ޥ
ñäΤʤ...ʤ®Ȥͽ۳Ǥ
PowerBook G4 727ڡ 0:10:06 MAIL:masaru-y@sansu.org HomePage:27914
̥ʡ ֥꡼
סΣ̡̻٤ᡩ
Ǥ̥󥭥󥰤񤷤ΤˡǶꥢ륿໲äǤ뵡񤬸ꡢޤޤʤäƤΤǡ򤷤ҤȤǤ
727ڡ 0:11:26 27915
ä
991
9998911892
9999998901092793

( o)ϥ

920180
Ȥޤ
Ǥ
Ϥ 727ڡ 0:12:45 HomePage:Ҳʲ27916
sugitakukun
#27916
⤽ʴޤμ890019989145ʤǤ͡
ơϲͤƤޤˡġ
727ڡ 0:14:23 27917
ӥ塼ƥե롼WITHॿɲ
ʤȤʤӤߤ礦ä͡

ʤǤʤ뤫狼äƤʤ

⴪ǤȤޤ
ӥ塼ƥ硡 727ڡ 0:14:45 27918
ӥ塼ƥե롼WITHॿɲ
ʤǤ

21ܤˤ顢ԤäȤȤǤ
ӥ塼ƥ硡 727ڡ 0:15:41 27919

ϣǤϡ
727ڡ 0:15:55 27920
㡼ߡ
ȤꤢäݤޤΤ
ޤˤ褯狼ޤ󡣺ͤޤ
727ڡ 0:20:58 MAIL:ojamaru@amber.plala.or.jp 27921
DrK
ȤˤƤäݤǤ
٤Ͼγڱࡡ 727ڡ 0:23:43 MAIL:satoka@star.odn.ne.jp 27922
㡼ߡ
n Ĥ¤ 9n ˤʤΤȤ顤ľŪˤʤȤƤޤޤ
ȤΤ⡤n ĤȡѤ 10*10^2**10^n=10^{n(n+1)/2} ˶᤯
Ϥۤ n 2 ˤʤϤǤƷ¤ n 1 ȤȤϡ
n ζ˸¤ n ĤγƷοʿ 0 ˼«ޤ
ˤԹΤ褤ȤϵʤǤ礦
Ȥ櫓ǡ180 Ȥ顤 n=20 ΤȤޤ 180 ˤʤ ϤǤ
727ڡ 0:43:15 MAIL:ojamaru@amber.plala.or.jp 27923

ºݤ˷̵¤Ƿ׻ȡȤǤʤˤʤäƤޤΤǤְ㤤Ǥ礦
727ڡ 1:07:54 HomePage:27924
ޥ
𤷤ޤ줫ͼǤ

⤷Ƥޤߥä.......äơߥä褦ʵƤ...äȸڤޤ
PowerBook G4 727ڡ 0:51:51 MAIL:masaru-y@sansu.org HomePage:27925
ޥ
㤤Ǥäפʵm(__)m
PowerBook G4 727ڡ 0:53:13 MAIL:masaru-y@sansu.org HomePage:27926
ݤ
Ǽ
ܤ꤫餺褦ʵ⤹ΤǼ򤬡ޡɥޥ
٤ߤʤ
Ƹɡ 727ڡ 0:56:27 MAIL:hopes@mba.ocn.ne.jp HomePage:ȣϣУţӤ褤Ȥ٤Ϥ27927
mhayashi
990 ˤʤäͤޤ󤫡
727ڡ 1:00:47 HomePage:M.Hayashi's Web Site27928
ޥ
ޤäְ㤨ޤm(__)m

ߣߣߡߣ

ȤȡǸοʣ³ˤ򤫤ľˤǤˤʤȴ㤤Ƥޤ...m(__)m
PowerBook G4 727ڡ 1:03:31 MAIL:masaru-y@sansu.org HomePage:27929
ޥ
#27928
990ȤϷ빽ϤƤޤʥץ޷Ϥ¿ΤǡʤΤ
PowerBook G4 727ڡ 1:09:26 MAIL:masaru-y@sansu.org HomePage:27930
㡼ߡ
#27928 ʬ⤽ʤޤ
̵Ƿ׻ͤƻ˷׻餽ʤޤġ
𡡡 727ڡ 1:09:59 MAIL:ojamaru@amber.plala.or.jp 27931
ޥ
ԲǽäݤǤ...ΤϤ줫ͤޤm(__)m
PowerBook G4 727ڡ 1:14:39 MAIL:masaru-y@sansu.org HomePage:27932
kasama
եȤˤ餻ޤ

9*99*999*9999*99999*999999*9999999*99999999*999999999*9999999999*99999999999*999999999999*9999999999999*99999999999999*999999999999999*9999999999999999*99999999999999999*999999999999999999*9999999999999999999*99999999999999999999
=
890010099998999000001088910111109988888789998900113232311998767666566681133536564421986643232355881354768674530785532323466902244656353310766656667679002132323109988999878888900111110088910999998999000010099891

Ȥʤäơ990Ǥϡ
727ڡ 1:23:13 27933
ڤڤ
ǥ󥭥󥰤Ƥ

ʤߤ˷׻990ˡϤޤ
727ڡ 1:28:08 27934
kasama
ץˤ餻Ƥߤޤ

import java.math.*;
public class Question551 {
public static void main(String args[]) {
String s = "";
BigInteger b = BigInteger.ONE;
for (int i = 0; i < 20; ++i) {
s += "9";
b = b.multiply(new BigInteger(s));
}
int count = 0;
String sb = b.toString();
for (int i = 0; i < sb.length(); ++i) count += Integer.parseInt(sb.substring(i, i+1));
System.out.println(count);
}
}
727ڡ 1:29:50 27936
ޥ
Ȥꤢ990Ȥ褦Ǥˤϡʼ׻ʳǡ˻ǤƳˡʬޤ󡣡ʤɤʤʬˤʤäСйǤ
ʤ櫓ǡȤꤢɽϡ990ȤƺʤޤβˡĤޤǤϽդϤʤȤˤޤʾ̤˿ޤˤޤԷǼĤˤϥѥɤʤ֤ˤޤΤǡոȽ񤭹ǤФȻפޤ

Υߥϡ#27929˽񤤤褦ĶŪʴ㤤Ǥ¤ϽľޤǤϡߣߣߡʣġˤˤʤäƤơ֤ȥĶƤޤפȵդƺηѤȤ˥ޥ̥äǤ𤱤ʤ...ʤ⤽⣲ʲʤ...ȤΤְäƤ⤦ɤ褦ʤǤȻפΤǤ...
PowerBook G4 727ڡ 1:30:28 MAIL:masaru-y@sansu.org HomePage:27937

ٷ׻Ƥ990ˤʤΤDzȤƤΤȻפä
727ڡ 1:51:35 27938

ٷ׻Ƥ990ˤʤΤDzȤƤΤȻפä
727ڡ 1:53:03 27939
sugitakukun
ߥǤ
#27926ǡ1ʬޤǸڤǤƤΤǰɬʲˡΤȡġ

ʤ󥭥󥰤ΤۤϡNo ContestפǽƤޤ
727ڡ 1:59:02 27940
Taro
ƤƳϻ֤˿ᤴޤ
Τ999999890109180ͽۤΤ³Ǥ
ĤΤ褦˥ץǥå褦ˤ⡤ϤĤäƤޤǤ
727ڡ 2:09:33 27941
̥ʡ ֥꡼
#27915
̤ӤäΤǤ...褹ȻפäƤΤǤäǼ(^^;
ѥ̵ǤȤϿڤǤ͡񤭹ɤٶƤޤ
727ڡ 2:37:47 27942
ߥƥ
ʤȤ˸¤äƥΡƥȤȤϡ(^^;;

ͤˤʤ뤫ɤʬޤ󤬡
44ޤǵƤߤޤץǤ
918275454
99108153189216
315333396495603
639702810864990
9901161120612781422
14851647168319351926
22322385255627182889
29613150341135553600
3852411344104446

100 x=1
110 for i=1 to 20
120 x=x*(10^i-1)
130 s=0
140 x0=x
150 for j=1 to i*(i+1)/2
160 p=mod(x0,10)
170 s=s+p
180 x0=int((x0-p)/10)
190 next j
200 print i,s
210 next i
220 end
727ڡ 3:40:14 27943
ߥƥ
for i=1 to 20 Ƚ񤭤ޤ
20ϾͤʤΤǡŬʿǹޤ(^^;;
727ڡ 3:43:17 27944
錄
󤫤黲ͽäΤǤsugitakuΤȤǤǤ
Ȼפǣ졩ñʤȻפäɤߴְ㤤ʾǤꤹ
ǡߥɥޥǤ
󤳤򤷤ʡ
727ڡ 4:09:20 27945
uchinyan
ϤϤ褦ޤѤǤ͡
īĤΤ褦褿顢Ĺ֤ͤӡפ
ͤƤߤޤϤ褯ʬʤǤ
ץǥå 990 Ǥǽ顢ץߥDzΤ 180 ˤʤäΤϡʥ (^^;
ͥνȡ 727ڡ 9:08:03 MAIL:uchi@sco.bekkoame.ne.jp 27946
ӥ塼ƥե롼WITHॿɲ
äѤꤽʤ˴ŤФȤʤ͡

ϡ
ӥ塼ƥ硡 727ڡ 11:41:08 27947
Σ
ޤ͡
727ڡ 13:04:16 27948
ֿء׾ι
Mathematica ǥץ򤷤Ƥߤޤ

z = 1;
Do[z *= (10^i - 1), {i, 20}]
m = IntegerPart[Log[10, z]] + 1;
n = 0;
Do[{k = IntegerPart[z/10^(m - i)], n += k, z = z - k*10^(m -i)},{i, m}]
n

ǡǤ뤳Ȥʬޤ
727ڡ 13:15:25 27949



727ڡ 14:56:49 MAIL: 27950

727ڡ 14:58:16 27951

727ڡ 14:58:27 27952
ޤ륱
ϡ̤η夬ȰꤷƤΤǡߡʣܣˡࣲˤʤäƤȤϤ狼ޤʤ줬ߣߣߡȤϤʤ
ʤΤǡĤοޤǤȡߣࣲᣲǣοˤʤ롣
飹ޤǤοŬˤȤơ֤Σ򤫤ƣ
ޡΡοˤʤʡ
ʤߤˡΣϣܿʤΤǡ­⣹ܿˤʤ롣

ȡ錄ƬǤ狼ΤϤդޤǡ

ºݤϡˤäRubyǥץ񤤤990ޤ

m=1
(1..20).each do |i|
m*=("9"*i).to_i
end
p eval(m.to_s.split("").join("+"))

ꡢïäȤäơ˴Ƥʡ

Ruby
Τ褦 920¤ϡ
("9"*20).to_i
äɽǤ㤤ޤ
꤬С̵¡
͡Ǥ硩
727ڡ 15:26:00 MAIL:take4310@mobile.email.ne.jp HomePage:ޤ륱27953
⡼ޥ
ʤ
ϤƤޤʤν뤵ʤΤˡϲ򤱤ʤ



9*99*9999*99999999*9999999999999999

ʤ򤱤ʵޤ󤫡
錄ϤޤǧƤޤ󤱤ɡ(^^;
Ƭ뤵Ϥʤǡ
727ڡ 15:50:27 27954
uchinyan
#27953
ʤۤɡRubyäǤ͡äȻȤäƤߤʤޤ ^^;
ͥνȡ 727ڡ 17:32:38 MAIL:uchi@sco.bekkoame.ne.jp 27955
DrK
¤ϡ7ޤdzݤ碌ƸΤǤ礭ͤˤʤäƹԤޤ줾οˤˡ§ꤽʵΤǡˡϤȻפΤǤϰϤĶƤȻפޤ
٤Ͼγڱࡡ 727ڡ 17:34:59 MAIL:satoka@star.odn.ne.jp 27956
ޤ륱
#27955
͡褵Ǥ硩ե꡼
ʤߤˡΥץηιפƤȤ
p eval(m.to_s.split("").join("+"))
mʸѴФФˤƤ"+"ǤĤʤ򼰤ȤƷ׻äƤȤƤޤ
Ǥ⡢ȡɴȤŸʤФʤ餺θ³
ʤΤǡPCǤϡ"9"184ĤޤǤ׻ǤޤǤ
ʤߤˡ184ΤȤγƷιפϡ76050Ȥʤޤ

ǡιפ eval Ȥ鷺롼פǥĥķ׻褦񤭴Ȥ(߷׻ʹǤ)20090387250140922ȴĥäƷ׻³Ƥޤ

def ketack(m)
s=0
while(m > 0)
s += m%10
m /= 10
end
return s
end
727ڡ 18:33:57 MAIL:take4310@mobile.email.ne.jp HomePage:ޤ륱27957
ޤ륱
θ塢
300塧202941
350塧276264
ʾ׻̣äơ
⤦ʡ
727ڡ 18:59:57 MAIL:take4310@mobile.email.ne.jp HomePage:ޤ륱27958
CRYING DOLPHIN
#27954
ʤۤɡΰ̤򶴤ǿ¤Ӥ˵§ꤽǤ͡
Ƥΰ̤οɽ(10^n)ݣɽ뤳ȤѤ
Ǥʵޤ֤ͤȵϤޤ̡
727ڡ 22:17:50 HomePage:ȤƻȤ27959
⡼ޥ
#27954
ƻ˿ơ
+ : 1,2,4,7,8,11,13,14,16,19,21,22,25,26,28,31
- : 3,5,6,9,10,12,15,17,18,20,23,24,27,29,30
Ĥޤꡢ
1-1-1+1-1+1+1-1-1+1+1-1+1-1-1+1-1+1+1-1+1-1-1+1+1-1-1+1-1+1+1-1
0 8 9 0 9 1 0 8 9 1 0 9 0 8 9 0 9 1 0 9 0 8 9 1 0 8 9 0 9 1 0 9
ȤʤϤ顢
9*11+(8+1)*5=9*16=144 ˤʤΤʡ

ʾηˤʤäȤϲ餫ε§ʤȷ׻Ѥˤʤꤽ
ΤȤ§Ϥ錄ˤʬޤ󡣡(^^)

916ϢͤȤ9*16 ʤΤǡҤäȤ顡9 2^n ϢͤȤϡ
9*2^n ˤʤΤʡ
727ڡ 23:38:18 27960
uchinyan
#27954, #27959
ΤˡʤСܡǤ͡
9*99*9999*99999999*9999999999999999
ξγݤη̤γƷο¤ϡ144 ʡ
̤ˡ
9 * 99 * 9999 * 99999999 * 9999999999999999 * ... * 99(2^n )99
ξϡ9 * 2^n ˤʤꤽǤ

ϡ
Ф餯񤫤ʤǤޤ礦 ^^;
ͥνȡ 727ڡ 23:30:06 MAIL:uchi@sco.bekkoame.ne.jp 27961
Ȥ
äƤ
727ڡ 23:43:17 27962
weapon
a_n=a_(n-1)-1+9*(2^n)-a_(n-1)+1
728ʶ 0:00:42 27963
ƣ
֤ҤȤǤȤ
728ʶ 0:30:03 27964
ä


728ʶ 0:30:32 27965
⡼ޥ
#27961
ޤϻפĤʤǤ
ǼˡǤΤʡ
9*99*9999*99999999*9999999999999999 ʤǤȻפäΤϡ
(10-1)(10^2-1)(10^4-1)(10^8-1)(10^16-1) ʤ顢ŸƤƷͤ٤ 1 -1 ˤʤ뤫׻ǤȻפäǤ(^^)
728ʶ 1:42:17 27966
uchinyan
#27966
䤬פĤƤˡϡʤʥ֤ʷ׻αĹΤ褦ˡǤǼˡȤȤޤ
weapon#27963ȡ̤Ʊˤʤޤʤˡ֤äפǤϤʤǤ (^^;
ݥȤϡ⡼ޥ󤵤񤫤Ƥޤ99(2^n )99 = 10^(2^n) - 1 Ǥ͡

ȡ񤭤ޤ⡼ޥ󤵤ήˡǤʾǤǤ
ͥνȡ 728ʶ 10:45:12 MAIL:uchi@sco.bekkoame.ne.jp 27967
⡼ޥ
#27967
ޤʤǤ

(10-1)(10^2-1)(10^4-1)(10^(2^n)-1)
ιϡn+1ࡣ
n+1फȤˡϡ
n λn+1 ϴʤΤǡ
(n+1)C1+(n+1)C3++(n+1)C(n+1)=2^n+1
(n+1)C0+(n+1)C2+(n+1)C4++(n+1)Cn=2^n-1
ƱǡϡƷο¤ϡ 1-11-11-11-1 ƱϤ顢
09090909 ξ9*2^n ƱϤ

ºݤοϡ
1-1-11-111-1-111-1-11-111-1
ǡ089091089108909109 ʴοˤʤäƤ롣

n λn+1 ϶Ϥꡢ
(n+1)C1+(n+1)C3++(n+1)Cn=2^n-1
(n+1)C0+(n+1)C2++(n+1)C(n+1)=2^n+1
ƱʤΤǡϤƱ9*2^n
ºݤοϡ
1-1-11-111-1-1-1-111-11-1-11
ǡ089091089891090891 ʴοˤʤäƤ롣

ޤʤ(^^;;
728ʶ 15:47:14 27968
ä
ĤijݤƤäƤ狼äȤϣܤޤǤ䤱ɡפϣܿ
728ʶ 22:36:09 27970
uchinyan
#27968
⡼ޥ󤵤ήΥץǤξϡƱǤ
ľŪˤ餫ʤɡäˤǤ͡

#27954γĥ
9 * 99 * 9999 * 99999999 * 9999999999999999 * ... * 99(2^n )99
= (10 - 1) * (10^2 - 1) * (10^4 - 1) * ... * (10^(2^n) - 1)
= (10^3 - 10^2 - 10^1 + 10^0) * (10^4 - 1) * ... * (10^(2^n) - 1)
= (10^7 - 10^6 - 10^5 + 10^4 - 10^3 + 10^2 + 10^1 - 10^0) * ... * (10^(2^n) - 1)
= 10^(2^(n+1) - 1) - 10^(2^(n+1) - 2) - 10^(2^(n+1) - 3) + 10^(2^(n+1) - 4)
- 10^(2^(n+1) - 5) + 10^(2^(n+1) - 6) + 10^(2^(n+1) - 7) - 10^(2^(n+1) - 8)
+ ...
+ (-1)^(n-1) * 10^3 + (-1)^n * 10^2 + (-1)^n * 10^1 + (-1)^(n+1) * 10^0
ǡݤͤϡ˥ޥʥȤͶɽˡǤθܤϡ2^(n+1) γƷ夬
+1, -1, -1, +1, -1, +1, +1, -1, ..., (-1)^(n-1), (-1)^n, (-1)^n, (-1)^(n+1)
ȤʤȤߤʤȤǤޤ
ϡĤĤ +1, -1 ȤˤʤäơŬ -1 äΤ֤Ƥޤ
֤ϡ2^(n+1) * 1/2 = 2^n ˤʤޤ
ο -1 ȤΤϥʥ󥻥󥹤ʤΤǡºݤˤϡ10 - 1 = 9 ΰ̣ˤʤޤ
ˡºݤη׻Ǥϡ10 - 1 = 9 Ԥݤˡľη夫 1 ڤƤ뤳Ȥˤʤꡢ
-1 ηľηο -1 ޤ
(äơºݤˤϡ2^(n+1) - 1 οˤʤޤϡεǤϡפǤϤޤ)
ơ᤿ΤϡݤͤǤϤʤͤγƷοäͤǤ
ǡΤ褦ʥȥåȤޤĤޤꡢäϡ
ο +1 λϡ+1 Ȥʤޤ
ο -1 λϡ10 - 1 = 9 Ȥʤޤľηο -1 ɬפΤǡ䤦 9 - 1 Ȥޤ
٤Ƥη¤ȡ9 - 1 -1 ϡ礦ɰľηο 1 ȤбƤޤ
ΤȤͤ碌ȡͤϡ
+1, -1 ȤǤϡ(+1) + (9 - 1) = 9 ˤʤޤ
-1, +1 ȤǤϡ(9 - 1) + (+1) = 9 ˤʤޤ
ºݤη׻Ǥϡ-1, +1 ȤƬ 9 - 1 -1 ϡȤκǸǤäΤǤ
ʬǤϡ +1 Ǥä׻ڤˤʤΤߥǤ
ΤƷο¤ʤΤǡΤ褦ʥȥåȤޤ
ǡɡͤϡȤη֤ 2^n äΤǡ9 * 2^n ˤʤޤ
ͥνȡ 729ڡ 9:15:01 MAIL:uchi@sco.bekkoame.ne.jp 27971
uchinyan
#27971³
ʥ֤ˡǤ
ܤ¤Ӥޤ¤ϡ

9 * 99 = 9 * (100 - 1)
0900
-009
----
0891

9 * 99 * 9999 = 891 * (10000 - 1)
08910000
-0000891
--------
08909109

9 * 99 * 9999 * 99999999 = 8909109 * (100000000 - 1)
0890910900000000
-000000008909109
----------------
0890910891090891

ʤɤ򡢰̲Ǥ

#27954γĥΤʥ֤ʾ
a(n) = 9 * 99 * 9999 * 99999999 * 9999999999999999 * ... * 99(2^n )99
Ȥޤ
99(2^n )99 = 10^(2^n) - 1
ʤΤǡޤ
a(n) = a(n-1) * 99(2^n) )99 = a(n-1) * (10^(2^n) - 1) = a(n-1) * 10^(2^n) - a(n-1)
ǡ̤ˡn >= 0 a(n) 2^(n+1) - 1 ǡa(n) β 0 ǤϤޤ
˴ؤƤ餫
˴ؤƤϡa(0) = 9 餫
̤ a(n) ϡa(n-1) 2^n - 1 Ȥȡ
a(n-1) * 10^(2^n) (2^n - 1) + 2^n = 2^(n+1) - 1 ʤΤǡ餫Ǥ
ƤΤȤˡa(n) γƷͤƤߤޤa(n) i a(n,i) Ƚ񤯤Ȥˤޤ
i ϡ0 ϤޤΤǡ0 <= i <= 2^(n+1) - 2 դޤ
ȡa(n) ϡꡢ
a(n)
= 10^(2^(n+1) - 2) * a(n,2^(n+1) - 2) + ... + 10^(2^n + 1) * a(n,2^n + 1) + 10^(2^n) * a(n,2^n)
+ 10^(2^n - 1) * a(n,2^n - 1)
+ 10^(2^n - 2) * a(n,2^n - 2) + ... + 10^1 * a(n,1) + 10^0 * a(n,0)

a(n)
= 10^(2^(n+1) - 2) * a(n-1,2^n - 2) + ... + 10^(2^n + 1) * a(n,1) + 10^(2^n) * a(n-1,0)
- 10^(2^n - 2) * a(n-1,2^n - 2) - ... - 10^1 * a(n-1,1) - 10^0 * a(n-1,0)
= 10^(2^(n+1) - 2) * a(n-1,2^n - 2) + ... + 10^(2^n + 1) * a(n,1) + 10^(2^n) * (a(n-1,0) - 1)
+ 10^(2^n)
- 10^(2^n - 2) * a(n-1,2^n - 2) - ... - 10^1 * a(n-1,1) - 10^0 * a(n-1,0)
= 10^(2^(n+1) - 2) * a(n-1,2^n - 2) + ... + 10^(2^n + 1) * a(n,1) + 10^(2^n) * (a(n-1,0) - 1)
+ 10^(2^n - 1) * 9
+ 10^(2^n - 1)
- 10^(2^n - 2) * a(n-1,2^n - 2) - ... - 10^1 * a(n-1,1) - 10^0 * a(n-1,0)
= 10^(2^(n+1) - 2) * a(n-1,2^n - 2) + ... + 10^(2^n + 1) * a(n,1) + 10^(2^n) * (a(n-1,0) - 1)
+ 10^(2^n - 1) * 9
+ 10^(2^n - 2) * (9 - a(n-1,2^n - 2))
+ 10^(2^n - 2)
- 10^(2^n - 3) * a(n-1,2^n - 3) - ... - 10^1 * a(n-1,1) - 10^0 * a(n-1,0)
= 10^(2^(n+1) - 2) * a(n-1,2^n - 2) + ... + 10^(2^n + 1) * a(n,1) + 10^(2^n) * (a(n-1,0) - 1)
+ 10^(2^n - 1) * 9
+ 10^(2^n - 2) * (9 - a(n-1,2^n - 2))
+ 10^(2^n - 3) * (9 - a(n-1,2^n - 3))
+ ...
+ 10^1 * (9 - a(n-1,1))
+ 10^0 * (10 - a(n-1,0))
ǡa(n-1,0) 0 ǤϤʤȤƤޤ
ǡ
a(n,2^(n+1) - 2) = a(n-1,2^n - 2)
a(n,2^(n+1) - 3) = a(n-1,2^n - 3)
...
a(n,2^n + 1) = a(n-1,1)
a(n.2^n) = a(n-1.0) - 1
a(n,2^n - 1) = 9
a(n,2^n - 2) = 9 - a(n-1,2^n - 2)
a(n,2^n - 3) = 9 - a(n-1,2^n - 3)
...
a(n.1) = 9 - a(n-1,1)
a(n,0) = 10 - a(n-1,0)
ȤʤΤǡƷ¤ s(n) Ȥȡ
s(n)
= a(n,2^(n+1) - 2) + a(n,2^(n+1) - 3) + ... + a(n.2^n + 1) + a(n,2^n)
+ a(n,2^n - 1)
+ a(n,2^n - 2) + a(n,2^n - 3) + ... + a(n,1) + a(n,0)
= a(n-1,2^n - 2) + a(n-1,2^n - 3) + ... + a(n-1,1) + (a(n-1.0) - 1)
+ 9
+ (9 - a(n-1,2^n - 2)) + (9 - a(n-1,2^n - 3)) + ... + (9 - a(n-1,1)) + (10 - a(n-1,0))
= 9 + 9 * (2^n - 1)
= 9 * 2^n
Ȥʤޤ
ͥνȡ 728ʶ 23:48:08 MAIL:uchi@sco.bekkoame.ne.jp 27972
⡼ޥ
#27971
ͤǤ͡
Ʊͤǡ-1-1-1 ʤɤ³ʤ뤫ɤ̤ˤơˤϡʬɤˡ¶б+1 櫓Ǥɤͤ餤ΤǤ礦

#27972 Ϥ錄ˤϻǰʤĤƤޤ(^^;
729ڡ 11:08:16 27973
uchinyan
#27973
ȡ
Ʊͤǡ-1-1-1 ʤɤ³ʤ뤫ɤ̤ˤơˤϡ
ʤȤ⡢ξϡϤޤ
̩ˤϿŪǼˡǾΤǤ礦Ÿλ餫Ǥ礦

䤽ʬɤˡ¶б+1 櫓Ǥɤͤ餤ΤǤ礦
⤷ʤä顢б +1 뤫ɤõʤФʤʤΤǡ
ΥåϡʤȤ⺣ΤޤޤǤϡޤʤȻפޤ
(10 - 1) * (10^2 - 1) * (10^4 - 1) * ... * (10^(2^n) - 1)
ȤüݥȤǤ͡

#27972 Ϥ錄ˤϻǰʤĤƤޤ(^^;
ͤƤߤСʤȤ⡢ǼϤǤȻפޤ㤨С
9 * 99 * 9999 * 99999999 = 8909109 * (100000000 - 1)
0890910900000000
-000000008909109
----------------
0890910891090891
ǤβγƷη׻򤿤ɤäƤߤȡ
1ܡ10 - 9 = 1
2ܡ1ܤη׻ 1 ƤΤ 9 - 0 = 9
3ܡ2ܤη׻ 1 ƤΤ 9 - 1 = 8
4ܡ3ܤη׻ 1 ƤΤ 9 - 9 = 0
5ܡ4ܤη׻ 1 ƤΤ 9 - 0 = 9
6ܡ5ܤη׻ 1 ƤΤ 9 - 9 = 0
7ܡ6ܤη׻ 1 ƤΤ 9 - 8 = 1
8ܡ7ܤη׻ 1 ƤΤ 9 - 0 = 9
9ܡ8ܤη׻ 1 ƤΤ 8 - 0 = 8 = 9 - 1
10ܡ0 - 0 = 0
11ܡ1 - 0 = 1
12ܡ9 - 0 = 9
13ܡ0 - 0 = 0
14ܡ9 - 0 = 9
15ܡ8 - 0 = 8
15ܤǽˤ­碌ȡ
8 + 9 + 0 + 9 + 1 + 0 + (9 - 1)
+ (9 - 0)
+ (9 - 8) + (9 - 9) + (9 - 0) + (9 - 9) + (9 - 1) + (9 - 0) + (10 - 9)
= (9 - 0)
+ (8 + (9 - 8)) + (9 + (9 - 9)) + (0 + (9 - 0)) + (9 + (9 - 9)) + (1 + (9 - 1)) + (0 + (9 - 0)) + ((9 - 1) + (10 - 9))
= 9
+ 9 + 9 + 9 + 9 + 9 + 9 + 9
= 9 * 8
= 9 * 2^3
ȤȤǤ
ƱȤ򡢰̤ n ǤäƤޤ
äľŪˤϡɽˡȤȡ
9 * 99 * 9999 * 99999999 = 8909109 * (100000000 - 1)
= 890910900000000 - 8909109
= 890910(9-1)90000000 + 10000000 - 8909109
= 890910(9-1)90000000 + 9999999 + 1 - 8909109
= 890910(9-1)90000000 + (9-8)(9-9)(9-0)(9-9)(9-1)(9-0)(9-9) + 1
= 890910(9-1)9(9-8)(9-9)(9-0)(9-9)(9-1)(9-0)(9-9) + 1
= 890910(9-1)9(9-8)(9-9)(9-0)(9-9)(9-1)(9-0)(10-9)
->
8 + 9 + 0 + 9 + 1 + 0 + (9-1) + 9 + (9-8) + (9-9) + (9-0) + (9-9) + (9-1) + (9-0) + (10-9)
= 9 * 8
ȤäƤ⤤Ǥ͡
ͥνȡ 729ڡ 13:25:15 MAIL:uchi@sco.bekkoame.ne.jp 27974
ˣţ
񤷤ä
729ڡ 13:47:16 MAIL:mochida@pop16.odn.ne.jp 27975
⡼ޥ
#27974
>Ʊͤǡ-1-1-1 ʤɤ³ʤ뤫ɤ̤ˤơˤϡ
ʤȤ⡢ξϡϤޤ
̩ˤϿŪǼˡǾΤǤ礦Ÿλ餫Ǥ礦

2ˡ3ܤϡڤؤޤ͡(^^;
10ˡǤϤϤʤ񤷤Ǥ͡
ȾζŪʡuchinyanΤäȤΥɽˡ錄ˤʬ䤹äǤOrZ

729ڡ 16:50:59 27976
???
¿׻ϤϤubasicǤ礦
10 'asave "sc511.ub"
20 Seki=1
30 for J=1 to 20
40 A=10^J-1
50 Seki*=A
60 next J
65 Str_seki=str(Seki)
70 Kotae=0
80 for J=1 to len(Str_seki)
90 Kotae+=val(mid(Str_seki,J,1))
100 next J
110 print Kotae
120 end
731ʷ 10:55:14 27977
󤫤
궵ơ
81ʲС 13:11:39 27978
ɤޤ
ñˤ
81ʲС 13:19:24 27979
⡼ޥ
ȤǤ褯ͷӤˤȤǤ
ɤ޷Ǥ빽ޤޤ(ݥݥ(^^)
ϥ󥸤С

Ҳʲ - livedoor Blogʥ֥
http://blog.livedoor.jp/mathematics_cocoa/
ʤߤ˺30Ϥ錄ˤϤäѤꡦ(^^;;


http://math.katuyou.net/
81ʲС 15:41:03 27980
hidy
>⡼ޥ󤵤
꤬Ȥޤ
äȤӤäꤷޤ
ʬΥȤƤ롪
ɤ

http://math.katuyou.net/
ʡ 81ʲС 22:25:07 HomePage:hidyλ27983
uchinyan
hidyͷӤ˹Ԥޤ ^o^
ͥνȡ 81ʲС 22:42:52 MAIL:uchi@sco.bekkoame.ne.jp 27984
²ۻҤ˥򤫤ƥޥ͡⤫
Ϩ£뤫
ܡ 82ʿ 13:07:13 27985
ФˤʤäƤⶵοǥϡϡФȥ
ऺऺ
ܡ 82ʿ 13:14:05 27986
Ω䤷ƥʤ
Ǿä
ܡ 82ʿ 13:16:58 27987