3 4 ' 4 1 G . & & 1 & ' 3 4 H - 4 4 4 &2 *
m mod 3
m
3
* &
(
* " 4 . . 1 2 3 - 4 4 4 &2 " 4 . I . . 1 2 3 & . 1 7 &
- 4 4 4 &2 ! " 4 . I . . 1 3 4 & . 1 2 &
- 4 4 4 &2
4.2
Winning Strategies
8 4 . 4 5 . 4 & - 4 4 4 . . I 4 4 4 & & . 4 /. .
. 4 . Æ
4.2.1
Assumptions
-
. 4 4
-
. A - 4 &
" A & / &/ . .
. A " & 4I 4 ' & . 4 & A& 4 + $% , $ 1
4
1 2 3
4
- $ 1
4%
4%
((
)
4.2.2
Labelling Positions
" A 4
+ ( . . (
0
1
2
3
4
5
6
7
8
+ ( J # , & & 4 . . %
- 4 & & 4 .
" A ( &
. + 0 8 .
4 + 1 ' & 0 + 4 & & + 4 + n 4 n 2 n−1 n−2 " .
4
. 2 4 4 = 4 4 4 & 4 A . 4 & . & . & 4 A . 4 4 . " 4 . . 4 J
. . 4 . .
A & & I . A A . 4 4 8 9 =4 4 & 4 " . . K& . 4 L 8 . G/'H 4
* &
()
" . . . . K& x & p L 4 " # " # 4 . K x L 5A 8 K L G &H A ( 0 KL . 8 . .& & 4 ?' 1 2 K4L . 0 4 4 4 ? 4 A . . 4 ?4 3 KL . 3 G 1 2 H 4 & K4L + 4 3
& . 4 4 & A . 4 & " 4 . ? 4 5 K4L 6 KL 7 8 K4L + ( 4 . 7 8 " 4 4 I 4 . 4 4 4 " 4 .
0
1
2
3
4
5
6
7
8
+ ( J 6 - & & . " 4
. . 3 " 4 I 4 & 4
4
. . 3
4.2.3
Formulating Requirements
" & 4 4 & K L &
. . 3 8
(*
)
4 ' & = 6 n
. " . 4 & ' & .4 J {
n . 3 n = 0
if 1 ≤ n → n := n−1 ;
{
2
}
2 ≤ n → n := n−2 fi
n . 3 }
n := n − (n mod 3) {
n . 3 }
" A . " A " ' 4 . 4
. . 3 /< " / & K if / fi L - / C & K 2 L &
C
. /
. b → S 4 b / ' S ' &
4 true ' & 8. true & .
8. . true ' 8 4& if / fi & / & 4 . 1 ≤ n I n := n−1 KL
& & 4 C & n := n−2 C KL & 2 ≤ n . & . n = 0 " .
K n . 3 L " A .
.
. 3
.
& 4 .
. 4
. 3 " . . 5 . 4 &I ./ & n mod 3 " A. A 4 . ' . 4 &
. 4
. 3 !
. ( (( ! ! (
* &
(
8
& . 4 n . 3 & 4 n . 3. 5& n mod 3 4 n . 3 8 4 & & & & losing & " G 4 H
.& & losing " 4 .& losing + 4 .& 4& . 4 I # . & 4 4 & .& .4 A { }
& GH ;
{ 4 }
& 4 & { }
8
& 4 & 4& . 4 & 4 4& .4 J
% + & 4
+ 4 4& & 4 & /
8. & . 4 & 8. & 4 @ / . 4 A & 4 . & 1 . 4 + & 4 & 5& G
. . H 5 G H & 4
. < 4
5& 4 4& & & . &
(!
)
-
& 4 & / 4 &
. . 1 M & "
M
A' -
& & & A M 0 " & & & . ?' M 1 " & & ?4
4 4 M 2 4 4 D 3 & 2 8. & 1
& & & 4 M 3 G31 4 H " M 0 1 2 G ' 0 1 H & A +& . . &
& 5 . 4 . M 2 Exercise 4.1 (31st December Game) "4 & & " 4 & 4 3
0& % . ' F . & 4 + 4 & 4 & 4 . 4 & & =J & . . *) & G ** . &H 4 & . & 8 & 4 & & .& & 3
H G%&H & . ' & .
& & G+ ' A & & + & 0& H
H G=H & & & . ' 2
4.3
Subtraction-Set Games
. . GAH .
I m 4 m .
.
& +& )
($
" 4 D ' . I . 1 M & {1..M} # ' & 4 + & 4 4& / - ' .& & 4 4 4 J
.
8 4 {1 , 3 , 4} &
. - .
K 0 L 4 K 1 L 4 D 4 0 4 / &/ 4 .& 4 4
. & . 4 4 . .
" " ( 4 4 < . 6 " 4 4 4 4 4 G-H G6H 8 4 4 4
. . + ' 2 & . 2 1 I 3 4 4 .
0 ? & . 4 + ' . 3 4 4 C 3 0 1 2 8 Æ D
4 . 0 "& 6 #
1 1
2 6
3 3
4 4
5 3
6 4
" ( J - G-H 6 G6H . {1 , 3 , 4}
)
) 7 "& 6 #
8 1
9 6
10 3
11 4
12 3
13 4
" ( J - G-H 6 G6H . {1 , 3 , 4} 4 ' J ( ( ( 4 . 4 . > 4& 4 . - & . . {1 , 3 , 4} 4 4 & r & .
. & 7 8. r 0 2 > 4 4 " 4 & 1 . r 1 3 . r 3 5 4 . r 4 6 " . 4 ' & . / 4 5 . 4& . & 4 4 G . 4 4 H " .4 4 & A 6 M " . M + k W.k true . k 4 false 4 - k M W.k & & 5 W.(k−1) W.(k−2) . . . W.(k−M) 5 s.k ?4 & 2M F 5 . . M 5 5 s.(M+1) s.(M+2) s.(M+3) . . . & 4 2M " . j k 4 M ≤ j < k < M+2M 4 s.j = s.k 8 .4 W.j = W.k 5 W . k 4 + ' & -/6 4 .
20 4 8 . ,& 4 & . 4/ 4 2M+M " 4 & 4 . k . . k s.k 8. R . & k W.k 5 W.(k mod R) Exercise 4.2 . 8 2 5 6
& G" {2 , 5 , 6} H
& +& )
)
(a) + n 0 ≤ n < 22 4 . n 4 (b) 8.& 4 .& & . . . n 4 . n
4
@.& & 4 4 . 1 4 2
+ ( 4 . 8 Exercise 4.3 D " 4 & . .4 " 5 1 A 5 25 I 4 A A . . . . 4 . I . . .
.
21
22
23
24
25
20
19
18
17
16
11
12
13
14
15
10
9
8
7
6
1
2
3
4
5
+ (J 6 & . .4
)
)
(a) 6 G" 5 " .& 5 & H (b) 8.& 4 ; . 4 & 4 4 .
4 (c) . A 4 4& %' 4 & 2
4.4
Sums of Games
8 4 4 ' . 4 & F & " ' . ( .
- . A I
& . 4 & F . "
. 4 I 4& .
8 4 4 4 . . .4 + & 4 4
. . . + (( ' . . 4 % 4 & & 8 . " .
& & 4 . 8 K L . 4 . 4 . " K L . & :' 4 K:L . K'L I F . ' & . K:L K'L . A (( I 5& / . ( 4 4 =4 . A (( 15 F
& % )
)
O
M
N
J
K
F
L
G
H
D
A
k
h
I
B
f
C
j
g
c
E
i
d
a
e
b
+ ((J , " . & 4 :' 4 K:L . . K'L . ' & . : ' 11 I . 4 15×11 F + . . /. & 8 4 & 4 4 & . . 4 - A F G . .
H . F 5 4 . - A 4 Æ 4 D 4 & . 3 . . &
4.4.1
Symmetry
A Simple Sum Game
- 4 & ' . . 4 4 . 4 & .
)(
)
. > 4
.
& 4& 4 & " K L . 4 . & & J GH . . 8 4 & 4 4 & " 4 & 8 5 & 4 4 & . Æ 4 . 8. & .
C 4 & . C 4 & " &
& 4 . 4 & 4 & 4 m n
. 4 8 5
. & 0 " m = n = 0 " & m = n " + 4 m = n G 1 ≤ m 1 ≤ n H & 4 4 m = n 5& 4
. ' . 4 & m = n + & . 4 & ' & .4 / 5 . { if
m = n ∧ (m = 0 ∨ n = 0)
}
1 ≤ m → m
2 1 ≤ n → n fi ;
{ if
m = n
}
m < n → n := n − (n−m)
2 n < m → m := m − (m−n) fi {
m=n
}
" / 4 m 1 ≤ m n 1 ≤ n & . " . m n m = n . .
& % )
))
" & m = n . 4 & %5& m < n n < m 8 m < n 4 . 1 ≤ n−m ≤ n n−m . 4 n n−(n−m) A m . n := n−(n−m) & m = n 4 " n < m &
" .4 5 .
/
D . & . 4 & ? 4 4 4 " '/ & . I losing & & 5 {
m = n
{
m
if
} }
m < n → { 1 ≤ n−m ≤ n } n := n − (n−m) { m = n }
2 n < m → { 1 ≤ m−n ≤ m } m := m − (m−n) { m = n } fi {
4.4.2
m=n
}
Keep It Symmetrical!
" (( ' . . &
&I 4 & 4& . . &
& 4 4 . / - & 4 4 / 4 " & ' . 4 &
& & 4 = " . . The Daisy Problem & 16 &
& " 4 & 4 D " 4 4 - 4 4 4 &2 , & & n . 4 1 M D G4 M A' . H The Coin Problem "4 & 4 &
" & & . . & " & &
)*
)
+ ()J 16 / & & " 4 4 - 4 4 4 &2 G! H - . & & D.& & 42
4.4.3
More Simple Sums
6 ((
. K . G4 K A' H " F . 4 4 8. . m n
. 4 4 m−n
4 K < m−n 5& & m = n + ' . K A' 1 4 4
4 J & . 4 I 4 A F . & . &
& - . 4 &
& . & F 4 J . ' 4 M
. & . . N
.
& . 4 M = N & . & F . ' . & 8. 4 & /
&2 ? . . K&
&L & 4 &J &
&
& % )
)
&9 - 4 ( 4& 4 / 4 M & & .
. & M+1 0 " . . m
m mod (M+1) 4 4 " 4/ K&
&L 4 . & m mod (M+1) = n mod (N+1) .
G M '
. . . N '
. H " 8 4 0
& A & 4& .4 & & & .4 { if
m mod (M+1) = n mod (N+1) ∧ (m = 0 ∨ n = 0) } 1 ≤ m → m & M
2 1 ≤ n → n & N fi ;
{
m mod (M+1) = n mod (N+1) }
if m mod (M+1) < n mod (N+1) → n := n − (n mod (N+1) − m mod (M+1)) 2 n mod (N+1) < m mod (M+1) → m := m − (m mod (M+1) − n mod (N+1)) fi {
m mod (M+1) = n mod (N+1) }
G?J 4 . . 4 H
4.4.4
The MEX Function
" . A K&
L K 5L & . 4 G l,r H 4 l . r
F D I & GH
)!
)
l := l G. l H . GH r := r G. r H " A 4 . L R & . / & 4& G l,r H ' & 4 L.l = R.r " 5 J 4 . .&2 8 4 4 4 .& . L R 2 " & . 4 & 4 A + G l,r H . ' & 4 l . . r . L R 5 & 4 . C G l,r H .& L.l = R.r C 4 C / G l,r H .& L.l = R.r C " { if
L.l = R.r ∧ (l ∨ r ) } l → l
2 r → r fi {
L.l = R.r }
.
" & 4 & . 4 C G l,r H .&/ L.l = R.r C C G l,r H .& L.l = R.r C " {
L.l = R.r }
& 4 & {
L.l = R.r }
.
- .& A 5 . 4 A L R . 4 .
5 J
+ l r . L.l = 0 = R.r + & l . l l . L.l = L.l
& . & r . r r R.r = R.r
? .
. . . 0 . 1 5 & " . . 5 8. L.l R.r F
& % )
)$
L.l < R.r R.r < L.l " 4 A . & 4 & & 4 L.l < R.r . 4 R.r < L.l G 4H {
L.l = R.r }
if
L.l < R.r → r
2 R.r < L.l → l fi {
L.l = R.r }
.
+ 4 4 5 J
+ &
m R.r . r r
R.r = m & . &
n L.l . l l L.l = n
" 5 A . 4 A . L R / K 'L . " A . . .4 6 p G " mex . p mexG.p A
n
" G . p q .& mexG.q = n
+ &
m n G . p q .& mexG.q = m
K#'L . K ' L . . . '
. p
' . '
. q 4 . p
4.4.5
Using the MEX Function
- A (( . '
+ (* 4 '
. . " & & I 5& & 4& '
& /. . " &
& " '
0 5& '
4 & '
G . p q . p q H "
*
)
2
1
1
2
0
0
0
0
0
1
0
3
1
2
0
0
0
0
0
1
1
2
1
0
1
0
+ (*J #' ?
" '
.
'
.
& A
'
. + ( 4 & " . A '
4 '
8 4 '
2 .
4 '
0 1 ?4 4 & 6 K>L " 4 '
. K>L F . '
. KL " G 3 2 H 4 & KL 4 '
K>L " . 4 '
F . 2 " A & & . '
5 & . ? . . . 4 15 . . 11 . '
. 8 . 4
& % )
*
0
+ ( J
1
4
5
'
" '
2
26 F D G . 4H . 15 11 165 " F # 4 < . Exercise 4.4 H / 4 . . 4 2 5 6 & '
. &
H 4 . 4 8 . 1 2
& 8 2 5 6 & 8 & & . & " 4 4
. F & .
J
. .
.
6. , , KL 4 2 2 ) ) 2 * $ 2 ( 2 " (J + K2L + 4 4 + 4/ 4 . : m 4 K:L . K6L G. K. LH KL G. H m
. /
2
*
4.5
)
Summary
" 4 4/ " & . # - 4 4 A . A . /. 4 4 /. & . " & . K L . 4 ' A 4& ' . 4 A & . . K&
&L
4 . 4 4 A . K 'L . " . ' . & F 4 K L . 4 /. , & 4/' . # 4 4 & 8 & & / > . .4 & . & . 4 . .4 & . ' ' . K / '&LI & . & . 4 Æ 4 & 4 #'
4 & ,& K? L '
K/,&L
. / ? 4/4 . - & Æ & 4 4 ? . 4 '
G- 4
'
. . 4 H
4.6
Bibliographic Remarks
" 4/ K- -&L M ,! N & 4& ,&
. & " 3
G' ( H . M3- N
Chapter 5 Knights and Knaves " . A . 1 & & " 4 & . K L 4 4& KL 4 4& 6 << . . & 4 4 4 & " & & C n 2n F & C & & 4& . 8 << & ' . 4 4
5.1
Logic Puzzles
= & . // << 8 B . 4 " J K" 8 L " (a)
4 2
(b)
4 2
& 4 . B . 4 - & 4 2 " & K - .
. 2
*
&L
*(
, , ( C & K 4 L - 5 & 4 2 ) 3 5 4 & 4 * - 5 & 4 2 - 5 & 4 & G H2 ! B 4 4
. B & &P 5 & . - 5 & 2 $ . 4 1 . . + &P 5 4 4 & . . . 4 4 4
5.2 5.2.1
Calculational Logic Propositions
" 4 4 ' 4
- . ' 4 ' m2−n2 4 . (m+n)×(m−n) & . . m n - & m2−n2 (m+n)×(m−n) " 4 m2−n2 = (m+n)×(m−n) .
" . . 64 & & 5 4 ' " & K L &
4 4 & KL & & . . & ' -
"4 ' . ' < J n+0 = n ,
n−n = 0 ,
-
*)
. 4 4 . n - & & K. n L " 4 . 4
& + ' K & . L 5&J (m+n)+p = m+(n+p) ,
4 . m n p G H 4 ' 4 / K L C true false %' . ' K &L G4 true false 4 4 KL .H n = 0 G4 true false . n H n < n+1 G4 true .
n H / ' $
4 " ' / 4 . ' m < n < p 4 4 / %
. m < n n
C KL KL K.L
" . ' . 4 G. . . . . . . H %5& . C . A & ,. - 6 < 4 . *(* * 4 4 A & . C 5& . ' - & 5& . & . 4 & & 4 . 4
5.2.2
Knights and Knaves
%5& . << 4& 4& 8. . K L true false & K .L . A K L S. " . 4 " A=S .
**
, ,
+ ' . & K .L A=L ,
4 L . K .L 8 4 . . ; . & K8 L 4 A=A .
" 1 & 9 1 A 4 4 ' 4 & 8. &P 5 Q 5 . A = Q " 4 K&L . 4 & & 4 & > 4 4 KL + ' 5 K & L 4 4 K&L A = A 5 K 2L 4 K&L . & & G A = B H 4 KL " 1 K&L KL .& . A = B 5 . 5& & . << . A ' . 4 4 1 A = B . 1 B = A 5& &
I . 4 4 4& ? & & . F . A B " . 5& . '
. 4
5.2.3
Boolean Equality
%5& C & . C
. + & " x = x 4 G &H . x " x = y y = x " " . x = y y = z x = z +& . x = y f & . f.x = f.y G4 A' . H " " " ' %5& & - & E'& &
& & 4 . %5& . 8 . 4 true false - 4 & .
-
*
. 4 . & &
& + ' J . x y z x + (y + z) = (x + y) + z
x × (y × z) = (x × y) × z .
" & &
J . x y x+y = y+x
x×y = y×x .
&
& . 5& 4 . D &
& . 5& 4 4 & . 5&2 8 5& 2 " 4 5 1 & . & . & . . 4 . " ' (p = q) = r D 1 4 p q r
5 " ' 5& . - p q r
p = q 4 r . 5& " (p = q) = r . & p = (q = r) 8 4 . 5& 8 4 4 5& . C & " . p q r G) H
[Associativity]
((p = q) = r) = (p = (q = r)) .
B & & . (p = q) = r . p = (q = r) B . 4 (p = q) = r true . 4
. p q r true 8. 4 . false (p = q) = r false " & . 5& & 4. & . & . ' - 4 ' I & ' .4 " E'& . 5& ' & (p = p) = true .
*!
, ,
" . p 4 & G
H . p 4 & & . 5& J p = (p = true) .
"
& .& ' & K true L . ' . . p = true - 4
5.2.4
Hidden Treasures
- 4 . 6 A ) - 4 . & K8 5 L2 6 A . K L G . K L " 1 A = G 4 A = (A = G)
true =
{
1
}
A = (A = G) =
{
5& .
}
(A = A) = G =
{
(A = A) = true
. 5 . 5
}
true = G =
{
5& &
}
G = true =
{
G = (G = true)
}
G .
- 4 4 . & 4 4 . & .4 . . B 4 .
-
*$
5 & 4 K&L . . . .4 KL . . .4 4 4 6 Q 5 " 4 4 5 4 A = Q 6 L K . & .4 . .L " 5 L 5 " 4 5 L = (A = Q) L = (A = Q) =
{
5&
}
(L = A) = Q .
5 Q L = A " 5 K8 . Q . & .4 . .1 5 . Q& 1 L ? & & . 4 L 8 & 4 4 4 & 4 8 . 5 4 P . 5 P = A 8 .
' P 5 & A
5.2.5
Equals for Equals
%5& . & 6 <1 J . 4 '/ 5 ' . = 4 ' . . 6 <1 . & K &L + 5 4 4 " 4 A B C G & H - Q 4 5 " 4 4 C & & ) ( Q = (A = C) 1 A = B 4 4 C = (A = B) 5 . 5 Q = (A = (A = B)) A = (A = B) A B 5 K 2L = . Q 4 4 Q =
{
. . 5
}
, , A=C =
{
.
1 C = (A = B)
. 5 . 5
}
A = (A = B) =
{
& . 5&
}
(A = A) = B =
{
(A = A) = true
}
(true = B) = B
}
true = B =
{ B .
5.3
Equivalence and Continued Equalities
. & & A' " A
8. & ⊕ G (x⊕y)⊕z = x⊕(y⊕z) . x y z H 4 4 x⊕y⊕z 4 . .
& " '
. . # & ' I 4 & .& x⊕y y⊕z 4 8. &
G x⊕y = y⊕x . x y H .
' 4 .& u⊕w . & . ' u w 8A' . . & - 4 . ' 0 ≤ m ≤ n = %
J 0 ≤ m m ≤ n 8 4& . G m 4 4 H # / & 4 . 0 ≤ n " & & . / 8. 4 m n m < n m ≤ n 4 4 0 ≤ m < n 4 & . 0 < n = . ' 4 . . . 8 . 5& . 4
3 4 / $ + % $ × % )
!' !'
5& ' . .
x=y=z
5& . . x y z 2 > 4 K &L (x = y) = z ,
5& x = (y = z) ,
D 4& 4 4 x+y+z 2 " 4 .& G. ' true = false = false false A true H " D 4 . . 8 4 & . x = y = z & 4& x = y y = z I 4 . 5 . ' & 5& &
4 & . ' D G. &H & C. C 4
5 & F "
4 F &
5& .
C &
K = L 4 & . 5& &
K ≡ L 4 & ' & 4 4 p = q p ≡ q - p q ' ' p≡q≡r ,
4 ' & K ≡ L &
C (p ≡ q) ≡ r p ≡ (q ≡ r) 4 C 4 ' p=q=r
%
C p = q q = r C # & " . .
p1 = p2 = . . . = pn
. p1 p2 . . . pn 5 4 " . .
p1 ≡ p2 ≡ . . . ≡ pn
, ,
& .& ' G & 4& 4 F H ' & # 4
K ≡ L &
K5LI . 4 4 & 4
. 4 5& " & & 5 A ' E'& G) H
[Reflexivity]
5.3.1
true ≡ p ≡ p .
Examples of the Associativity of Equivalence
" . . ' F . & . 5 Even and Odd Numbers " A ' .4 & . even
G
' & 4 . 4H m+n ≡ m ≡ n .
8 4 . 4 . 4
.
" . 4 m+n ≡ (m ≡ n ) ,
m+n ' & 4 . m n (m+n ≡ m ) ≡ n ,
.
n
m & . m ' & 4 n 4& . . 5/ p ≡ q ≡ r ' & 4
. p q r & . F J
GG m+n GG m+n GG m+n GG m+n
H H H H
Gm Gm Gm Gm
H H H H
Gn Gn Gn Gn
HH HH HH HH
!' !'
" & . ' . & " .
. 4 K m L K n L ; & . 5 . K m+n L ' . 4 & . ' . F . & F Sign of Non-Zero Numbers " .
& 4
+ /<
x y x×y . . x y 5 8. . x y F x×y x y /< ' x×y ≡ x ≡ y .
0 . &
. F & 8 DA . x×y ≡ (x ≡ y ) .
" C4 . x 4
& y ' & 4 y C K. .L . & . 5&
5.3.2
On Natural Language
#& 4 5& . I . 4 . 4 F . . . 5& # 5& K. & .L 8 5& .
& A / G ≤ H / G ≥ H A K/ /L " ' . & 4 . . KL KL KL & . 5 GK8. 8 4 &
LH " 5& &
4 A & )) 4 & . 5 I 4 5& KL 4 ? 5 " ' 5 K 4 & 5 & 5 &L & & . & 9
(
, ,
" . . 5 G & & H 4 & . " 4 & + '
4& ' 4J K5 L K L 5 G 4 4 . H E 4& & . ' $J() J && 4 1 A & J 9 B 4 4 & 4 4 4 F 4 $() J 8 . 4 . & . . 4 . 4 4& 4& " . . " KL F
5.4
Negation
.4 // " 4 ? & K 5 8 L - & 2 " / J . KL ? & G . 4 ' & H & &
K ¬ L 4 A' 8. p ' K ¬p L K p L ; . S 4 4 A ≡ S 4 . J A ≡ B ≡ ¬A .
G- 4 . K = L K ≡ L ' &H " .& ' 8 & & . . + & p 4 ¬p J G)H
[Negation]
¬p ≡ p ≡ false .
¬p = (p ≡ false) ,
.
)
. A . 4&J (¬p ≡ p) = false
4& . .& ' 8 &
& . 5 4 5 & 4 4 &J p = (¬p ≡ false) .
// 4 J A ≡ B ≡ ¬A .
" A ¬B .4J A ≡ B ≡ ¬A =
{
}
¬A ≡ A ≡ B =
{
4 G)H 4 p := A
}
false ≡ B =
{
4 G)H 4 p := B
}
¬B .
? 4 G)H 4 F 4& " 4 G)H D 4 &
& & . 5 4& . .& 5 4 P . ' 4 4 .& ¬p ≡ p ≡ q ≡ ¬p ≡ r ≡ ¬q .
- & . K p L K q L " 4 ¬p ≡ ¬p ≡ p ≡ q ≡ ¬q ≡ r .
?4 4 G) H G)H
. . K p L K q L G & H 8 ' 4 true ≡ p ≡ false ≡ r .
*
, ,
+& 4 G) H G)H " . A ¬p ≡ r .
0 . 4 A . ' 4 4 & " ' p + (−p) + q + (−p) + r + q + (−q) + r + p
A q + 2r
& . p q r . −p . p F & " 4 4 G) H G)H A 4& 4 5 I 4 4 4 4 ' . 4 4 4
. ¬false = true J ¬false =
{
4 ¬p ≡ p ≡ false 4 p := false
}
false ≡ false =
{
4 true ≡ p ≡ p 4 p := false
}
true .
5.5
Contraposition
4 4 & . 4
G)(H
[Contraposition]
p ≡ q ≡ ¬p ≡ ¬q .
" . . . (p ≡ q) = (¬p ≡ ¬q) - . & / G H . . . 8. 4 n
. l 0 $ % (
K . . L .
& J n , l := n+1 , ¬l .
8 4
. & " . even.n ≡ l
" (even.n ≡ l)[n , l := n+1 , ¬l] =
{
.
}
even.(n+1) ≡ ¬l =
{
even.(n+1) ≡ ¬(even.n)
}
¬(even.n) ≡ ¬l =
{
}
even.n ≡ l .
- & . <
4 even.n ≡ l & true 8 4 . 5
. ' .4 5 5 4& & 5 4 4 G A ) H =4 & & & 90◦ . . 8 2 8. 42 8. 4 & 2
+ ) J # & " 4 & / ' E & 4 4 5 4 . 5 < G + ) H & 5 " 5 . / 5 / 4 5
!
, ,
?4 col . 5 G& true . false . 4 H dir . G& true . / false . /4H " & & J col , dir := ¬col , ¬dir
" . . col ≡ dir .
. ' 4 5 4 & 4& 5 " . col ≡ dir & & 8 4 . 90◦ 5 ≡ . / 4 . 4 4 5 . /
90◦
+ ) J 8 4 &
Exercise 5.5 (Knight’s Move) 8 . 1 4 4 . 4 . 4 8×8 . 5
/ 0
$
4 . /. . /
/ 4& & 5 ' & =J =4 & 2 # . F
. . 5 4 I .& 4 4 2
5.6
Handshake Problems
6 . . / " ' . J & 1
. & 4 4 4 4 . " I & 8 K & L . & 4 C0 0 &C 0 4 0 . G8 & / . & . . 4 H 8 K&
L . & 4 C0 0 &C 0 4 0 5 0 4 0 +& K/E'L / 4 - 5 4 G H 4
. 6 ' 5 . . / 4
. n " & 4 n =4 /E'& & 4 - & 4 4 0 n−1 " n
0 n−1 " . K4
. L K&
. L 8 0 n−1 " &
& . / 4
K L S 4 x y . 8 4& xSy K x 4 y L D x 4 y " &
& . K L . x y xSy ≡ ySx .
!
, ,
" . . x y ¬(xSy) ≡ ¬(ySx) .
8 4 x 1 4 y 5 y 1 4 x ?4 a 4 b 4 & " a 4 b ¬(aSb) b 4 a bSa 5 . 5 4 ¬(aSb) aSb 4 . " & 4
. 4 4
. ? .& 4 &
& /E'& . / - 4 F F + ' . 4 K L & . . K 4 L 4 &
& G 4 4 H =4 . K L & K L & 6 K L K L &
/E' Exercise 5.6 = 81 Æ
. J K L &
K1 L
. G 4.H & = 4. > K L & 4 & & ./ . 4 & =4 & 1 2 2
5.7
Inequivalence
8 // . )( K F . &.L " . B = A ¬(B = A) " . & K 5 8 L .4 4 ? 4 4 . K = L K ≡ L ' & / 1 2 " 2
1 '
!
¬(B ≡ A) =
{
4 ¬p ≡ p ≡ false 4 p := (B ≡ A)
}
B ≡ A ≡ false =
{
4 ¬p ≡ p ≡ false 4 p := A
}
B ≡ ¬A .
- . p q G) H
[Inequivalence]
¬(p ≡ q) ≡ p ≡ ¬q .
? 4 & . 5 & ? 4 & . 5
& . 4 . . " A &J ¬(p ≡ q) = (p ≡ ¬q) ,
. 4 &J (¬(p ≡ q) ≡ p) = ¬q .
" ¬(p ≡ q) & 4 p ≡ q " " G
H 85 J (p ≡ q) ≡ r =
{
A . K ≡ L 4
}
¬(¬(p ≡ q) ≡ r) =
{
G) H 4 p,q := ¬(p ≡ q) , r
}
¬(p ≡ q) ≡ ¬r =
{
G)(H 4 p,q := p ≡ q , r
}
p ≡ q ≡ r =
{
G)(H 4 p,q := p , q ≡ r
}
¬p ≡ ¬(q ≡ r) =
{
G) H 4 p,q := p , q ≡ r
}
¬(p ≡ ¬(q ≡ r)) =
{
A . K ≡ L 4
}
p ≡ (q ≡ r) .
!
, ,
4 4 " p ≡ q ≡ r 4 . .
/ & ? & 4 4 p ≡ q ≡ r p ≡ q ≡ r 5 A 4 ' 4 4 5 4 5 J (p ≡ q) ≡ r =
{
' A . p ≡ q
}
¬(p ≡ q) ≡ r =
{
¬(p ≡ q) ≡ p ≡ ¬q
}
p ≡ ¬q ≡ r =
{
&
& . 5 4 G) H . ¬(p ≡ q) ≡ ¬q ≡ p 4 p,q := q,r
}
p ≡ ¬(q ≡ r) =
{
A . q ≡ r
}
p ≡ (q ≡ r) . Exercise 5.8 .& .4 G? 4 & ' P 1 & F H (a) false ≡ false ≡ false (b) true ≡ true ≡ true ≡ true (c) false ≡ true ≡ false ≡ true (d) p ≡ p ≡ ¬p ≡ p ≡ ¬p (e) p ≡ q ≡ q ≡ p (f) p ≡ q ≡ r ≡ p (g) p ≡ p ≡ ¬p ≡ p ≡ ¬p (h) p ≡ p ≡ ¬p ≡ p ≡ ¬p ≡ ¬p
p = q = r
!"
2 & (
!
2 Exercise 5.9
¬true = false
2 Exercise 5.10 (Double Negation)
.
¬¬p = p . 2 Exercise 5.11 (Encryption)
" . 5
(p ≡ (q ≡ r)) ≡ ((p ≡ q) ≡ r) ,
& " & b . & a & . . b a ≡ b " &
c " & a a ≡ c 4 . b & & 4& b & . & a 2
> . & 4 Exercise 5.12 - 5 & 4 F &2 2
5.8
Summary
8 4 << 5 C 5& . C . %5 & . . 5& %' . & . 5 F & . << $ (% 3 ( " !" "
1
" 4
!(
, ,
" & . 5 Æ & . ' =4 ' . " . / . <
&
K 0 L KL . D F " . KL KL 4. " . 5& . 4 $ 1 & . " 3 4 & 0 64 < G K> . L M")*NI " . H ? . 4
. & %- 3D 4
G M3$ NH " . << & &1 K- 8 " ? >. " 2L M !N " 4 . << . ' ,R1 & &1 . & & " ' . & . 5 . << , - M-! N + . 4 . D GKLH D GKLH .4/. GK.LH GK& .LH M N M,$N
Chapter 6 Induction K8 L / 5 . . " 4 4 K<L . . + ' . 4
. I . . <
. 5 < / 4
C 0 1 2 3 - . / 4
;& 4 < . . 5 . = 4 < 4 4 + 4 . < 0 " . & & & G" & . KLH 4 4 4 . &
n . < n+1 . < n "
& 4 . < 0 - 4 4 . < 1 I 4 & . < 1 .
4 . < 0 " 4 4 4 . < 2 I 4 & . < 2 .
4 . < 1 - 4 . < 3 . < 4
6.1
Example Problems
.4 & 8 A <
. 8 ' & & n Warning
#
0
0
!)
!*
/
.
. 8 & B - Cutting the Plane
. 4 . ' 4& . . A * 8 4&
. 4
4 4 D G 4 . & F H
+ * J -
Triominoes 5 . . < 2n×2n 4 n
" 5 " > 5 . 6/ . 5 + * 4 . 8×8 4 5 > 4 5 4 G/H G+ * 4 H ?J " n = 0 & Trapeziums 5 4 . 2n .
n
. 5 " 5 / < . 5 A *( *
0 1 2
/ !"
!
+ * J "
+ *J " A * 4 4 G/H < A *) . n 2 ? 8 n = 0 & ( Towers of Hanoi " "4 . = . << !! & + S
% 6 & # " << 4 4 A' > A . . 41 , '& . . F < . < G A **H " 4 . & . & " 1 4
4 & . A .
!!
/
+ *(J & . %5 "
+ *)J A *( & & 4 4 9 " 4 G- &
=4 . / H
/
!$
+ **J "4 . =
6.2
Cutting The Plane
. J
. 4 . ' 4& . . A * 8 4&
. 4 4 4 D G 4 . & F H +
. . K<L . " K &
. L " 4 4 4 4 < C K L . C 4 4 4
4 n+1 4 n &
4 4 n C C + & 4 . 4 & 4 D
$
/
" 4 < & " . 4 . G . D H + 4
. 4 . F 4 4 D "
- 4 4 " 4 . ' 4I . 4 ' . & + * ' " 4 & A 4 5 4 . & " F . . . 4
. & . > . %4 D F " 4 4 .& . &
+ * J
4
" & . & . & / G 4 /H . & ?4 . 4 6 G& . 4 & . . 8 D &
4& . H ? . & . . . 8 . & . " . & . . G . . & / . & H 8 . & . G
/
$
1 4 . & &H . D & . . . & & . F + *! 4 F ' . .
+ *!J " . G 4 4 H " 8 & . 4 & 4 & 4 . " / &/ % ' A " / 4& " . . G 4 H A " . G4 A 4 ' H A +& 4 K.L 4 K L A " A
& . F 4& 1 " A K. &L 5 A Exercise 6.1 & & - & 5 2 =4 ' 4 " 4 . 8 . 4 . . . . 2 - . / =4
& F 2 2
$
6.3
/
Triominoes
' . * . 5 . . < 2n×2n 4 n
" 5 " > 5 . 6/
. 5 4 5 4 G/H " . K<L . .
n - & n. " 4 n 5 0 " < 20×20 1×1 " ' & 5 " 5 & 5 8 0 59 " 4 ?4 4 . < 2n+1×2n+1 -
& . < 2n×2n 4 . A & 5 - 4 4 ' & . < 2n+1×2n+1 . 4 5 . < 2n+1×2n+1 4 . < 2n×2n &
& 4 < . 6 .
> 5 & " 5 4 . . / - & /. G8. H " /. . < 2n×2n . 4 5 & & 5 /.
4 " 4 . / /. / 4 ? . 5 & - & & . D 5 . " & D . 4 A *$ ?4 & 5 . / /. / 4 > . 2n+1×2n+1 4 Exercise 6.2
< *
2
/ - #
$
+ *$J " 8 " . / " 5 4 A / 4 D . " & . / 4
6.4
Looking For Patterns
8 * * 4 4 . K<L " & 4 K LI K L & K8 L . ' ' . . 4 & 4 . . . . ' . 34 . & . & . . . 4 8
64 . & . & 4 4 & & 4 4 8 ' 4 4 & I & & &
& . . 8
. . 4 . ' 4 4 &
4 4 & 8 4 . ' 4 . 4 & & . . ' 4 (
. 81 / . . . . %
4 D . ' 4 ,//.& . 4& .
G, . . D I A . 4
$(
/
H . ( ' . / . ' ( J . . 4 4 4 %' 4 4 0 3 6 4 1 2 4 5 7 8 4 "
J 4
.
. 3 4 " D . D =4 4 .& D & 4 & 8 4 K<L . . &
. &
. & 3 4
K<L . . 0 1 0 K<L . . 3 4 5 1 " & . 3n . 3n + 1 3n + 2 4 " . 4 n 5 0 . 0 & A 4
. 1 2 4 & ?4 . 4 . 3n . 3n + 1 3n + 2 4 - 4 . 3(n+1) . 3(n+1) + 1 3(n+1) + 2
4 3(n+1) " & 4 1 2 3(n+1) − 1 3(n+1) − 2 " . 4 3n + 2 3n + 1 & & 4 = 4 3(n+1) ?4 3(n+1) + 1 3(n+1) + 2 & 1 A 2 & 4 3(n+1) " 4 4 4 = 4 3(n+1) + 1 3(n+1) + 2 4 " . 4 A D ' & 4
.
. 3
/ . # %
6.5
$)
The Need For Proof
- & D & A 8 & ' . .4 #& D .I & & D . . 4 . & " / ' . . D n . . D " 4& . . " 5 4 & 2 + * 4 4 n 1 2 3 4
+ * J
"
. & 1 2 4 8 "
. . & n 2n−1 8 D & n = 5 G- 4 AH 8
. 16 4 25−1 =4 . n = 6
. 31 9 G A * H ? n = 6 A 4 4
5 = 4 & n = 0 D 4 & C 1 & 20−1 / 9 " & 5 4& 5 n F . 0 " . . .
. n 5 0 9
6.6
From Verification to Construction
8 ' . 4 . @A D 4 C . &
$*
/
+ * J
" n = 6 "
. 31 26−1
4 . A . 1 D 4 A . 8 A / .
. " .
. . " 4 4 . . . k 4 . A n
4/4 . .
. 1 n : 1+2+ ... +n =
1 n(n+1) . 2
"4 ' . ' .& 12 + 22 + . . . + n2 =
1 n(n+1)(2n + 1) 6
13 + 23 + . . . + n3 =
1 2 n (n+1)2 . 4
4 ' . . . ' 4 . A J 5 4 . 4 4 4 . 4 & 49 . ' & 4 . . . 4 4 . A n
14 + 24 + . . . + n4 = ? .
// # 3 4
$
=4 4 & 2 @A . 4 & 4 . 5 & 4 4 2 & 4 4
& k27 2 & 9 / " . 4 & // & 4 & / & 8 . .
" . .& " & & 31 /
6 4 . & . m 5 1 2 3 . m 4 . A n
& n . m+1 G" . A n
5 . . n . A n 5 . . n . A n 5 . . n H " A G./.H m 0 J 10 + 20 + . . . + n0 = n .
& . . . & . . 4 A. & n
Æ " 5
& 4 4 . . 1 + 2 + . . . + n G 4 & 4 4& . . 8. 4 " H - D 5 . & n & a + bn + cn2 Æ a b c = 4 + & S.n 1+2+ ... +n .
- P.n S.n = a + bn + cn2 .
" P.0 =
{
A . P
}
$!
/ S.0 = a + b×0 + c×02 =
S.0 = 0 G . & .
{
<H
}
0=a .
. 4 a Æ . n0 0 ?4 4 b c " 4 & 0 ≤ n P.n " P.(n+1) =
A . P a = 0
{
}
S.(n+1) = b(n+1) + c(n+1)2 =
. . &
{
S.(n+1) = S.n + n + 1
}
S.n + n + 1 = b(n+1) + c(n+1)2 =
J P.n a = 0
{
" S.n = bn + cn2
}
bn + cn2 + n + 1 = b(n+1) + c(n+1)2 =
{
}
cn2 + (b+1)n + 1 = cn2 + (b + 2c)n + b + c ⇐
Æ . 4 . n
{
}
c = c ∧ b+1 = b + 2c ∧ 1 = b + c =
{ 1 2
=c ∧
1 2
}
=b .
+ D . A n
5 n 4 1+2+ ... +n =
1 1 n + n2 . 2 2
%' .
' 1m + 2m + . . . + nm & . . &
m " J
& n 4 m+1 ; . 4 . S.0 0 S.(n+1) S.n + (n+1)m G4 S.n 1m + 2m + . . . + nm H
/1 #+ 5
$$
& . 5 Æ +& & . 5 ) J 8 . 1 + 2 + . . . + n 4& . & 4 4 5
1
T
T
T
n
n−1 T
T
1
T
n+1
2
& 4 n
T
" 4 4 J n+1 T
n+1 T
+ . n . n+1 4 1 n(n+1) =4 . 1m + 2m + . . . + nm 2 . m 1 * Exercise 6.3
; 5 D . .
10 + 20 + . . . + n0 12 + 22 + . . . + n2 . 2 Exercise 6.4 4 . . 4 m n & .4 ' G. ' m 1 n & m 2 n 3 H . . 4 4 4 4 4 & "& .& 4& 4 8 . 2
6.7
Fake-Coin Detection
" . **
K31 9
L -
/
4
. . < - K.L KL KL 4 4 K.L F 4 KL " 4 . & A . . ? . I 4 1 & 4 4
& K &L " >. .
& .& . > . . . KL K L
6.7.1
Problem Formulation
- 4 . 4 4 C 4
C 3 J & . & &
& & " 4 n 3n F " 4 . ?4 4 m . 4 . " 1 + 2m F 4 . J K 1 L & I 4 K 2 L 4& . K m L & . G & H " 4 n
. 4 . m 4 1 + 2m = 3n # & .
m . (3n−1)/2 . . 4 n - 4 4 & - D (3n−1)/2 . 4 . .& . G 4 H n + n 5 0 D & I . 4 + n 5 1 4 4 " G (31−1)/2 = 1 H 4 4 4 . . 4 2 > D
4 - .& D & 4 4 " 4 (3n−1)/2 4 4 4 ' . 4 5
/1 #+ 5
4 "
4 .& . . ' n
6.7.2
Problem Solution
> . . .
. n The Basis - < 4
& / . (30−1)/2 " n 5 0 Induction Step ?4 4 n < +
& c.n (3n−1)/2 & 4 & . . ' . c.n n - 4 4 A . . ' c.(n+1) n+1 A 8
. . " 4 & .
. 4 5 > 5 . . 4 . . . " 4 & A . .
4 & c.n . " c.n '
. 4 . . 4 n 8 4 & C F
4 c.(n+1) c.n ?4 c.(n+1) = (3n+1−1)/2 = 3×((3n−1)/2) + 1 = 3 × c.n + 1 .
c.(n+1) − c.n = 2 × c.n + 1 = 3n .
"
I & . 4 4 G 4 4 c.(n+1) 4 4 H - A c.n + 1 . 4 " ' 4 . A " . 4 4 & 8.
/
. c.n . " 4 . . 4 & 1 8 49 8. 4 . . 4 . 4 3n . 4 4 4 & 3n c.n - & &
. " 8. 4 4 & K & L 4 . & K & L 4 . . 4 4& The Marked Coin Problem 8 4& 4 4 F " 4
. . 4 K & L K & &L %' & . . 4 4 n . 3n & 8. n 5 0 4 . " 0 G n H . + 4 . 4 4 3n+1 8 A . . 8 & & 5&J 3n . 3n . 3n " 4 F 4& 4 4 -
.4 l1 & . l2 & & h1 & & . h2 & & " 4 & . 4 5
. . 5
" l1+h1 l2+h2 5 + & & 5 3n ?4 . . 4 . & & &
4 4 l1 &
/1 #+ 5
. h2 & & . G & & & 5 HI h1+l2 . & . h1 & & . l2 & & l1+h2 . & & 4 5
. 5 3n 4 . " 5 h1+l2 = l1+h2 = 3n " 4 l1+h1 = l2+h2 4 . l1 = l2 h1 = h2 - 5
. . " 5 & 4 . . . 3n 4& 4 4 " 4&
4& . 4 I & & 4 . 4 The Complete Solution " / / " . A . . 3n+1 & 3n 4& 5
. & . . . .4 '
8. 4 .
8. .
& . & & 4 & & . &
8.
& & & . 4 & & & .
" / 4
. (3n+1−1)/2 .4 3 . < (3n−1)/2 (3n−1)/2 + 1 (3n−1)/2 A . 4 3 . .4J
(
/
8. & / G &H
8. . #
. 4 ' . K & &L # K & L &
/ 3n
8. #
. 4 ' . K & L # K & &L &
/ 3n
- 4 . ? & & U J & & & . . " . . F Exercise 6.5 &
. D D 4 4 ' . " D 4 F 4 8 D B 5 4 5 D + & 4 . 4 & m 2×m .& 5 D 4
. D 3m G=J . 3m+1 D 3 . 3m D H & .& 4 5 D D 2 2 Exercise 6.6 , n D 4 1 ≤ n . F 4 . . & 4 . D 4 4 H =4 & A D 2
H 4 & n 4 4 D 2n − 3 2 ≤ n H 4 D 4 4 A B C D A < B C < D 4 4 A . . 4 4 ; 4 4 A . 4 D 4 G 5 & G HH
/2 & (
)
H 2m D 4 1 ≤ m 4 & m A D 3m − 2 G=J
. G HH 2
6.8
Summary
8 . / " .
. . & K<L . K<L . (a) . K<L 0 (b) . . K<L n . & n . K<L n+1
; . " & 4 4 D . " & 4 & . &
A
6.9
Bibliographic Notes
" ./
. 4 & % - 3D M3D$ 3D$ N
*
/
Chapter 7 The Towers of Hanoi " "4 . = " &
' . A . K L / & " "4 . = << 5 Æ 4 & / & 8 & 4& . A =4 & . 8 4&
. 5 J & . 5 + 4 4 . > . 4 4 I "4 . = I . . & . 4 Æ
. . .
7.1 7.1.1
Specification and Solution The End of the World!
" "4 . = . << !! & + S
% 6 & # " << 4 4 A' > A . . 41 , '&/. . F < . < G A H " 4 .
& . & " 1 4 4 &
!
1 6 % 0
. A . & & 4 4 9
+ J "4 . =
7.1.2
Iterative Solution
" & & "4 . = &
& ' " . 4 . # . & 4 4 - . ' 4 - &
. 0 4 > & 0 & 1 - .4 > & & A & " . " .
. 8.
. & 4 > 4 & 4
1 &
$
> & & C D & 8 & . ' & " 4 . /
& 4 // / "& ' &. & 4 / << " & /
& . . /
8. & & 4 A 4 3 4 & & ' C 5 / 6 /
7.1.3
WHY?
4 / 4 8. & & C 4 8 9C 4& # 4 4 . 4 4 . A . . " 4 4 9 8 4 A . " &
. 4 4 4 .
7.2
Inductive Solution
&
. & 6 4 / M . A A 6 G6 4 H . & -
. 0 + 4 4 n . 4 4 n+1 . = 4 9 " & . 4& & 4 J # n . . 4 ' /
. & n 4 4 & .
1 6 % 0 # . " n . > 4 ' . &
n 4 4 & .
" 4 A & " 4& & A . 4 G &H 4 ' &
& " &
& 4 . 5 4 A G" &
& & . H " 4 4 . " . &
& . " .& 4 4 & & 4 4 4 . . "4 . = 4 n . ' d 4 d 4 4 " . . 4 A . " & ?4 4 - . / & 8 Æ & A & " A ' & M - & n . M−n 8. . M−n n 5 & . & - & " & 4 n . d . & G 4 & . .H 8 n 0 5 . & 5 8 . n+1 4 4 . n . . 4 - 4 4 n+1 . d 4 d 4 4 + 4
. 1 4 4
1
1 &
, . ' & . 4 - & n d ¬d & . 4 . & . G. & & & . H n ¬d " n+1 d. G" & n+1 . =4 n n+1 4 n+1 H +& 4 & n ¬d " n+1 n+1 4 . d " .4
" A Hn.d 5 . k , d 4 n
. k
d d 3
. 1 4 1 3 true 4
false / 4 " k , d
k . d " 5 [ ] & 5 [x] 5 4 ' & x " . . 5 Hn.d 4 n / &/ . d .4 . . H0.d = [ ] Hn+1.d = Hn . ¬d ; [n+1 , d] ; Hn . ¬d
? H . 5 . Hn+1.d . 4 4 & 4. / 5 . " . . I K L . " 4& "4 . = . & . n C 4 & .// 4 . H + ' 4 4 H2.cw G- cw aw 4 4 true false &H H2.cw =
{
5 n,d := 1,cw
}
1 6 % 0 H1.aw ; [2,cw] ; H1.aw =
{
5 n,d := 0,aw
}
H0.cw ; [1,aw] ; H0.cw ; [2,cw] ; H0.cw ; [1,aw] ; H0.cw =
{
5
}
[ ] ; [1,aw] ; [ ] ; [2,cw] ; [ ] ; [1,aw] ; [ ] =
{
. 5
}
[1,aw , 2,cw , 1,aw] .
' & H3.aw 4& 8. & & 4 5 & . F " ' . 5 . n = 3 16 . n = 4 32 " & 9 " " & G &H ' & " . . 5 .
& ' " & . & 9 Exercise 7.1 "
. & . 5 H64.cw 6 Tn.d . 5 Hn.d 3 A . T . A . H GB A Tn.d . d H ; A To T1 T2 = 4 . D ' Tn . . n & D & n 2
; . .
. F Exercise 7.2 "4 . = ; 4 4 / 4 . n 4 4 ; 4
. 2
1 &
7.3
The Iterative Solution
8 4 J A & G . & 4 & 4H 4 8 4 4 4 . Cyclic Movement of the Disks
8 4 4 4& & 8 . 4 - 4 & 4 . . " & . k , d 5 Hn+1.d even.k ≡ d G 4& 4& .H " / 5 . . )) - . . Hn+1.d K n+1 L & K n L K d L & K ¬d L even.(n+1) ≡ ¬(even.n) . even.(n+1) ≡ d - even.k ≡ d . G. k , d 5 Hn+1.d H 4 . n d 6 N D " . k , d 4 even.k ≡ d ≡ even.N ≡ D .
" . 4 . d . k A & . 5
. 4 /
& 4 /
& 4 @ / . 5
. 4 /
& 4 /
& 4 8 G4 /
H & D . N D . N Exercise 7.3 ' 4 & . . & 4& 4 " 4 . 41 4 4 4 4 4 =4 & . 4 .
(
1 6 % 0
> ' & 4 . . 3
2 2 Alternate Disks
- 4 D . &
4 & ' << . Æ . 4 . 4. & 4&
4 . F . << - 4 4 . . 5 Hn.d A & 6 5 .
. 4 " A & 4 I & . 5 / & 4 - 4 alt.ks . 5 ks 4 " 5 . & 4 4 & diskn.d & A . . Hn.d 6 5 4& & diskn.d " . A . H 4 J disk0.d = [ ] diskn+1.d = diskn . ¬d ; [n+1] ; diskn . ¬d .
> alt.(diskn.d) " . & n " n = 0 & & 5 + & . 5 4 . . 5 ks
k alt.(ks ; [k] ; ks)
⇐
alt.ks ∧ ((ks = [ ]) ≡ (k = 1)) .
" . J alt.(diskn+1.d) =
{
A
}
alt . (diskn . ¬d ; [n+1] ; diskn . ¬d) ⇐
{
& . 5
}
1 & (
)
alt.(diskn . ¬d) ∧ ((diskn . ¬d = [ ]) ≡ (n+1 = 1)) =
{
& A D .4 & . diskn .
}
true .
" ' . & Exercise 7.4 > . 8 . & . '1 A . & " . 4 & 4 4 I & / 9 +& ' 4 4 // / 4 & 4
& . 41 " & 4 . - 4 ' 2 & 4 &
G=J " 4 4 B & 4
. 1 4 # . . 4 n even.n ≡ d ≡ even.k ≡ d
. d & k 4 n d H 2
4 Exercise 7.5 (Coloured Disks) " . I F & F& 3 4 4 B
& & 2
7.4
Summary
8 4 4 "4 . = "
*
1 6 % 0
4 . . n / " 4 D.& . " 4 . . . . 5 . . . " 4 J . 0 / . G 1 / H . & & . . . G 4 4H
7.5
Bibliographic Remarks
8. & . "4 . = . M$ N . . . 4 M6! N " . . M+ N
Chapter 8 The Torch Problem 8 4 . ' ( " &
. I A & 4 .4 N 4 8 & 4 & & 4 " 4 2 & "
. 1 N. i t.i I 4 4 & . 4
4 N + & 4 t.i < t.j 4 i < j G" 4 8. t.i = t.j . i j 4 i < j 4 4& (t.i , i) 4 i ' & K L 4 4 &H
8.1
Lower and Upper Bounds
" .4 5 & Æ & / A 4 5 81 4 & Æ " 4 F 4 K L K4 L
!
2
8 . 4 . 1 2 5 10 & 5 4 4 . 17 8 4 5 . / 8 & ' 5 . 5 4 17 4 4 4
- . . F
4 4 # . 4 . 4
4 A ' 4 . . 2+1+5+1+10 19 & ' 5 4 19
I 4 4 G8 H & & ' 5 . 17 3 D 5 8 . N " 4 5 4 & " F 4 & 4 "
. 5 4
I 4
& 8 !* 4 4 . 5 "
Æ 4 . &
8.2
Outline Strategy
> 4 . & " . " . . .
4 8 & . ' & / &/ 4 . ' 8. 5 4 4 ' & F "
. F (N−1)! 4 &
. 5 . N " 4& & . 4 4 K.4 L
2 7 &(
$
4 4& " 4 K.4L KL 4 8. & .4 4 4 .4 " . 4 . .4 5 & & 8 F 4 . & + & K L . .4 4 K L . . "
. K L G K L H 4 A & & 4 . A &
. + '
. 4 A & & 4 &
4 & - 4 4 . ' {1∗a , 2∗b , 0∗c} . a b c 4 a b 4 c + & 4 4 {1∗a , 2∗b} 8 F . 5 % 4 4 4 4 ' 4 . G {1∗a , 2∗b , 0∗c} . ' H A " ' {1∗a , 2∗b , 0∗c} {2∗b , 1∗a , 0∗c} & . . ' {1,3} 4 1 3 8. 4 4 .4 4 A' 4 K + L G. .4H K − L G. H +{1,3} .4 & 1 3 −{2} & 2 4 . 4 . .4 5 . - &
. . 5 . 4 - 5 . & 4 " - 4 & 5 5 . & A . ? E' G& 5 .H G. 5 a 5 b 5 b 5 c 5 a 5 c H " A 5 5 + & " .
4 .4 4 / J
" I 4
5 G4
. 1 H
2
>/
5 .4 I /
" . 5
" & 4 .4 4 .4 G T & i . i∈T H
8
5 . A 4 & A 5 J
"
. .4
. "
. .4 &
. &
.4
" 5 & . .4 " A G
! H 4 & 5 & 4 " A . .
"
. .4
N−1
. N−2 G N
. H
" & 5 4 & 4 .4
I 4 & 4 G74 . .4 & 4 & " .4 .4 8 4& . H
# & 4 D . .4 5
Æ 5 5 4 . A 5 . .4 & . A . .4
+ . .4 & . 4 - & 4 4 -
2 &'
8.3
Regular Sequences
KL 5 5 4 .4 4 - 4& 5 . .4
Lemma 8.1 %& 5 & & . 5 4 Proof 5 - 4 J A .4 A 4 8. A 4 & p & 8.& .4 & p 4 p . # . & 5 .
u +{p,q} v −{p,r} w
4 q r u v w 5 p 4 v G? .4 & p 4 A 4H 5 & u +{q} v −{r} w
" 5 . 4 5 G" 5 4
& 4 .4 " . p 4 & . p & p & . .4 " . & x y t.p ↑ x + t.p ↑ y ≥ x+y H "
. 4 .4
. / ?4 A .4 " 4 J & A 5 & A 8. A 5
& G- N 2 H " 4
&
. / 8. A .4 & A p 4 q 4
& .
2
.4 G" & . p 1 .4 A H " 5 .
u −{q} +{p} v
q s p q. " 4 J .4 8. .4 q p 4 p 4 q q 1 p 1 " .
u +{q,r} w −{q} +{p} v
G4 w p q H u +{p,r} w v .
" 5 # G . & x t.q ↑ x + t.q + t.p > t.p ↑ x H
. / 8. & & G" 4 p s .4 A 5 H " p 4 p 4 q q 1 p 1 " .
u −{p} w −{q} +{p} v
G4 w p q H u −{q} w v .
" 5 # G t.p + t.q + t.p > t.q H
. / - 4 4 . 5 / I . F . & 4 . 4 5 5 / 2
8.4
Sequencing Forward Trips
6
! A +
. .4 5 N−1
. N−2 "
& 5 .4 .4 &
2 &' #6
. &
& 4 " . A 2×(N−2) N−2 2 I . 2×(N−2) + 1 & & 5 . & . .4 5 4I 4 " . .4 4 & .4 =
. . 4 . + ' .4 5 .4J 3∗{1,2} , 1∗{1,6} , 1∗{3,5} , 1∗{3,4} , 1∗{7,8}
G" &
I 3∗{1,2} 1 2
3 .4 1∗{1,6} 1 6 .4 ? . 4 .4 5 H "
. . 1 4 .4 3 I & 2 3 .4 2 4 3 2 .4 1 " G 4 5 6 7 8 H 1 .4 " J 3 × (t.1↑t.2) + 1 × (t.1↑t.6) + 1 × (t.3↑t.5) + 1 × (t.3↑t.4) + 1 × (t.7↑t.8) +
3 × t.1 + 2 × t.2 + 1 × t.3
G" & .4 & H ?
. .4 7 G
. H
. 6 " & .
! D . .4 5 5 & 4 . .4 " / & . .4 4 F . .4 5 " F . 4 ' & 4
& " . F 4 4 .& 4
1 N
"
.
. .4 & - 4
. & & .4 I 4 . & .4 3 . 4
(
2
& F &
. 8. 4 & I . . 4 & # I A& . 4 & ?4 4 Vnomad Vhard Vfirm Vsoft
.
.
. A
. . & F "
. F Vhard + Vfirm + Vsoft . "
. 5
. .4 &
. .4 . 2
. .4 & A 1
. 2 × Vsoft + Vfirm − Vnomad .
. .4
. " Vhard + Vfirm + Vsoft = 2 × Vsoft + Vfirm − Vnomad + 1 . %5& Vhard + Vnomad = Vsoft + 1 . -
.4 A Definition 8.2 (Regular Bag) N 2 . . {i | 1 ≤ i ≤ N} N . .4 J
% . F < 4 G8. & F 4 H ∀i : 1 ≤ i ≤ N : ∃T : T ∈F : i∈T G8. & & . F H
8. Vhard.F Vsoft.F Vnomad.F &
. F 4
. F
. F
G!H
Vhard.F + Vnomad.F = Vsoft.F + 1 .
2
+ & 4 4 .4
2 &' #6
)
Lemma 8.4 , 5 . N 4 N 2 . .4 F . 5 & .
. N # & 5 . . F 6 VFT & . T F. "
G!)H
ΣT : T ∈F : ⇑i : i∈T : t.i × VFT + Σi :: t.i × rF.i
4 G!*H
rF.i = ΣT : T ∈F ∧ i∈T : VFT − 1 .
G rF.i
. i H 2
8
5 . G!H J G! H
Vnomad.F = 0 ⇒ Vhard.F = 1 ∧ Vfirm.F = 0 = Vsoft.F .
8. Vnomad.F < & A . A
& G!H Vhard.F 1 G!!H
Vnomad.F = 1 ≡ Vhard.F = 0 = Vsoft.F .
8. & 4 " Vsoft.F < 8 .4 . G!H Vhard.F 5 < "
. G!H G!$H
N > 2 ∧ Vhard.F ≥ 1 ⇒ Vsoft.F ≥ 1 .
8. N G
. H 2 Vnomad.F 1 8 .4 . G!H Vsoft.F 1 ?4 4 4 4 5 . F Lemma 8.10 , N G 2 H N 5 . . F N " & 5 & G!)H Proof " . & < . F - " 4 F . ' & G4 & H " 5 D " & 8 Vnomad.F = 1 8 & F A " . {n,s} 4 n s " 5 & & . F &
*
2
& n 4 . .4 . 4 A {n,s} . s " Vhard.F /< F 8 & G!$H Vsoft.F 1 8 .4 & A . soft Vnomad.F 2 & . F {n,m} 4 n m 5 4 4 {n,m} .4 & . n & . m
& . . F & 1 G" . . F H - 4 F 4
.
& . n m & 1
. & 2 & 5 F 4 2 Optimisation Problem 6
!( ! A 4 . 8 . 5 . . 4 A A ! & G!)H 8 4 4 8 . & 4 & G!)H " 4
. " . A
4 F 1 .
F 1 , F T F 4 ⇑i : i∈T : t.i × VFT T F G K F L . . 'H + i 4 . t.i × rF.i i G i 1 H
8.5
Choosing Settlers and Nomads
" 4 - 4 . # A & 4 2 1 4& . N 2 I & 2 & Lemma 8.11 %& & . 4 4 Proof N F . (p, q) . 4 F p q p . q
2 & .
& (p, q) q p &4 F - N # & & t.q − t.p " .4 . p q . " .4 . & q G t.p < t.q H " .4 . & p & t.q − t.p " A & 4 " A 4 p {p,q} 8 4 p q F 0 8 p {p,r} 4 r = q 8 Æ t.p ↑ t.r + (t.q − t.p) =
{
& . '
}
t.q ↑ (t.r + (t.q − t.p)) ≥
{
t.p ≤ t.q & . '
}
t.q ↑ t.r .
+& . .4 " F " .
. & " & . .& F 2 Corollary 8.12 %& N & N F 4 .4 J
8 & A F 1 %& . F {1,2} G?J & . &
0 H
" & . {1,2} F j . j 4 1 ≤ j ≤ N÷2
{k: 0 ≤ k < j−1: {N − 2×k , N − 2×k − 1}} G? & 4 j 5 1 H
Proof F N + ! 4 & 4 A F 4 i 4 i 1 i . & 1 " F .4
!
2
i 4 =4 G & t.i − t.1 H - N 4 F G .
!( . 5 . H >. < . F "
. & i & i F "
. & . 1 & G!H 8. i . & G!H & G A .H . =4 i & "
. & & 8. i . 4 F. " A F . F A 8 & G!H
?4 . F F . {1,2} 4 & {1,2} 4 & F - & 4 . G!H . & . {1,2} F j . j 4 j 2 j−1 - . A 4 4 1 & A I .4 1 & F . G!H " 1 2 . GA H F 8 4 4 j 2 . {k: 0 ≤ k < j−1: {N − 2×k , N − 2×k − 1}} .
G" & . 1 4 4 H
. . j 4 j 4 " 3 N 2×(j−1) . . N − 2×(j−2) . A & & & . 4 4 A . 4 "
&
. 2
2/
8.6
$
The Algorithm
- 4 Lemma 8.13 F .& & ! " . j & . {1,2} F & F
G! (H HF.j + FF.j + j × t.2 + (N−j−1) × t.1 + (j−1) × t.2 , 4 HF.j = Σi : 0 ≤ i < j−1 : t.(N−2i)
FF.j = Σi : 3 ≤ i ≤ N − 2×(j−1) : t.i .
G" A .4 4 H Proof " 4 8. . . j 1 8 Σi : 2 ≤ i ≤ N : t.i + (N−2) × t.1 .
HF.1 + FF.1 + 1 × t.2 + (N−1−1) × t.1 + (1−1) × t.2 =
{
A . HF FF
}
0 + Σi : 3 ≤ i ≤ N : t.i + t.2 + (N−2) × t.1 =
{
}
Σi : 2 ≤ i ≤ N : t.i + (N−2) × t.1 .
8. . . j 5
. . 2 8 HF.j .4 . F FF.j .4 . A F j × t.2 .4 . j . +& 1 1 (N−j−1) × t.1 2 1 (j−1) × t.2 G 2 . j .4 1 . j + (N − 2×(j−1) − 3 + 1) .4 ? . j−1 N−j−1 N−2 4 4 4 '
. H 2 + j 4 j 2 A OT .j & N 4 & . {1,2} j " OT .j & G! (H ?4 4
2 HF.(j+1) − HF.j = t.(N−2j+2) ,
FF.(j+1) − FF.j = −(t.(N−2j+2) + t.(N−2j+1)) .
5 OT .(j+1) − OT .j = −t.(N−2j+1) + 2 × t.2 − t.1 .
? OT .(j+1) − OT .j ≤ OT .(k+1) − OT .k =
{
}
−t.(N−2j+1) + 2 × t.2 − t.1 ≤ −t.(N−2k+1) + 2 × t.2 − t.1 =
{
}
t.(N−2k+1) ≤ t.(N−2j+1) =
t
{
}
j ≤ k .
" OT .(j+1) − OT .j j 5
. OT .j & . 4 F . . " 4& & 5 N 4 j 1. 2 × t.2 ≤ t.1 + t.(N−2j+1) . 8. true . {1,2} I .4 & . 1 {N−2j+2 , N−2j+1} . 2 8. . N−2j+2 N−2j+1 A 8 & 1 I 4 4 1 -
N
. & & . j " .4 {
2≤N }
i,j := 1 , N÷2 ; { Invariant: 1 ≤ i ≤ j ≤ N÷2 ∧
∀k : 1 ≤ k < i : OT .(k+1) ≤ OT .k
21 ∧
∀k : j ≤ k < N÷2 : OT .(k+1) ≥ OT .k }
do i < j →
m := (i+j)÷2 ; if
2 × t.2 ≤ t.1 + t.(N−2m+1) → i := m
2
2 × t.2 ≥ t.1 + t.(N−2m+1) → j := m
fi od {
1 ≤ j ≤ N÷2 ∧
∀k : 1 ≤ k < j : OT .(k+1) ≤ OT .k
∧
∀k : j ≤ k < N÷2 : OT .(k+1) ≥ OT .k }
.
> j & . {1,2} OT .j 5 " j−1
. & ! / A .
! A 4 & " N−2j+2 N−2j+1 A
8.7
Conclusion
8 4 . / &
. & " . 4
D
4 " . . . .4 @
.
4
. . . .4 4 5 #& . 4 5 KL C .4 & 4 & I KL G 4 H 4 K L G 4 H . 6 4
. K L G & 4 H
. 4 . " . . . 4 & & & . . .4 I . 4 5 . & & Æ % & A G H K L . 5
2
C K L G . & & H 4& 4& 4 . K L
8.8
Bibliographic Remarks
8 /% KE LI / K L " &/ / 4 & . M N 8 . & 2 I & & M N . K L " F/ A % . K L 4 D . 4 " F
! I & 5 . = & . / . 1 & G & = & K . . . & & 2t2 − t1 . ti 1L . 2t2 − t1 C 1 2 5 10 ' 9H & 4 - & . #& K L . . + '
. G8. N = 4 & . 3 1 1 4 4 4 8 4 5 " 3 9 H " . K&L .
4& 4 . .4 K.L G . & . H 1 . . . K L . 4
Chapter 9 Knight’s Circuit " & B & & /
4 F " A 7 1 . " A 5 .
4 7 5 . 5 ' & 5 4 " . I & & ' ' . 7 .4 /
C / =4 64 5 I 64 . 5 + . 5 . D 2 . . 16 5 . 8 G A $ HI . 5 4 6 " . .4 6 . & 4
4 & 5/
. & 5 ' & " 7 1 .
. 4 . " 5 & 7 1 & A & G4 5 <& & 5 H " . 4 4 '
9.1
Straight-Move Circuits
+ 7 1 Æ ' .
(
8 , 9
+ $ J 7 1 # 61 & 4 . /
& 4 . & 4 - . 4 & 2 " A 4 4 . & 5 & 5 8 1 4& . & 4 < C . & 5 5 & 5 ' I 5 D & 61 . . " 5 <& & G" 7 ' H - 8 . D 4 2 " 4 & 1
& B & A / & & A & & . I 8×8 & & 5 & . 4 / . 2 " & 5 ' & 5 K L 2 8 . & 4 .4 ' 8 & .4 Exercise 9.1 (a) - 4
. .
. 52 ; & 4 4 . . G=J
8 & +:
)
. 5 F 5 > 4 4 . 4H (b) + 4 . m / . . < 2m×1 2 G 2m×1 . 5I
. 5 2m H (c) 4 / . 2×n . GH . n G 2×n 4 4 4 n 5H 2
" . ' $ / & .
< 2m×n .
m n " 2m n G m n /<
' . 5H / 4& 4 < 2×n . n " 4 4 & / . 2m×n . m n & m 2×n . " 4 . < 2m × n 4 m 1 . 4 m 1 2m × n 4 . < 2p × n 2q × n & 4 p q m p+q 5 m - & & / . - D
4 " &
5 " 4 .
. 5 & / 8 < ?4 2m × n 2p × n 2q × n 4 . G" 4 A
. 4 . . H " /. . 2p × n
& /. . 2q × n / . 4 + $ 4
& " 4 < /. . < 4 4 . . 4 " . G>. . . . H ?4
. . < . /. /. & 4
& A $ "
*
8 , 9 n
2p
2m
2q
+ $ J
/ + 4
/ . n
2p
2m
2q
+ $J
4 4 + $( 4 4& . 6 × 8 %./ . & . / . 2 × 8 "
& < & 4 A $( Exercise 9.2 4
. 5 3×3 8 & / . 5
5 G A $)H 8 / . 5 . 5 =4 / . . . 5 C 5 D 5 . ' /. 5C %' 4 4 / . 5 . <
8 & '
+ $(J / . 6 × 8
+ $)J / G 4 H . 3 × 3 . 5 2
9.2
Supersquares
6 4 7 1/ " & ' 4 4 4 " 4& 8 × 8 4 × 4 & 2 × 2 5 K/ 5L 4 A $* 8. 7 1 A 4 &J + K L 4 5I 7 1 . . 5 5 & 4 <& . , 4 5I .
!
8 , 9
+ $*J
4 × 4 . 5
7 . 5 . 5 8 A $ . 5 & I . 7 4
0110 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 11111111111111 00000000000000 1010 1 0 0 1 0 1 0 1 0 1 0 1 1010 1 0 0 1 0 1 11111111111111 00000000000000 00 11 0 1 0 1 0 1 00 11 0 1 0 1 0 1 0 1 11111111111111 00000000000000 00 11 1010 1 0 0 1 0 1 0 1 0 1 0 1 1010 1 0 0 1 0 1 11111111111111 00000000000000 0 1 0 1 0 1 1010 1 0 0 1 0 0 1 1 0 1 0 1
+ $ J GH 3 GH 7 1 # + G H . 5 & + 4 4 + $! 4 . 5 C /. 5C & / 4 <& 4 " + ' . /. 5 /. 5 / 5 > . @ E ' 4 < E < ' G" / /&4 / / I < /'// 4 4 &4/'// 4H
8 & '
1111 00 00 0011 11 00 0011 11 0011 00 00 11 0011 11 0011 0011 00 0011 11 0011 0011 00
$
+ $!J . /. 5 6 & v . E . 2 × 2 5 G v . K LH & & h G . K <LH . E 4 . 5 ?4 A' . . ' v ; h . A E E 4 . 5 + E 4 E 4 " v;h = h;v .
5 2 × 2 5 1800 c G . K LH " & A . c G$H
v;h = c = h;v .
- 4 A 2 × 2 5 " . 4 / %4 4 skip = . & 4 n G . K LH + 4 & 4 <& 4 1800 " I G$(H
v;v = h;h = c;c = n .
. . & G$)H
n;x = x;n = x .
" 5 G$H G$(H G$)H 4 . . G x .4 & 4 y z A x .4 & y 4 z C &
x ; (y ; z) = (x ; y) ; z H 4 A . & 5 . + '
(
8 , 9 v;c;h =
{
v;h = c
}
v;v;h;h =
{
v;v = h;h = n
}
{
n ; x = x 4 x := n
n;n =
}
n .
8 4 E & 1800 E <& G? 4 & / & 4 A " . A' . K.4
&L . / 5H Exercise 9.6 4/ 4 F . ' 4 x y " . 4 . & . n v h c G; & . E 5 H ; .& . x y {n,v,h,c}
G$ H
x;y = y;x .
. x y {n,v,h,c} G$!H
x ; (y ; z) = (x ; y) ; z .
G8 & 43 64 F
" . 4& . 4H 2 Exercise 9.9 "4 2 × 2 5 90◦ 4 4 8.& . 2 × 2 5 ' ) 4 F . & . 2
8
9.3
(
Partitioning the Board
" A . . 5 A 4 7 1/ . 5 K n L " A. 5 5& n v h c 5 5 + $$ 4 4 4 /. 5 K n L 5 "
& K v L 5 . K h L 5 & D K c L 5 5 . 4& . 5 64 5 . . D 8 A $$ F & A & . 5 "4 5 5 & . & 7 1
G" 4 5 . . & 7 1 4 5 . F .
& 7 1 H
11 00 000 111 00 11 000 111 00 11 00 11 000 111 000 111 00 11 111 00 11 111 111 111 000 000 00 11 00 11 000 000 v 11 c 11 v11 c 11 111 111 111 111 00 000 00 000 00 00 000 000 00 11 000 111 00 11 000 111 00 11 00 11 000 111 000 111 n 11 h 11 n11 h 00 11 000 111 00 000 111 00 00 000 111 000 111 00 11 00 11 00 11 000 111 00 11 000 111 000 111 000 111 11 111 11 111 111 111 00 000 00 000 00 11 00 11 000 000 00 11 00 11 00 11 000 111 00 11 000 111 000 111 000 111 v c v c 00 11 000 111 00 11 000 111 00 11 00 11 000 111 000 111 00 11 000 111 00 11 000 111 00 11 00 11 000 111 000 111 00 11 000 111 00 11 000 111 00 11 00 11 000 111 000 111 00 11 000 111 00 11 000 111 000 111 000 111 00 11 00 11 n h n h 00 11 00011 111 00 000 11111 0011 00 000 111 000 111 00 11 00011 111 00 000 11111 0011 00 000 111 000 111 + $$J 6 5
. / $ " 4
(
8 , 9
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+ $ J . . K/ L K/&4L " /
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&4 C 4 KL & 4 KL %' &
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8
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1111 00 00 111 00011 0011 0011 00 000 111 000 111 00 11 00 11 111 000 00 11 00 11 00 11 000 111 000 111 00 11 00 11 000 111 00 11 00 11 00 11 000 111 000 111 0011 11 00 00011 111 0011 0011 00 000 111 000 111 00 00 00000 00 11 00 000 000 11 11 111 11 11 111 111 00 11 00 11 000 111 00 11 00 11 000 111 000 111 00 11 00 11 00 11 111 00000 11 00 11 00 11 000 111 000 111 00 11 00 11 00 11 000 111 00 11 00 11 000 111 000 111 11 111 1111 111 111 0011 00 00000 0011 00 000 000 00 00 000 00 00 11 00 000 000 11 11 111 11 11 111 111 00 11 00 11 00 11 000 111 00 11 00 11 000 111 000 111 0011 11 00 00011 111 0011 0011 00 000 111 000 111 0011 11 00 111 00000 1111 0011 00 000 111 000 111
+ $ J + K/ L K/&4L " 4 & 4 /
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8
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8 × 8 6 × 8 . *
4m × 2n m n m ≥ 2 n ≥ 3 . G" . . 8 × 8 . 6 × 8
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8. & Æ & 90◦ 4
.
H 2 Exercise 9.12 3 . . < (4m + 2) × (4n + 2) 5 & (2m + 1) × (2n + 1) KL
. GH 5 / & ' $ =4 7 1 . . < (4m + 2) × (4n + 2) 4 m n 1 & ' ' $ " & . / . . 5 G ' $ . 4 H " .
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& 4 C C 4 5 " A $ (
11 00 0011 11 00 00 11
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9.4
Discussion
8 . & & 7 1 & Æ 4 & ' . /
8 5
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A . K L KL / . 5
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4 . K '& &L . & 5.& 4 Æ &
9.5
Boards of Other Sizes
" 4
9.6
Bibliographic Remarks
. 7 1 & . . M# U 4 ;N A 7 1 . & / / & . 4 " . 5 7 1 % - 3D M3D$ N = . /< .
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5 0 %& & & k g 4& 5 " 1234−1 & =
. & 8 . p & . p−1 2 2.2 6 m n p
. D . " & m,n,p := m+1 , n−1 , p−1 .
m−n n−p p−m 8 & & . & G8 F & 2 H
. F . &
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& 1 2 G 4 .H 4 1 " 3 4 2 +& 1 2 " & t1↑t2 + t1 + t3↑t4 + t2 + t1↑t2 .
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