Nim
Transkrypt
Nim
3 5 100 ! "# $ %& ' () *) ! $" + , - - . / 0 1 2 . 3 + 1 100 1 0 3 5 4 2 1 . 5 - 6 3 - 2 - 27 1 0 2 08. + 20 4 08. 9 4 1 1 1 1 .1 4 1 - 100 0 1 2 3 . + : 8 2 4 1 + 0 4 1 18. 1 41 17 - 0 41 ; 12 14 8 0 8 < . > 8 13. 15. = 2 1 + ; 0 2 , - : 084 - . ? 1 0 4 1 @ 4 : 27 . A - 1 8 100 - 4 2 4 1 0 + - 1 - - 1 - 0 . n 8 2 n = 100 BC DEFGHDG I J 4 < 0 . = 9 84 - 7 K. 5 0 0 LMNO 2: 1 0 PQRST ; < = -. U8 0 @ 4 2 1 V ; 2: 2 0 -. 9 1 - 0 - ; : . J 7 -1 0 W X + 0 4 6 2: . 3 2 . Y 7 7: 0 1000 Z 22 4[. \ 84 2 7 ; 4 7 : 70 . ] - 1 0 :8 - 7 K . ^ :8 1 7 - :41 V84 . ? V - 2:7 K. ^ 01 0 1 7 0 4 2 1 . U8 @ 1 7 1 0 4 2: 4 - @ 2 02 7 V2 . A 1 :8 7 - . U 4 1 - + - @ 6 9 2 2:2 V2 _:` < 8 : - 2 1 + @ : . / - 0 2:1 7 2 - = 0 2 2:7 . + : @ : - 2: 1 24 @ - V 2 4 2: : _:`. 0 2: V 2:7 1 7 2 0 8 84 . a@ 4 1 0 1 2: : 7 . BC DEFGHDG I A 4 1 + 2 bcdefg higjkehigl d mnopmqrst joqiuvd g we xnoyzq zqnroe ejxne{|gy esnyhfq} jxq| wnpl xqs qzp tdowf~r|g} |m nywtk~ nygjt me xnopsnex|p medxvnoy|gt meopi g d joqiuqiu q rerqxys |gy vdgfghp xt |gi e wnqiu o nygjqg iue} xe qstnqx |gy jxdqnoq dg~s jopiu xntr|ehig W 2 2 . W. ¡¢£ ¤ ¤ ¡£¥ + - 1 8 - 0 - 4 1 7 2 46 ^ 7 1 7 0 4 2 . U8 1 : <6 8 100 =Z 7 - 1 6 9 - 100 110 - . \ 0 41 7 2 - . U :0 : 41 - 6 ? 8 0 4 4 ; ¦NM§ ¨© ª . Y 7 0 7 «¬ ªRT O ª §« 2:7 2 1 28 7 4 1 - V 1 § L Q® «¨ 2:7 . BC DEFGHDG I U8 2 < 2 + = . 1 0 U8 7 + - 1 0 4 :0 -. [ ¥ ¯ °±¡² ¤¡² ³ ´³ ¤ µ¶¡² · µ ¶¤ ¶¡¤³¸ ³ ´ ¹¶¤° ¤ µ¶¡² · ³¶¸º ¤ µ¤¶¤ ¶ µ¡ ¸ µ¡¶¸º ¸ ¥ »¤¼ ¤ ¶ µ¡µ±³ ³ ´³ ¤ ±³² ·¥ ½ ¤ µ¹¾ µ µ¶¤¸ ¼ ¾ ¹¶ ¤¸±²¡ ¶ ¶ ´¿³ ¶¡² ¤ ¾³ À ´¥ Á¥ ½¡ ¡ ° »¤ ¸±¤¡² ³ ´² µ¶±  ± ¡³ ¸ µ³¤³ [ ¥ / 0 1 @ 0 < @ 100 110 . ? - 1 0 0 4 . =à - :. /2 : - 72 2:7 - A 1 a2 <2 1 a 2: a1 a2 =Z (a1 , a2 ). 2 : - 2 2:7 1 2 a1 B 1 b2 b3 - 1 b 2: . (b1 , b2 )+ (b1 , b3 ) (b2 , b3 ) Z 1 9 2 - . 9 - 7 b 72 2 V 0 + 1 - 7 72 2 . \ A - 7 72 2 ; A B ; + B 0 -04 . ] :. 9 - L Q® T Ä § ÅÆ Ç ª § ¨Æ . 9 1 0 0 - < . : 0 = 4 7 1 - k = k0 +2k1 + È 1 1 . ^ 4k2 + 8k3 + ... + 2n kn ≥ 0 kj ∈ {0, 1} n 0 - 4 kn = 1 1 0 . j > nZ j = 0 U8 1k 2k = . k0 +2k1 +4k2 +8k3 +... k=0 kj = 0 U8 0 7 1 j0 84 2 a b 7 Á K 7 . c = a⊕b ? 8 . U8 4 _:` 1 c = |aj − bj | 0 _ -` 1 cj . 3 1 cj aj b j _ ` - @ ⊕ 7 É? < . ¨Ê R ¬ Q L«Ë¨ ÌÍ . / _ 7 ` - 1 0 8 = . cj aj b j 2 BC DEFGHDG I ^4 1 1 1 . 7 ⊕ 4 3 ⊕ 6 16 ⊕ 5 13 ⊕ 22 3 5 21 27 ¥ a 41 0 . ^ a⊕b = b⊕a (a ⊕ b) ⊕ c = a ⊕ (b ⊕ c) 2 1 1 . ^ - a ⊕ a = 0 a ⊕ 0 = a |b − a| ≤ a ⊕ b :84 . BC DEFGHDG I /2 . 9 7 . 0≤c<a⊕b a⊕c<b b⊕c<a ¯¹¶¤¸²¡ µ ¾¤¤¡ ¼ ¾ c d = a ⊕ b ¥ ´ n ¾± ¤³²² µ¶¸ ¹°¤º ¸ ¸ µ¡ ¸ ¤¥ ¤ ´ cn 6= dn ¼ cj = dj ±¼ j > n ¥ ¯¤¶° c < d ¡²³¸²¡ cn = 0 dn = 1 ¥ Τ ²¡ a0 = b ⊕ c b0 = a ⊕ c ¥ ϼ j > n ´± a0 = bj ⊕ cj = bj ⊕ dj = bj ⊕ aj ⊕ bj = aj ¥ j ¯±¾¤ b0 = bj ¥ Ð ±³¸ ¤¡ a0 = bn ⊕ cn = bn ¤¼ ¤ ¥ ¯¤¶° an 6= bn ¾¶² b0n = an n j £ ¶ an = 1 ¼³¾ bn = 1 ¥ ÑÒ¼ an = 1 a0 < a Ó ¶ µ ¶¤¡² ¶¡µ±³ b0 < b ¥ an ⊕ bn = dn = 1 ^7 - 2:7 1 8 2 1 2: @ - 7 . 9 2: V 1 0 2: - 0@ v 1 7 - 2: 1 7 - . BC DEFGHDGvI J 4 1 0 @ - 7 - K 1 @ 2: - . BC DEFGHDG I 94 2:7 2 2 72 7 W. Ô¡³¤ [Õ 1 µ¶ ¶ 0, 0, 0, 1, 1, 1, 2, 2£¥ Ô¡³¤ Á Õ Ö [ ¥ ×ØdqwqÙ gy q tjxqfe|ywe eo|qioy|gq |q jt~ rdv sedÚÛ |gysxvnop t{pdq Ú Ü l q +2 Ý d dgyft ~opsqiu mnewnqedq|gq t{pdq jg~ o|qst Ü Þ Ý ß > 8 4 : - 7 72 @ 2 . àC DGáâFGHDG ãI A -@ 7 - 72 2 1 2 2:2 . ^0 8 - 1 0 7 . 0 - 7 - 72 2 Z - :0 2:7 1 2 8 K . àC DGáâFGHDG ä I å 2:7 :0 7 2 2:7 2 2 . æ ç Cè â I 3 - . å @ :0 2:7 V2 8 7 1 = 0⊕0 - W 2. 0 :0 0 2:@ 7 2 :701 0 0 2 7 1 7 2ab 2. /2 . ? 8 0 1 . 9 7 0 ≤ c < a⊕b a = c ⊕ b b0 = c ⊕ a . / 4 :8 _ `1 2 a0 ⊕ b = c = a ⊕ b 0 . 3 - - 0 4 ⊕ 1 0 0 . a0 < a b0 < b a < a ? 1 0 - 2: 0 2: 0 1 0 2 2:7 a 2 0 . å < a b :0 0 1 . c = a ⊕b J 8 - 1 0 0 c < a ⊕ b 0 2 2:7 . ^ 41 0 0 4 2 c . 9 1 0 2 K ⊕b 2: a 70 1 8 8 1 a x⊕b = a⊕b . ^ - 2 x = (x ⊕ b) ⊕ b = (a ⊕ b) ⊕ b = a K. 9 7 81 0 0 V8 7 W. a⊕b 3 0 4 1 0 8 ; 1 1 + 0 @ 100 110 4. 9 1 2: 100 1 . 1 110 2 BC DEFGHDG I 9 é « ® < . 2 7 = . - . 5 2 1 . U 41 + ê < - - 6 U 4 = 26 ë¶ µ¶±ì ° ¡ µ¸±¡¤ ³ k ²¤² k ¥ Τ ° »¤ ² ¶¡¡¶¸º º ¶±¡ ¡¼ ¶±¡ ±¡ ³² ±¶¹¸¶ ¼ ¾ ²¤ ¤ µ ¹¼¤¡ ´ ´ ¸ ¤¶¥ ½¡¡¶ ³ ´ ¹¡ ³¸ ³²¥ ¯¶±²¡ ° º ¡ ¡ ¼ ¤Ò a, b, c ¥ Ф¸±³¸²¡ s = a⊕b⊕c ¥ ÑÒ¼ s = 0 »¤ µ ´¡¾ °  µ²¡¼£¥ ½ µ ¶¤¡² ¶¡µ±³ ´± ¤¸²¤¸ ¸±¤ ¤¹¶¤Ò Õ a ⊕ s < a ¼³¾ c ⊕ s < c ¶¡¤ ¶±¤ Á ¼ ²°¤ ° ³±¶±¤ì ¾µÒ±¤£¥ ÑÒ¼ ¤µ¥ b⊕s < b »¤ ±¸²³¸ a − (a ⊕ s) ²¤ µ¶ ³¥ a⊕s<a BC DEFGHDG I í / 70 - 0 1 0 1 7 0 4 2 1 . U - 0 6 Î ±¶ ²¤ ¾± ¤¶¼ Õ ¡ ¤³ ¸ ¡ ¸ ² ¸ ¶ ¡î ¤¼¤¸ ² ´¡¾ ° °±¡ ¼ ¡ ¤¸¶¡°¸ ¸±¤ ²¥ ½ ±¡ ¡º ¸ ³² ±¶¹¸¶ ¼ ¾ ²¤ ¡¼ ¸±¡¤£ ¸ ±±¶¸ ¸±¡¤¥ ï ¸ ¶ ±¤¡ ¤ ° ± ²î ²¤³ ±¡ »¤ ¤ ¡¼ ¸±¤ ¤¸²¤¸ ±¶² ²¤²¥ ½ ¹¶ ¸Ò¼ ¶ µ¶¤¸ ¶¸ ¡ ² »¤ µ¹°¤¿¾¡ ¶ ²¤¤¸ ¶ ¸±¤ ² ¤ ±¶¹¥ ϼ µ³î ¤ º ¸³° ¶¡²³¤¥ ¡ ¤ ¶¡¤ ¤ ¸ ¸³° ¾µÒ±¤º ¤¶¤ ¸º ¶±¤ Á¥ ^ W. 0 @ - 7 / 7 . 2 - 1 - 7 . Y- 2 0 4 <24 1 5 = 2 1 < / . 37 1 7 308 = . í - 8 - @ - < 2 - . ^ = 4 1 : . 2 2 :2 7 0 4 < . < Z / . 2 . = = 3 5 U 41 6 / W. 7 2 <8 . ^ = 4 - . n ] : . 3:70 - 1 0 0 2 =0 7 n 8 0 1 . ^ - n ð n > 0 ñûñü òûñü óûòü ôûòü õûóü öûóü ÷ûôü øûôü ùûõü úûòü òñûòü òòûöü òóû÷ü òôûôü òõûôü òöûóü ò÷ûóü òøûòü òùûòü òúûñü ûñüñ ññ òò ó ôôó õò ö÷ò ôô óó ò ñò ûñüò ñò óóò ôô õò ö÷ò ôô óó ò ñò ûòüó ûòüô ûóüõ ûóüö ûôü÷ ûôüø ûõüù ûòüú ûòüòñ ûöüòò û÷üòó ûôüòô ûôüòõ ûóüòö ûóüò÷ ûòüòø ûòüòù ûñüòú ññ ôô óó ññö õø ó ôôó ññ ò ñô ôó óö ññ õø ó ôôó ññ ò ññ ñ òò òò ÷ô ô÷ ôø ôø õò õò ññò ññò ôô ôô ó ó > ññ øó ÷óö ññ òò ó ôó ñø ñ ó ö ñ ÷óö öóò ñõø ñõø ññ øø óó óó òò ÷÷ ôô ôô ó ö ñ ñ ôó õö ñò ñò > 7 ñô ñ ÷÷ öö øø õõ õõ øø ö ÷ ññ ñ òò òò ññ ó ó ô ôó ôó ôó _` ñô ñ ôó ñò ñò ñ W. 2 7 1 7 2 0 84 8 . 9 1 8 0 4 2 1 7 n72 7 8 1 8 1 - - a b a < n b < n ; 7 2 7 2 . , - b a . n 31 : : . 3 : :8 . ^ 1 0 0 4 : . 19 2 4 å 7 :8 0 1 8 0 . ^ 2 0 :8 4 1 - . 3 : 3 :8 1 - 0 - . 3 7 1 0 4 1 1 1 2 1+1 < . :8 1 1 0 - 1 =Z 0 1 0 . 3 0 4 < 4 3[1] -2 1 . å - . 3 8 0 5 2 4 = 2 +1 1[1] 11[0] 1 1 - . í 6 3 . 4[2] 3 + 1[1] 2 + 2[0] 2[1] 1 + 1[0] \ 0 4 - . ? @ i @ - :8 < @ i i ý . / -i @ @ i 7 = 2 . + 4j - :@ 8 1 : 7 7: -2 k (0, k − 4) : 7 . å (k −4, 0) k −2) (k −2, 0) :8 1 7 (0, 2 2 . k þ ] -8 : :- . / :04 - . 9 + :84 < 70 :8. /- 1= 1 . 90 ; 7 ; 27 85 285 0 - . \ 2 7 1 : 0 :4 - @ + , . ^ 1 0 + : : . , 1 0 84 5 + 12[4] . ^0 0 1 - , 5[2] @ ⊕4 = 6 2 84 . 12[6] 6⊕4 = 2 ^ 1 , 1 0 0 4 84 4 1 4 5 + 7 + 3[0] . 9 : 1 8 + 2[0] 5 +6 W. + 2 4 +1 :8 6 > 2 - 841 0 4 4 8 20 1 - 0 2 , 4 @ - 8 . BC DEFGHDG I ] 2 + - @ 1 8 n 4 2 :2 76 1×n 2 6 ½¡ ¡ ¾¶¶ì ¤µ¤ ³±¶±¤ì£ ° ±¼ n ≥ 3 ¡º µ 1 × n ¸ ±µ¶±¤ 2 0 3 1 ¸Ò¼ n ±¸ ±¼¤ µ 4 ¹¶¤º 0 1 2 3 ¥ ¯ º¶ ¡¡ 0 0 1 ±¼ n = 0, 1, 2£¥ / 74 / - . 2 - - 7 . BC DEFGHDG I U + 6 ÿ ² ¸ ¶ ¡¤¼¤¸ ² ¢