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2006 D r. H e n e r 18 4 1

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1 2 1 1.1 2 10..................................... 1 1.2 2...................................... 3 1.3 2...................................... 5 1.4......................................... 7 2 9 2.1...................................... 9 2.2................................. 12 2.2.1................................... 12 2.2.2 Day Camp.............................. 14 2.2.3........................ 15 2.3 vs.................................. 16 2.4................................. 18 3 19 3.1 ( )................................ 19 3.1.1................................... 19 3.1.2..................................... 20 3.1.3..................................... 20 3.2......................................... 21 3.2.1................................ 21 3.2.2................................ 23 3.3............................... 23 3.4....................................... 24 3.4.1 RSA.................................... 26 3.4.2..................................... 27 4 28 4.1.................................. 28 4.2....................................... 29 4.3...................................... 31 4.4.................................. 32 4.5....................................... 34 ii

5 37 5.1........................................... 37 5.2........................................... 39 5.3.............................. 41 6 43 6.1........................................... 43 6.2........................................... 44 6.3.......................................... 45 6.4................................. 46 6.5........................................... 48 6.6.......................................... 49 7 53 7.1......................................... 53 7.2.................................... 55 7.3....................................... 57 7.4........................................... 61 7.5...................................... 63 A 66 A.1 2,.................................... 66 A.2........................................... 70 A.3.......................................... 74 B 77 B.1..................................... 77 B.2..................................... 78 B.3...................................... 79

1 2 1.1 2 10 Magic 1 ( ) 16 17 18 19 20 21 22 23 8 9 10 11 12 13 14 15 4 5 6 7 12 13 14 15 24 25 26 27 24 25 26 27 20 21 22 23 28 29 30 31 E 28 29 30 31 D 28 29 30 31 C 2 3 6 7 10 11 14 15 1 3 5 7 9 11 13 15 18 19 22 23 17 19 21 23 26 27 30 31 B 25 27 29 31 A ( ) 6 15 26 31 1

1.1 2 10. (1) (3) 16 8 4 2 1 0 0 1 1 0 16 8 4 2 1 1 1 0 1 1 (2) (4) 16 8 4 2 1 0 1 1 1 0 16 8 4 2 1 0 1 1 1 1 1.2 10 2. (1) (3) 6 26 16 8 4 2 1 16 8 4 2 1 (2) (4) 13 31 16 8 4 2 1 16 8 4 2 1 10 2 16 0 0 0 1 1 1 2 10 2 3 11 3 4 100 4 5 101 5 6 110 6 7 111 7 8 1000 8 9 1001 9 10 1010 A 11 1011 B 12 1100 C 13 1101 D 14 1110 E 15 1111 F 16 10000 10 10, 2, 16 1.3 ( ). 1 1101 1101 2 (16 ) 2 101 1001 1010 1101 2 (16 ) 3 2DA 16 (2 ) 4 AC8F 16 (10 ) 5 239 10 (16 ) 2

1.2 2 ( ) Basic : CIRCLE(200,150),100,5 ( ) G R B 0 0 0 0 0 1 0 0 0 1 2 0 0 1 0 3 0 0 1 1 4 0 1 0 0 5 0 1 0 1 6 0 1 1 0 7 0 1 1 1 G R B 8 1 0 0 0 9 1 0 0 1 10 1 0 1 0 11 1 0 1 1 12 1 1 0 0 13 1 1 0 1 14 1 1 1 0 15 1 1 1 1 1.4 7. 1 3000. 3000 7...,. 1.5 ( ) 1g, 2g, 3g, 4g, 1g 15g. 1 g,. g. X 40g g. 3

1.6 ( ) (1) 1 8. 1 1 1g, 1 1 2g 1. 1, 1, 1.. 1 2 3 4 5 6 7 8 (2) 1 8. 1 1 1g, 1 2g 1. 1, 1, 1.. 1.7 (1997 V(2)) 1. 1,, 1. 2,, 1. 3.,..,. 4

1.3 2 1.8 ( ) 1 2, 2 2 4. 1, 15., 1.9 ( ) 20, ( ). 1.10,,.,...,, 1, 1. 2 2, 3 4, 81, 2,.,.., (??? ). 5

1.11 ( ), 64,.,. 1, 1. 2. A B C 64. (1) 5? (2) 6? (3) 64,?,?.. 1. 1. 1. 10. 100. 1000. /* */ #include<stdio.h> void hanoi(int k, char x, char y, char z); int cnt = 0; /* */ void main(void) { int n; } printf("? "); scanf("%d",&n); hanoi(n, A, B, C ); void hanoi(int k, char x, char y, char z) /* */ { if(k > 0){ hanoi( k - 1, x, z, y); printf("%5d: %d %c %c Y=n", ++cnt, k, x, y); hanoi( k - 1, z, y, x); } } 6

1.4 Magic 2 (, ) 1,. 2, 2. 3 1. 4,. 2 5? K A Magic 3 ( ) 1. 2. 3 7 9. 4 4. 5 3. 6. 7 6 8. 8 8. 9 4. 1 2 3 4 5 6 7 8 9 10. 11 1 5. 12 2. 7

Magic 4 ( )..,,.,. 3, ( ). 1.12 ( ) 40,., 1, 2. 20.. 1.13 ( ) 7, 1 8. 0 1, 8 1 ( ). 7 1 16,, 16. 1 00 2 3F 3 7F 2, 1.. 0F. 80. 8F 8

2 2.1 2.1 ( ),., 3. 10,. ( ) 2 3 4 ( ) 1 3 6 2.2 ( ),., 4, 11., 10,. 2.3 ( ). 10. 1 2 3 9

2.4 ( ). 11. 1 2 3 4 5 6 : : 1 2 3 5 4 6 : : 2.5 ( ) 4 9 1, 2, 3, 5,. 500. ( ) http://www.cong.ac.jp/ hener/lecture/c1/loops/looptop.html cong 1.. 2.. 3.. : ( ).? 10

2.6 ( 2, 3 ; ) (1), 25. 4.,. (2),.,. 2.7 ( 2, 3 ; ) (1),., ( ). (2),., ( ). 11

2.8 ( : ),. 1. 2. 3. 2.2 2.2.1 2.9 ( ),,.,.,.,,,.,,. 2.10 ( ),,,,,,,.,,, 3, ( 1 ).,,.,.,. 12

2.11 ( ) 4 A, B, C, D. 4.. A B C D 1 3 4 7 2,.. 2.12 2.11,. (1) A: 1, B: 2, C: 4, D: 8 (4 ) (2) A: 2, B: 6, C: 9, D: 10, E: 11, F: 12 (6 ) (3) A: 2, B: 6, C: 7, D: 9, E: 10, F: 11, G: 12 (7 ) 13

2.2.2 Day Camp 2.13 ( ) Day Camp. 10. A (8 ) B (2 ) C (3 ) D (3 ) F (7 ) H (18 ) G (7 ) E (2 ) I (8 ) J (8 ) (1), MRC, Hener 3. (a). MRC Hener 10 20 30 40( ) (b). 10 20 30 40( ) MRC Hener (2), MRC 2. MRC 10 20 30 40( ) 14

? ( ) 1.? 2.? 3.? 2.14 ( ), 1 2. 10. 3. 2.15 ( ) 3. 1. 1, 30. 2. 1. 30. 3. 30.,. 2.2.3 2.16 ( ) 7. 7,,,,.,.,., A B, A, B. 15

2.3 vs Fermat x, y, z? x n + y n = z n (n > = 3) 2.17 (Euler ) x, y, z, t? x 4 + y 4 + z 4 = t 4 2.18 ( ) (1), 2, 1 6 1.. (1) 6 2 5 4 3 1 (2) (1) 10. (3) (1) 15. (4) (1) 21. (2) (3) (4) 16

2.19 ( )? 3 A, B, C., C, A B. C,.,,. 2 ( 2 ). ( ; ) 1956 12 3 ( ) ( 15) ( ) 1961 12 3 ( ) ( ) ( ) 1965 12 3 ( ) ( ) ( ) 1990 13 2 ( ) ( ) ( ) 1993 13 2 ( ) ( ) ( ) 1994 12 3 ( ) ( 12) ( ) 1996 11 4 ( ) ( ) ( ) 1996 5 ( ) ( ) ( ) ( ) ( ) 2.20 (Collatz ), 2 3 1. 4 2 1 4 2 1.? 17

2.4 2.21 ( ) 2.1, 10 1.5km 4.0km 5.,., 5,. 1 2 3 4 5 3.0km 1.0km 5.0km 1.5km 4.0km 2.1: 10 1 2 3 4 5 A 4 55 5 10 4 57 5 04 5 09 B 5 15 5 10 5 05 5 05 5 02 C 4 58 5 01 5 05 5 03 5 05 D 4 44 5 07 5 06 4 46 5 04 E 5 16 5 23 5 22 5 25 5 20 F 4 40 5 00 4 52 4 55 4 59 G 5 01 5 14 5 15 5 21 5 26 H 4 54 5 06 5 02 5 07 5 09 I 5 09 5 03 5 15 5 31 5 12 J 5 22 5 21 5 17 5 20 5 19 1.5 km 1 2 3 4 5 A 13 07 13 14 13 31 13 38 13 39 B 14 32 14 14 14 06 13 59 13 51 C 13 37 13 30 13 41 13 40 13 36 D 14 04 13 31 14 28 13 48 13 43 E 13 43 13 56 13 44 13 38 13 48 F 14 20 15 12 14 18 13 52 13 42 G 16 26 16 03 15 39 15 05 14 55 H 14 41 15 06 14 37 13 41 13 47 I 15 38 13 35 15 41 14 51 14 06 J 13 39 13 37 13 42 13 41 13 45 4.0 km 1 2 3 4 5 3.0km 1.0km 5.0km 1.5km 4.0km 18

3 3.1 ( ), 26.,. [ ] ( ) ( ) : KSDKS : aitai : aitai : KSDKS ˆ ˆ.. 3.1.1 3.1 ( ), K = 9. (1). computer (2). WRQXW 19

A B C DE W X Y Z F GH T UV +10 S I R J Q K P O N M L K = 10 A B C DE W X Y Z T UV S R Q P O F GH I J K N M L 10 W X Y Z T UV S R Q T UV S R Q P O W X Y Z P O A B C DEFG H I J K N M L A B C DEFG H I J K N M L 3.2 ( ),. JDNXLQ 3.1.2., 3.1. a b c d e f g h i j k l m X N Y A H P O G Z Q W B T n o p q r s t u v w x y z S F L R C V M U E K J D I 3.1 3.3 ( ), 3.1. (1). hener (2). AXZVUWZ 3.1.3 ( : ) ( a b c d ) ( x y ) ( ) ( ) ( ) ( ax + by a b x z ax + by az + bw = = cx + dy c d y w cx + dy cz + dw ( ) 1 a b = c d 1 ( d b ad bc c a ) ) 20

3.2, = ( 5 4 0 2 ). a b c d e f g h i j k l m 1 2 3 4 5 6 7 8 9 10 11 12 13 n o p q r s t u v w x y z 14 15 16 17 18 19 20 21 22 23 24 25 26 3.2 ( ) x = 24. ( 5 4 0 2 ) ( 2 4 ) = ( 5 + 4 0 + 2 ) = ( 26 8 ) (26, 8). ( ) (26, 8). 1 10 ( 2 4 0 5 3.2, x. ( ) 1 ( ) 5 4 1 = = 1 ( ) 2 4 0 2 5 2 4 0 10 0 5 ) ( ) 26 = 1 ( ) 2 + 4 = 1 ( ) 20 8 10 0 + 5 10 40 3.4 ( ), 3.2, = (1) love. (2) (11, 12), (8, 11), (7, 4). ( 3 1 1 2 ) =. ( 2 4 ) 3.2 3.2.1 a.082 h.061 o.075 v.010 b.015 i.070 p.019 w.023 c.028 j.002 q.001 x.001 d.043 k.008 r.060 y.020 e.127 l.040 s.063 z.001 f.022 m.024 t.091 g.020 n.067 u.028 3.3 26 (Beker, Piper) 21

ˆ (1) e : 0.127. (2) t a o i n s h r : 0.060 0.091. (3) d l : 0.040. (4) c u m w f g y p b : 0.015 0.028. (5) v k j x q z : 0.010. ˆ 2 30 th he in er an re ed on es st en at to nt ha nd ou ea ng as or ti is et it ar te se hi of. ˆ 3 12 the ing and her ere ent tha nth was eth for dth. YIFQFMZRWQFYVECFMDZPCVMRZWNMDZVEJBTXCDDUMJ NDIFEFMDZCDMQZKCEYFCJMYRNCWJCSZREXCHZUNMXZ NZUCDRJXYYSMRTMEYIFZWDYVZVYFZUMRZCRWNZDZJJ XZWGCHSMRNMDHNCMFQCHZJMXJZWIEJYUCFWDJNZDIR A 0 H 4 O 0 V 5 B 1 I 5 P 1 W 8 C 15 J 11 Q 4 X 6 D 13 K 1 R 10 Y 10 E 7 L 0 S 3 Z 20 F 11 M 16 T 2 G 1 N 9 U 5 3.4 26 A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Step 1 = Z e? Step 2 3.5 C t, D a, F o, J i, M n, R s, Y h, N r. Step 3 22

ˆ DZ, ZW (4 ), WZ, W. ˆ NZ, ZU (3 ). ˆ RZ, HZ, XZ, FZ, ZR, ZV, ZC, ZD, ZJ (2 ). ˆ ZRW RZW, RW, R. W d, R n ------ --------- ---- --- ------------ YIFQFMZRWQFYVECFMDZPCVMRZWNMDZVEJBTXCDDUMJ -------- ---- --------- -- --- ---- ---- NDIFEFMDZCDMQZKCEYFCJMYRNCWJCSZREXCHZUNMXZ - --- ------ ------ --- --- -- - - - -- NZUCDRJXYYSMRTMEYIFZWDYVZVYFZUMRZCRWNZDZJJ - ----- ----------- ---- ------- --- -- XZWGCHSMRNMDHNCMFQCHZJMXJZWIEJYUCFWDJNZDIR ˆ,. A B C D E F G H I J K L M y a s p r b c u t v i N O P Q R S T U V W X Y Z h x f n k g w m d l o e 3.6 ( ) ( ) Our friend from Paris examined his empty glass with surprise, as if evaporation had taken place while he wasn t looking. I poured some more wine and he settled back in his chair, face tilted up towards the sun.,.,,. 3.2.2 3.5 ( ), 3.2. 2,. K (3, 8) O (7, 20),. 3.3 ˆ. ˆ,. 23

: muzui : kyhtb ( ) : a b c d e f g h i j k l m n o p q r s t u v w x y z Q R S T U V W X Y Z A B C D E F G H I J K L M N O P a b c d e f g h i j k l m n o p q r s t u v w x y z C D E F G H I J K L M N O P Q R S T U V W X Y Z A B a b c d e f g h i j k l m n o p q r s t u v w x y z T U V W X Y Z A B C D E F G H I J K L M N O P Q R S a b c d e f g h i j k l m n o p q r s t u v w x y z a b c d e f g h i j k l m n o p q r s t u v w x y z 3.6 ( ). = modnar. AQKQOR a b c d e f g h i j k l m n o p q r s t u v w x y z a b c d e f g h i j k l m n o p q r s t u v w x y z a b c d e f g h i j k l m n o p q r s t u v w x y z a b c d e f g h i j k l m n o p q r s t u v w x y z a b c d e f g h i j k l m n o p q r s t u v w x y z a b c d e f g h i j k l m n o p q r s t u v w x y z,., ( ).,.? 3.4 3.7 ( ). 24

(1) 147573952589676412927? (2) 193707721 761838257287. [ ] ( ) ( ) : : 68,24,70,24,100 : 17,40,49,40,53 H e n e r : H e n e r 17,40,49,40,53 : 68,24,70,24,100 ˆ. ˆ.., ( ). RSA 1,. 1 (Ronald L.Rivest), (Adi Shamir), (Leonald Adleman). 1977 4. 25

3.4.1 RSA ˆ. RSA 1.. 0 1 2 3 4 5 6 7 8 9 A B C D E 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 F G H I J K L M N O P Q R S T 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 U V W X Y Z a b c d e f g h i 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 j k l m n o p q r s t u v w x 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 y z,. ;!? + - * / < > = @ 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 3.7 2.. 1 2 p, q. 2 n = pq( > = ). 3 p 1, q 1 E. 4 (E, n). : 2. 1 p = 7, q = 17. 2 n = 7 17 = 119 > = 75 3 5 6 = 7 1, 5 16 = 17 1, E = 5. 4 (E, n) = (5, 119). 3.. = E % n. 17 5 % 119 = 68 ( H = 17). 4.. 1 f(n) = (p 1)(q 1). ( ) 2 D. E D % f(n) = 1. 3 = D % n. D E p 1, q 1. 1 f(119) = (7 1)(17 1) = 96. 2 D = 77., 5 77 % 96 = 1. 3 68 77 % 119 = 17 (17 = H ). 77. 26

77 = 1 + 4 + 8 + 64 68 1 % 119 = 68 68 2 % 119 = 102 68 4 % 119 = 102 2 % 119 = 51 68 8 % 119 = 51 2 % 119 = 102 68 16 % 119 = 102 2 % 119 = 51 68 32 % 119 = 51 2 % 119 = 102 68 64 % 119 = 102 2 % 119 = 51 68 51 102 51 % 119 = 17, 68 * 51 % 119 = 17, 17 * 102 % 119 = 68, 68 * 51 % 119 = 17 ( ). 3.4.2 RSA,.. 1.. 1. 2. 2.. 1. 2. ( ) :(E, n) = (5, 119) ( ) :(E, n) = (11, 95) D : p = 7 q = 17 : 17,40,49,40,53 H e n e r p = 5 q = 19 63,70,64,70,2 : D : 105,49,64,49,32 27

4 4.1 4.1 ( ). A B C D E F G H I J K L M N O P Q R S T U V W X Y Z 4.2 ( ),.. A B C D E F 4.3 ( )... A B C D E F 2 2 4 4 3 1 28

4.2 4.4 ( )..? (, 2. ) (1). (2). (3) 2. ˆ ˆ : ( ). : v.,. =. =. 1 ( ),. 2 ( ),. 3 ( ), 2. 4.5 ( ) (1),,,,.,.. (2). 29

ˆ ˆ :. :,. 4 ( ) (1) 4. (2) 5. 4.6 ( ) A, B, C,,,. 3. 4.7 ( ), 2.,, 2 1.. 2 3 4 5 4.8 ( ),.. A B C D D A C B 30

4.3 4.9 ( ) 3 3 10 100 2. ( ) ( ). 10 100,. 10 10 10 100 100 4.10 ( ) A, B, C, D, E, F, G 7.. 1. 1 7 7. 2. A G 3. 3. A G. 1 5 ( 1 B C E B C E, B C E ). 6 7. (1999 ) 1 2 3 4 5 6 7 BCE DEG BFG AEF ACG 4.11 ( ) A, 3 ( 4 )..,,., A,., (, 0 ), A? (1997 ) 31

4.12 ( ) (1). 1 (2). 5 ( ). 1878, 100. 1976,, 1200. 4.4 4.13 ( ), 1,. c C d g A e D a B b f 1. 32

4.14 ( ) (1),. (2) (1),. 6 ( ) 4.15 ( ).,? (1) (2) (3) 4.16 1? 33

4.17 ( ) 8. (, )., 1, ( ). v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 4.18 ( ),. (1) (2) (3) (4) 4.5 4.19 ( ) V 1, V 2, 1,,.,. (1) V 1 = {,,,,,, ( )} (2) V 2 = {,,,,,,,,,,, } 34

ˆ : ( ). 4.20 (2 ) (1) 0 1 4.,, 2 4, 2 2.. 0 1 1 0 01, 10, 01, 10 00, 01, 10, 11 ( ) ( ) (2) 0 1 8, 3 8, 3 2.. 000, 001, 010, 011, 100, 101, 110, 111 ( ) 35

00 01 10 11 4.21 ( ),,, 1 4 16 ( ). 4, ( ). 4, 1., 5, 2, 6, 3, 7, 4 ( ).? 1 2 3 4 A 2 3 4 A 2 3 4 A 2 3 4 A 2 3 4 000 001 010 100 011 110 101 111 36

5 5.1 5.1 ( ) 1, 2. (1),,. P (2),. P (3),. P 37

5.2 ( ) 5.1 (1) (3), P? (1) (2) (3) 5.3 ( ) XY, A B 1, AB.,. XY, PQ. XY, PQ, M N., MN. MN AB,., P M N Q X A B Y 38

5.2 5.4 ( ) (1) 2. 1 2 3 4 (2) 4. 1 2 3 (3) (2) 3, 5. 39

5.5 1 64. 5.6. ABC A B 5.7 ( ) C 1.,. 2.,. 5.8 ( ),............,. 40

5.3 5.9 OD = 5cm, DE = 4cm, ABCD CD. A 5.10,,. B O D E C 5.11 2 3. 5.12,. 5.13. (1) (2) (3) ( ) 41

5.14 OA 4 1 AOB, OA, OB. OA = 2cm,. B A O 5.15 2cm, 1cm 4. (4 ). 5.16 1 4cm, 8.. 5.17 ( ), 1 1cm, 1 12.. 42

6 6.1 1 ( ).,.. 6.1. (1). (2). (3). (4). (5). (6). (7). (8) 1 3 = 0.333, 3 10. (9) 3.1415926, 9 10. (10).. A, B, C,..., p, q, r,.......,.,. 2 A, B, A B 43

., A. A 1, B. B 0. 6.2 (1), (0)., (0), (1). A A. 2 ( ) A, B, A 1 B 0, A 0 B 1 A A 1 0, B A 6.2. (1). (2)., B A. 6.1 : (3) Dr.Hener. (4) Dr.Hener. (5). (6) ( ). p, p = ON = ( ) ( ) p. 6.1: NOT 1. 2. 44

,,. (1) 1, (1 ) 0, (2) A 1 A, (2 ) A 0 A, (3) A B A B, (4) A, (5) A B A B. 6.3 P : Dr.Hener.. 2. p :. q : Dr.Hener. P p q,. (p q ). 3 ( ) A, B, A 1, B 1 1, 0 A B, A B. ( ) A B A B 1 1 1 0 0 1 0 0 6.2: 6.3 a a W, a S, J,. (1) a. (2) a. (3) a,. (4) a,. (5) a OL. p q ON ON ON OFF OFF ON OFF OFF p q 6.2: AND 45

,,. (6) A B, (7) A B C A B C, (8) A A, (9) A 1, (10) A 0. 6.4. P :. Q :. P { p1 :. p 2 :. Q { q1 :. q 2 :. P p 1 p 2, Q q 1 q 2. p 1 p 2 P 1 1 1 0 0 1 0 0 q 1 q 2 Q 1 1 1 0 0 1 0 0 4 ( ) A, B, A 0, B 0 0, 1 A B, A B. ( ) 5 ( ) A, B, A B 0, 1 A B, A B. ( ) A B A B 1 1 1 0 0 1 0 0 6.3: A B A B 1 1 1 0 0 1 0 0 6.4: 46

6.1 ( 2!!),,. 6.4,,,. (1),. (2). (. = ) (3). (4). 6.5 1000., 2,. 1., Dr.Hener. 2.,. 1000,. p q L R L R p q 6.3: OR 6.4: EOR p q 6.3 6.4 ON(L) ON(L) ON(L) OFF(R) OFF(R) ON(L) OFF(R) OFF(R),,. (6 ) A B, (7 ) A B C A B C, (8 ) A A, (9 ) A 1, (10 ) A 0. 47

6.5,.,. P : Dr.Hener. Dr.Hener. 1 Dr.Hener.. 2 Dr.Hener.. Dr.Hener Dr.Hener 3 Dr.Hener.. 4 Dr.Hener.. A B Dr.Hener Dr.Hener 1 2 3 4 A B. 6 ( ) A, B, A 1, B 0 0, 1 A B, A B. ( ) A B A B 1 1 1 0 0 1 0 0 6.5: A 1, 0 B. 48

6.6 (1) P :,.. P. 1,.. 2,.. 3,,. 4,,. (2) P :,. 1,. 2,. 3,. 4,. 6.7. (1). (2). (3). (4). ( ) B A B. 6.8 ( ) (1) X. (2),,.. 6.6 A B A A B A B A B A B 1 1 1 0 0 1 0 0 6.2 ( ),,,,. 49

6.1. (1) A B (2) ((A B) C) ( A B) A B A B A B C ((A B) C) ( A B) 6.9. (1) (A B) (2) A A (3) A (A B) (4) A B (5) A B (6) (A B) (7) A A (8) A (A B) A B (A B) A A A (A B) A B A B A B (A B) A A A (A B) 6.10. (1) A (B C) (2) A (B C) (3) (A B) (A C) (4) (A B) (A C) A B C A (B C) A (B C) (A B) (A C) (A B) (A C) 50

7 ( ) 1 0.. 8 ( ) A, B,. A A, A A, (A B) A B, (A B) A B, A (A B) A, A (A B) A, A (B C) (A B) (A C), A (B C) (A B) (A C). 6.1,. A B C A B ((A B) C) ( A B), A B ((A B) C) ( A B). 6.11 (1). (2). 1 A B 2 A B 3 A B 4 (A B) 5 A B 6 B A 7 B A A B A B A B A B (A B) A B B A B A 1 1 1 0 0 1 0 0 A B A B (A B) A B 51

6.3,, Q :, R :. 6.12. 1 A B 2 A B 3 (A B) B 4 (A B) (A B) 5 (B A) A 6 A (A B) B 7 A A B 8 A (B C) 9 A B C 10 (A B) ((A C) (A B C)) A B A B A B (A B) B (A B) (A B) 1 1 1 0 0 1 0 0 A B (B A) A A (A B) B A A B 1 1 1 0 0 1 0 0 A B C A (B C) A B C (A B) ((A C) (A B C)) 1 1 1 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 52

7 7.1 p q 1 1 1 0 0 1 0 0 p p p q p q p q p q p q L R L R p q 7.1, x. p : 1100 q : 1010 (1) (2) (3) p p p q x q x q x 53

7.2, x. p : 1100 q : 1010 (1) (2) (3) p p q p q x q x x 7.1 L, x. p q r x p : 11110000 x : q : 11001100 r : 10101010 7.1 L, L = 6.1, L. p q x 7.3,, 7.2 ( ). p q x 54

7.2 7.4 ( ) 3.,,, 3.,... 7.5 4 1, 1, 1, 1. (1),. (2),. (3),. (4),. (5),. 4? 7.6 A, B, C, D 4. 4,. 4.. 1: B A. 2: A C 1. 3: A B 2. 4: B D 3. A B C D 1 2 3 4 55

7.7, A, B, C, D, E 5 3. 1, 1, 1., ) A, B, C 3 2 1. ) A, B, D 3 1. ) A, C, E 3 2 1.,. A B C D E 7.8 (1995 ) A E,,,,., A E. ) A 3,. ) B E. 2. ) D.. ),. ( ).. A B C D E 56

7.3,.,,., ( ).,,.,,,,.. ( ) N(x) : x. : x.,,,. 2, 2. 2 a, b. a, a.,,. 2. 2. a. a : : 1 1 1 0 0 1 0 0, 2.. 57

7 ( ) x P. ( P? ) Proof. [1] x P. N(x) P (i) x. (ii) x. [2] N(x) P x P. (i) x. (ii) x.,,, a, b. 58

. b, g. 7.9 2 1. 2. : : :,. 1 1 1 0 0 1 0 0 7.10,....,.,,.,. h, K. : : : 1 1 1 0 0 1 0 0,. 59

7.11,,., (a) (b),. (p). 1. 2,,.,,.,.. 1 1 1 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0,. 7.12 (1998 ) c, d..,.,. c. d. 1 1 1 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 c, d,. 60

7.4, A : ( )Dr.Hener. B : ( ). Dr.Hener (Dr.Hener ).., ( ) Dr.Hener..... [ D] A B (Dr.Hener ), A (Dr.Hener ). B ( ).. [ D] A A B B 9 ( ) A 1, A 2,..., A n B, A 1, A 2,..., A n, B,, A 1 A 2... A n 1 B 1, A 1, A 2,..., A n B. A 1 A 2... A n B A 1, A 2,..., A n, B., [ D]? 10 ( ) A 1 A 2... A n B. (A 1 A 2... A n ) B. 7.13 [ D]? A B 1 1 1 0 0 1 0 0 61

7.14. (1) A B B A (2) A B B C A C (3) A B C A C (4) A B C A C A B 1 1 1 0 0 1 0 0 A B C 1 1 1 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 A B C 1 1 1 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 A B C 1 1 1 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 7.15 (1)? 1. Dr.Hener.. 2 Dr.Hener... (2) ( ) (1),. A B 1 1 1 0 0 1 0 0 A B 1 1 1 0 0 1 0 0 62

. A B C A C 1 A B B C A C 2,? A B C C D A D (A B C) (C D) (A D),,. A B C A C 1 A D C D 2 7.16 A C.. A :. B :,,. C :........... p :. q :. r :. s :. t :., A C. ( I) 7.5 7.17 ( ),. X : Y :,.,. A, B,. A : B :,.,. 63

(1) X, Y A, B. X : Y : (2) X, Y.. 1,. 2,. 3,.... A B 1 1 1 0 0 1 0 0 7.18 (1) (4).. (1),. (2),. (3),. (4),. 64

7.19 1, 2, 3. (1). 1 : 2 : 3 : (2). 1995 : 1 : 2 : 3 : 7.20.( ) (1),. (2),. (3),. (4),...?. 65

A A.1 2, Magic 5 ( ) 1 ( 16 ). 2 1,. 3. 4. 5 (15 ). 6. 7.. 8,.. 9,, 2, 3,,,. 10 ( ). 11,. 1. 12 1.? 13 1,. A.1 7,,.,. 66

Magic 6 ( 1) 1 10, 10. 2. (i),. (ii),.,. (iii),.. 3. (10 ). 4 1. 5,., 1. 6. ( ) 7.. (i),. (ii),. (iii),,. (iv),,. 8,., 10. Magic 7 ( 2) 1, 2.. 1. 2 10,. 3 10. 1 10 1 1. 4 10 1,. 67

5 10., 2,.. 6 1,. Magic 8 ( 3) 2 6, 2,. 3., 1.. 1 20. 2 1. (i) 2 (1 2 1 ), 20. (ii) 2, 2. 3 20,. 4. (i) 1. (ii). (iii) 1, 1 10. 5.,. 6 10. 7 10 2,. 10. 8. ( 1.) 9., 1, 1. 10 1. 11?, 10. 1 Bob Hummer. 68

A.2 Magic 8 4, 1, 1,.,, 1..,,. 1 1 1 1 1 2 2 2 2 2 3 3 3 3 4 4 4 5 5 6 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 A.3,, 2, 3,. A.2,, 1. 1 1 1 1 1 2 2 2 2 2 3 3 3 3 4 4 4 5 5 6 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 69

A.2 Magic 9 ( ) 1 4. 2 9. 3 25. 4. 5?.. A.4? Magic 10 ( ). 13 13 1. 2 1,.. 3.,. 4,. 5,. 6,,. 7.. 8.. 9, 13.. 10,. 11 1. 1. 70

12... A.5. Magic 11 ( ) 1 ( 12 ). 2 4. 3. 4. 5, (J, Q, K) 10 2, 10. 10. 6 4,. 7. A.6, 4,,? Magic 12 ( ) 1 ( 9 ). 2 1. 3,.,. 4, 1, 10, 9, 8, 7, 1.,., 1,,,.. 5 4,. 6, 4.. 7,. 2 1 10. 71

Magic 13 (13 ) 1 52. 2, 13. 3 2., 13. 4. 5 3, 3,. 6, 3, 4 13 ( 10 ). 7 3 2,. 8 ( 2 ). 9, 1. A.7. Magic 14 ( ) 1 1 16 4 4. 2. 3,. 4. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 5. 6 1,. 7 4. 8., 34. A.8,. 72

Magic 15 ( ) 1. Magic 14. 2, 16. 3. 4,. 5. 2005 3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 6,. 7. 8. 9 1,. 10 4.? 11,. A.9. 73

A.3 Magic 16 (,?) 1., 12. 2 3 3. 3, 6,?,,.?? Magic 17 ( ) 1,.. 2 7. ( A.1) 3.,. ( A.2) 4,, 7! 1!! ( A.3) 5.,. ( A.4) 6,, 7! 1!! ( A.5) A.1 A.2 A.3 A.4 A.5 74

Magic 18 ( ) 1. 2. 3 ( ),. ( ),. 4,. 5, ( ). 6. ( ) Magic 19 ( ) 1 3,. 2,. 3. 4,.,!,. A.10.,,. 75

Magic 20 ( ) 1 6. 2,.,. 3,. 4, 1. 5,. (,.) 6 6, 1! A 2 3 76

B B.1 2, 3 1.. 2.. 3. 1. 77

B.2 (1) 6 2. (2). 6. 2... 78

B.3 79

B.1 ( ),. 80

Hener,.,,...,,.,.,.,,,.,.,.,..,.... Albert Kurt Hener (Dr.Hener) hener@cong.ac.jp God Door university graduate course takaharu.hirai@nifty.ne.jp 81

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