L
졩ѥʤ㤦
83ڡ 0:13:08 27988

³뤹
ʪɤ߾ 83ڡ 0:14:59 27989
ä
ĨԨĥ󡡥⥪ġʡ)
Ϥ 83ڡ 0:17:14 HomePage:Ҳʲ27990
ҥǡ
2709Τ͡
ʤ䡡 83ڡ 0:34:52 27991
Melvy
۾˽ŤǤ͡
ˡѥɤɬפʤ褦ᤷƤʤγǧǤʤΤǤ ()
83ڡ 0:18:59 HomePage:رMelvy27992
L
ڤʤޤ
83ڡ 0:21:07 27994

ؤ󡢽ŤäǤ⤦ޤ
83ڡ 0:22:10 27995
fumio
ҥǡҤޡᤤᤤϤϤϡ
83ڡ 0:24:04 27997
CRYING DOLPHIN
ܤ˲鸫ȤοޤˤǶʬȳƮ
岼ΤȤ狼äʬηʤʤ
ĤĤҤǺǤ褦äȬΤ⤫Ǹִ֡
Ǿǡ١ԡȿ

äƣǯ餤Σ۳رƱ
ᤤorz

ơʬΩΤľܵˤڤˡ
ꤽʵ⤹뤱¿˻ƬƯʤġʤޤΥѥ󤫡
Ȥ櫓ǡȤϳˤޤ(^^;

83ڡ 0:56:50 HomePage:ȤƻȤ27998
إ饯쥹ԡ
󡩡ѥʤ줿ޤϽ񤫤ʤۤ褵ʡ
軲̻ԡ 83ڡ 0:26:51 HomePage:ɥԥ27999
kasama
ä̾ܤޤ()
83ڡ 0:27:20 28000
ڥ󥸥
ʤǥѥʤ줿Ǥ礦
83ڡ 0:32:27 28001
ڥ󥸥
ʤǥѥʤ줿Ǥ礦
83ڡ 0:32:33 28002
kasama
ѥɤʤޤ͡ǧڤʤ狼ʤ()
83ڡ 0:36:54 28003

ޤ
83ڡ 0:40:07 28004
Melvy
ѥɬפˤʤä褦Ǥ
ߥꡤˤĤƵȤ뤿ᡤ 1 ֤ϥѥʤ褦ˤʤäƤΤǤ줬ĤäƤΤ < #28001
83ڡ 0:41:34 HomePage:رMelvy28005
ޥ
޵𤤤ޤ

#27998
ɤäˤʤΤФƤǤorz

ϡǤΤǡ֤ȤˤФΤפȤˤǺޤǥΤϤ⤫٤ΤΤʤΤǤäȸڤ˻֤򤫤ޤ

줫顢ФŤ֤ˤʤäƤޤؤǤ򤪤ޤʬäƤޤΤǡ轵ˤн褤ޤm(__)m
PowerBook G4 83ڡ 0:42:05 MAIL:masaru-y@sansu.org HomePage:28006

нŤ
83ڡ 1:02:13 28007
ޥ
㥵ФŤ

ȤꤢˤǤк򤤤ޤm(__)m
PowerBook G4 83ڡ 1:04:46 MAIL:masaru-y@sansu.org HomePage:28008
CRYING DOLPHIN
#28006
Σ¤򡢤줾ΣĤĺߤ¾̤濴
פ褦ˡ̤θߤ˵դޤˤʤ褦˽Ťͤ롣
¤ζʬȣ衣(2000 ۳رʸ)
83ڡ 1:09:11 HomePage:ȤƻȤ28009
ޥ
#28009
ΤˤΤޤޤ....Ʊˤ񤷤Τ˽ФȤ...

ʤߤ˺2^9=512ˤʤΤäꤷޤ
PowerBook G4 83ڡ 1:14:39 MAIL:masaru-y@sansu.org HomePage:28010
إ饯쥹ԡ
512󡡢2^92/9orz

Ȼǹͤ衦ȼʬˤĤäǤߤ롣
軲̻ԡ 83ڡ 1:23:05 HomePage:ɥԥ28011
إ饯쥹ԡ
#28010
äѤꤽǤ͡
ʤȤʤȲ1/9Ĥ­2/9512ߤʤäΤǤ
軲̻ԡ 83ڡ 1:17:16 HomePage:ɥԥ28012
CRYING DOLPHIN
#28010
ǤȤơ̤θߤƱˤʤ褦˽Ťͤޡ
ʬΩΤȿ޼ΤĤƤΤǡֲ򤭤䤹
ȤϻפޤǤ⾮ˤϡʤȤԤǤ餫
Ǥ͡

ˤƤ⡢ǿͤ碌Ȥޤޤ

ʤȥХ˸ƿʤzzz
83ڡ 1:25:14 HomePage:ȤƻȤ28013
ȥȥ
СŤƤĤʤʤΤǤƤޤ

Ҥι۳ر̤ϻǤϤʤʤ䤷餷ȸäƤޤ
ˤʬʤΤǤη뾽¤ͳ褹ΤǤ礦
ʼ˸лԡ 83ڡ 1:30:08 MAIL:h-sakai@zb3.so-net.ne.jp 28014
ȥȥ
ʤߤ˻䤬Dz򤤤ϤȤƤĹǤ
ʬ褦˿ڤ˺äĤ꤬ؤˤʤä⡣
¿ʬ̤ʿԤ˥饤Ȥˡ礦֤ι⤵ζʬ
ϻѷˤʤ뤳ȤŬʬ䤹Ǥ͡
ʼ˸лԡ 83ڡ 1:33:05 MAIL:h-sakai@zb3.so-net.ne.jp 28015
sugitakukun
Ҥꥢ륿ĩǤäȲ򤱤ʤ
䡢ήФ˵ٷĤĤǤ

Ȥꤢ岼1/3ʬʤΤǤ1/3ʬǤ⤿Ĥ⤿Ĥ^^;
夫1/3ڤäơ2/3Ǥڤäơˤ1/2Ǥڤäơޤʬʤ麸ȾʬȱȾʬʬƹͤơġϵޤorz

ޤɾ岼ʻơ2/9cm^3פȤ񤤤ƤǥȤǤorz
83ڡ 2:44:12 28016
y.kobayashi
ϻι⤵HȤȣH/ޤǤΤѤϻΣ/H/飲H/ޤǤȬΤѤϻΣ/H/HޤǤϣH/ޤǤȹƱΤѤϻΣ/äƹפϣ/ĤǤäƤǤ礦
83ڡ 3:06:35 28017
uchinyan
ϤˤϡϡͤޤǤ⡢ڤäǤ ^^/
ޤʤΤŤ餤ΤǤˡǤ

ޤ褯褦ˡʿAGJDǤ̤ͤƤߤޤ
ȡOJ ȹѷθAP ֤ѷθ򡢤줾 T, S Ȥȡ
δط顢OTTJ = 12, ASSP = 21, OT = PS ʤɤʬޤ
ޤΩΤϡդ OT ξΤȲ˰ĤĤȡ˶ޤ줿ʬʬޤ
Τϡ礭Τǡ 13 127 Ǥϡޤ줿ʬǤ
ϡ̿ޤ޷оʤɤθȡդ OT Ʊѷ 8 Ľ̤ˤƺäΩΤʬޤ
פˡȬΤǤ͡˵դΤ˾ϫޤ
ѤϡϤ̿޵ڤоʤɤ顢ξΤΰ̤ȬΤΰ̤ȤƱʿ̾ˤ뤳Ȥθȡ
դ OT 2 ܤΤ龮Τ 4 ĤΤʬޤ
ǡΤ 2 * 2 * 2 - 4 = 4 ʬǤ
ʾꡢɡΩΤϡΤ 1 + 4 + 1 = 6 ʬǤ
ǡΤѤϡΤ 1/27 Ǥ
ˡΡѿϡϻȡ̤Ⱦʬ⤵Ʊ좪 1/3 ʤΤǡ
ϻ 1/2 * 1 * 1/3 = 1/6 Ѥˤʤޤ
ǡѤϡ6 * 1/6 * 1/27 * 6 = 2/9 cm^3 ˤʤޤ
ͥνȡ 83ڡ 11:09:20 MAIL:uchi@sco.bekkoame.ne.jp 28018
uchinyan
Ǽɤߤޤ轵ΤäƤ㴳Υȥ֥뤬ä褦Ǥ͡

#28010
ʤߤ˺2^9=512ˤʤΤäꤷޤ
ʥȥåäȤϡ

#28009¾
ࡢȤϡΤ˻Dz򤱤ޤ˲򤯤ΤѤʤ

񤷤ǤꡢǤޤ򤱤ơȤäƤⵤǤ ^^/
ͥνȡ 83ڡ 11:26:17 MAIL:uchi@sco.bekkoame.ne.jp 28019
Melvy
ʿ OLH ڤȡ礦ɽŤʤʬΤҤȤĤ̤ФƤơ
ϢOLH 3:1 ѷ岼 2 Ĺ碌Ҥˤʤޤ
ʿ OLH ʿԤʿ̤ǡʿ OLH ʿ EPC δ֤ڤȤɤƱ̤򤷤Ƥޤ
Ťʤʬι⤵ (ʿ OLH ˿ľ) ϡ A-EPC ι⤵
(= ϻι⤵) 1/3 Ǥ

äƽŤʤʬѤ
(ϻ) 1/2 (ޤǤǢOLE) 2/9 (ޤǤǤҤ)
(ϻι⤵) 1/3 (Ťʤʬι⤵)
= (ϻ) 1/27
= 2/9 cm^3
Ȥޤ

Τι⤵ȤȤ鷺˻Dz򤯤ȤǤޤ
̩˸СѤ () (⤵) ǵޤΤϥХꥨ顤
ǤʤȸФʤΤǤɡġ

#28014
ʤǤƤߤСФʤ󤫤פΩΤǤ͡

С0:07 餤˲᡼ΤǤɡ 0:31 ˤʤäƤޤ
ϤꥵХȥ֥αƶǤ͡
եʤä餷褦ǤʤΤ⤷ޤ󤬡
83ڡ 12:21:51 HomePage:رMelvy28020
uchinyan
#28020
򤤲ˡȻפޤٶˤʤޤ
äȡʬʤȤΤǡĺйǤ

ʿ OLH ʿԤʿ̤ǡʿ OLH ʿ EPC δ֤ڤȤɤƱ̤򤷤Ƥޤ
ƤߤСΤо餽ʤȤǤԥʤǤ
٤ƢOLHΤҤƱˤʤäƤ롢ȤȤǤ͡
ݡ
ʤǼǤޤ (^^;

(ϻ) 1/2 (ޤǤǢOLE) 2/9 (ޤǤǤҤ) (ϻι⤵) 1/3 (Ťʤʬι⤵)
(ޤǤǢOLE)סݡ(ޤǤǢOLH)ס

丷̩˸СѤ () (⤵) ǵޤΤϥХꥨ顤
֥ХꥨפȤΤΤʤΤǤѤθΤȤǤ礦
ݡ
Ĵ٤ƤߤޤäȰŪäʤΤǤ͡䤬ȤǤϡ
ĤʿԤ̤ΤˤϤޤ줿Ĥο޷Ǥäơ
ʿԤʿ̤ڤäڤѤĤǤСξѤ
Ȥޤ褯ʬʤɽʤΤǤξϡפˡ
Ѥľ(ȤΤʡ)Ѥ
ȤȤǤ͡äȡ礦ɡʿԻշƱ⤵ĹѤΤ褦ˡ

ξϤȤ⤫ľŪȻѤ餤ϡǤϤʤΤʡ
ͥνȡ 83ڡ 15:35:59 MAIL:uchi@sco.bekkoame.ne.jp 28021
fumio
ϤϤϡդäƤȻפ㤤Ƥޤ
ǤȤۤۤǤäѤꡢҤϤ
83ڡ 15:45:45 28022

OP3ʬ̤ʿ̤ǿʿڤäƹͤޤ

AE1/3Ĺ1ȸƤ֤Ȥˤȡ
ʬΰ־ʬϡ1դ1
ֲʬ⡢1դ1
δ֤ʬϡ1դ2Τ4飱դΤ
ڤäˡʤޤ
ǡפϡդ16ʬ
ޤιΤѤϡ1cm^3 顢1դ1ΤѤϡ1/27cm^3
ɡ61/272/9cm^3 ˤʤޤ

ν񤭹ߤɤޤǡʬȬΤȤϵŤޤǤ

83ڡ 16:03:00 28023
ե륳
#28014
κǤ礦͡
ϸͥǤߤޤ<(__)>
83ڡ 16:23:59 28024
ޥ
#28020
> С0:07 餤˲᡼ΤǤɡ 0:31 ˤʤäƤޤ
> ϤꥵХȥ֥αƶǤ͡

<Melvy>ΥᥤǤޥ󤫤Date: ˤ 00:07:56ǡץХSMTPФϤΤ 00:08:12sansu.orgϤΤ 00:31:01Ǥ뤳Ȥϡ긵ΥᥤʬäƤޤפˤƤޤΤ...ߥޥ
ɽϡ֣ĤλספȽǤʤȥޥǤ..ƥץХλ郎ΤǤȤϸ¤ޤʤȤ񥺥ƤޤˤƼPCλפϰտŪǽǤΤǡΤ褦٤ˤƤޤm(__)m

ʤߤ˺ΥᥤٱʤȤŪ˥ФŤʤäˤθǤĤͳޤơĤSPAMкˤ륪إåɤá⤦ĤϲեǤܥϢǤˤ̥ᥤǡ餬ޤƱ˵äȯޤԤʬ˴ؤƤϺդΤнѤߤǤʾܤȤϽ񤱤ޤ󤬡˸ԤˤĤƤϡܤкǽʬɤ򸫶ˤƤȤȤˤʤޤˤäƤϡ֣ٲä顢ƱIPɥ쥹ϣô֤٤ʤ֤Ȥ뤫Τޤ󡣡ʤޤꤿϤʤΤǡȤꤢڥǥ󥰤Ƥޤ
PowerBook G4 83ڡ 17:25:54 MAIL:masaru-y@sansu.org HomePage:28025
ӥ塼ƥե롼WITHॿɲ
ٶפФǤ

򤤤Фޤơ

*4Τ˺ƤΤ䤷
ӥ塼ƥ硡 83ڡ 18:35:50 28026
Melvy
#28021
> (ޤǤǢOLE)סݡ(ޤǤǢOLH)ס
ϤñʤǤ

#28025
> ƼPCλפϰտŪǽ
ˤϵŤƤޤΤǡǼƤޤ

Ȥǡߤʤ޿ʿڤäƷ׻ƤäǤ͡
ʤʤ᡼񤷤ΤǤȬΤȤʤΤϤʤȤǼǤޤ
о餹ȡŤʤʬΤϡ٤Ƥ̤ƱҤʿϻΤʤǤ͡
83ڡ 21:24:26 HomePage:رMelvy28027
ߤγ
줾λѿ⤵Ⱦʬڤäơѿɤ碌ޤȡλѿɤ碌ޤʬƹͤȥ᡼䤹Ȼפޤ
83ڡ 21:59:05 28028
BossF
̤ѷν¿ΤǤ
ԣ 84ʶ 7:49:28 MAIL:fv2f-ftk@asahi-net.or.jp HomePage:£ơǣ󡡣ԣ£28029
⡼ޥ
Ǥ(^^)
βˡɤǤ⡢ޤäޤ᡼Ǥʤ錄ˤޤ͡
Ǥ⡢Ψ10ʤ麹ϤĤʤʡ
84ʶ 12:03:25 28030
ʤ
#28029
>̤ѷν¿ΤǤ

ΤĤεˤ⤢褦ˡ
Τˤϡ̤ҤʿԣΤˤʤޤ
84ʶ 18:06:03 MAIL:naka@sansu.org 28031

Ԥ⾯ʤΨ㤤꤬񤷤Ǥ礦
84ʶ 18:52:29 28032
ޥ
#28032
Կ˴ؤƤϡ٤Ƚ˻Ȥ뤫ȻפޤʤߡǣפȤäüʴ֤ơΨ...ͤŪΨ򲼤⤢äꤹΤǤʤȤ...(^^;
PowerBook G4 84ʶ 19:35:23 MAIL:masaru-y@sansu.org HomePage:28033

ͤŪΨ򲼤⤢äꤹĤΤǤǼĤäƤ뤫ǧƤФΤˡġˤƤ⺣㤤Ǥ͡
84ʶ 20:45:02 28034
BossF
#28031

Ȭ+ΣĤȻפäְ㤤Ǥ͡ĤȤۤ

Τ˺äƤߤ顢ʿϻΤФƤޤʤԻ׵(=^^=)
ԣ 84ʶ 23:56:16 MAIL:fv2f-ftk@asahi-net.or.jp HomePage:£ơǣ󡡣ԣ£28035
BossF
PS 椦Τϡplydrono 夦ȤäȤΩƤȡ᡼ǧǤޤ̣
ʲѤۤȤɤ
ԣ 85ڡ 0:01:28 MAIL:fv2f-ftk@asahi-net.or.jp HomePage:£ơǣ󡡣ԣ£28036
⡼ޥ
Ʋ

ͧǤ򸫤Ƥ褯ʬޤ󡣡(^^;


nϼǡ򽽿ˡɽƷο¤Ȥ
1000Ǥꡢn^2γƷο¤Ȥ1000^2ˤʤ
ȤΤ褦n¸ߤ뤫ݤꤻ衣

87ʷ 21:41:31 28037
ߤä
ȤäƤ⸶ŪǤ̻ޤϷĤä㤤ޤ
ˡϤ󤵤ƱǤ
ʿϻΤˤĤƤϤߤʤν񤭹ߤ򸫤ޤǤĤޤǤ
Ʊ礷Ƥߤޤ轵⻲äƤߤǤ
88ʲС 0:09:07 28038
ޥ
ȡ勵ȤʤǤäȸѤ򤷤ꤷƤޤơ֣֤ϰŤˤ褦ˡפȸäꤷƤޤ

ѤƤϡץιܿäƤĤǡ礦ɲβιäơʬ˿դ롢Ȥʤץǥβ¤ߤʼѤǤ̾Ǽ»ܤΤǡ빽Τؤô礭ΡȤΤȤǤ

....ȤϸäƤʤΤǤޤʤ櫓ǡ٤Τʤ褦ǤʤȤĤġդAppleWWDCΣͤѤ򸫤ƤޤޤˡְšפˤƤʤȴΤΥϥʤդ٤Τ褦ʾ֤ˤʤäƤޤˤΤҤ٤뤽ǤΤǡλιϤ٤ߤȤƤפޤޤ⤽⤪ߤʤΤǵ٤ߤˤ褦ʡȻפäƤȤǤϤΤǤ

Ȥ櫓ǡޤǿޤ󤬡򤤤йǤm(__)m
PowerBook G4 88ʲС 9:48:13 MAIL:masaru-y@sansu.org HomePage:28039
uchinyan
#28039
λιϤ٤ߤȤƤפޤ
ޥ뤵ѤǤ͡ʹǶʼѤǡ
λǤ
Թ⡼ޥ󤵤񤷤ФƤ褦Ǥ
äȤܤ
äȤ⡢ϲƴֽѤʤǤ礦͡
ͥνȡ 88ʲС 11:05:17 MAIL:uchi@sco.bekkoame.ne.jp 28040
⡼ޥ
#28039
Υå䥹åפȤΤȻǰ
Ǥ󤬶ʡ
Ρʻˤޤ
椪񤤤ͤޤơ
88ʲС 13:10:44 28041
uchinyan
#28037

nϼǡ򽽿ˡɽƷο¤Ȥ
1000Ǥꡢn^2γƷο¤Ȥ1000^2ˤʤ
ȤΤ褦n¸ߤ뤫ݤꤻ衣
Ϸ빽Ż˻ΤǡľŪäǶ̤ǤϤʤʵޤ
ˤϤʤäƤϤʤΤǤ

n k οȤޤĤޤꡢ0 < n < 10^k Ǥ͡
ȡ0 < n^2 < 10^(2k) 顢n^2 2k ʲοǤ
̤ x 10ʿɽǤγƷ¤ s(x) ǽ񤯤Ȥˤȡ٤Ƥηο 9 ΤȤ s(x) ϺʤΤǡ
0 < s(n) <= 9k, 0 < s(n^2) <= 9 * 2k = 18k
Ȥʤޤ
s(n) = 1000 ʤΤǡk >= 1000/9 > 111 k >= 112 Ǥ
s(n^2) = 1000^2 = 1000000 ʤΤǡk >= 500000/9 > 55555 ǡk >= 55556 Ǥ
ǡk >= 55556 ǡĤޤꡢn 55556 ʾοǤ
ΤȤs(n) = 1000 ʤΤǡn γƷοʿѤϡ1000/k <= 1000 * 9/500000 = 9/500 Ȥʤꡢۤ 0 Ǥ
󡢤٤ 0 ϤꤨʤΤǤ0 ¿Τϴְ㤤ޤ
ʤˤ1000 ̵٤˼ФȤˡ1 ʾο 18 İʲʤΤǤ顣
褷Ƥ⡢ϤۤȤɤη夬 0 ˤʤꤽǡ
­ s(n^2) 1000^2 Ĥޤꡢs(n) 1000 ܤˤʤ롢ȤΤϡäѤʵޤ

¿ʬľ򤦤ޤǤФ褦˻פΤǤ
褯ʬʤ
ͥνȡ 89ʿ 7:54:42 MAIL:uchi@sco.bekkoame.ne.jp 28042
kasama
#28039

μѤȤϸΤؤô礭褦ѤǤ͡Ĥ˻ƤΤǡϲƵ٤ߤͤơɤäܤʤäƲ
ˡd(*ޢ*)b
88ʲС 20:58:40 28043
⡼ޥ
#28042 uchinyanͤƲäƤ꤬ȤޤOrZ
#28037
ǵդͤƤߤΤǤ
s(1)=1,s(1^2)=1^2
s(11)=2,s(11^2)=s(121)=4=2^2



s(111111111)=9,s(111111111^2)=s(12345678987654321)=45+36=81=9^2
ȡ
ޤǽ񤤤Ƥ顢ΰ̣狼ޤ(^^)

Ф餯åפޤ͡
88ʲС 22:37:51 28044
toby
#28039 ޥ뤵󡡤ˡ
88ʲС 23:50:21 28045
ma-mu-ta
ޥ뤵󡢤꿽夲ޤ
ɤˤʤäƤ
89ʿ 2:00:31 28046
uchinyan
#28044
ΥץͤΤǤ
s(1111111111) = 10, s(1111111111^2) = s(1234567900987654321) = 2 * 45 - 8 = 82 < 10^2
ݥȤϡñ n = 10a + b ǽ񤯤ȡ
s(n) = a + b
n^2 = 100 * a^2 + 10 * 2ab + b^2
Ǥa^2, 2ab, b^2 ϰ̤ʾˤʤΤǷη꤬ޤ
θƤ⡢षθȡ
s(n^2) <= a^2 + 2ab + b^2 = (a + b)^2 =(s(n))^2
ǡϡ夬꤬ʤʤɤΤʤȹͤƤ

٤ͤΤѤʵƤޤ
ƻˤäƤߤΤʡ
ͥνȡ 89ʿ 8:04:36 MAIL:uchi@sco.bekkoame.ne.jp 28047
⡼ޥ
#28047
ǤۤФƤޤ衪(^^)
10^2 褦ˤɽ뤫ʬФ褤櫓Ǥ
89ʿ 8:57:45 28048
ޥ
ʤȥ饯ˤʤäƤƤޤäΤǤ...㤢ɤäʤ

Ȥ櫓ǡͤӤ˾Ф̿ޤޤͥǤ顢ï˶ƤäƤ⡢ɤ˻ƤäƤ빽Ǥǡ(^^;

http://www.sansu.org/060810iiwake/
PowerBook G4 89ʿ 18:46:34 MAIL:masaru-y@sansu.org HomePage:28049
uchinyan
#28048
⡼ޥ󤵤ء
˻ޤƤǤƤޤ
뾯֤줽ʤΤǡ⤦ͤƤ
ͥνȡ 89ʿ 18:49:21 MAIL:uchi@sco.bekkoame.ne.jp 28050
⡼ޥ
#28050
uchinyanء
λǡ(^^)

#28049
ޥͤء
ϤƤºʤ㤤ˤ򤢤꤬ҸƤޤˤƤ⤪路ʥڤäͻҤǤ͡ѤɤʤΤˤߤӥӥΤ錄ȤޤƤϡ(^^;
89ʿ 21:56:48 28051
uchinyan
(ְäƤޤ#28055, #28063ʤβսޤ)

֤줿Τǡ#28047ΥץǵͤƤߤޤ
ĹʤäƤޤääŪˤ԰¤Ǥɤʤ

0 ʾ n 10ʿɽȤγƷ¤ s(n) Ȥޤ
ȡa, b 0 ʾȤơs ˤϼޤ
1) a 1 οʤ s(a) = a
2) s(10^b * a) = s(a)
3) b 1 οʤ s(10a + b) = s(10a) + s(b) = s(a) + s(b)
4) s(a + b) <= s(a) + s(b), a + b η׻Ƿη꤬ʤ
5) s(a * b) <= s(a) * s(b), a * b η׻Ƿη꤬ʤ

1), 2)ϡs 餫Ǥ礦

3)⡢b 1 οʤΤȡs ڤ2)餫Ǥ礦

4)ξ˴ؤŪǼˡȤޤ
a, b 1 ο0 9ξˤϡºݤˤäƤߤСޤΩޤ
a k ʲοb m ʲοǡΩȤޤ
ββǡa k+1 οȤƤߤޤ
a = 10c + dc k οd 1 ο
b = 10e + fe m-1 οf 1 ο
Ƚ񤱤Τǡ
s(a + b) = s((10c + d) + (10e + f)) = s(10(c + e) + (d + f))
ǡ0 <= d, f <= 9 ʤΤǡd + f ϡ1 ο 2 οˤʤޤ
d + f 1 οξ
1), 3)ꡢ
s(a + b) = s(10(c + e) + (d + f)) = s(c + e) + s(d + f) = s(c + e) + (d + f)
c k οe m-1 οäΤǡǼˡβ꤫顢
s(a + b) = s(c + e) + (d + f) <= s(c) + s(e) + d + f
ϡc + e ˷η꤬ʤǤˡ1), 3)ꡢ
s(c) + s(e) + d + f = (s(c) + s(d)) + (s(e) + s(f)) = s(10c + d) + s(10e + f) = s(a) + s(b)
Ȥʤꡢ
s(a + b) <= s(a) + s(b)
ϡd + f ˷η꤬ʤΤǡa + b ˷η꤬ʤˤʤޤ
d + f 2 οξ
d + f = 10g + h, g, h 1 ο
Ƚ񤱤Τǡ
1), 3)ꡢ
s(a + b) = s(10(c + e) + (d + f)) = s(10(c + e + g) + h) = s(c + e + g) + s(h) = s(c + e + g) + h
c k οe m-1 οg 1 οäΤǡe + g Ϲ⡹ m οǤ뤳ȤդȡǼˡβ꤫顢
s(a + b) = s(c + e + g) + h =<= s(c) + s(e + g) + h <= s(c) + s(e) + s(g) + h
ϡc + e + g ˷η꤬ʤǤˡ1), 3)ꡢ
s(c) + s(e) + s(g) + h = s(c) + s(e) + s(10g + h) = s(c) + s(e) + s(d + f)
d + f ˵ǼˡβȤä
< s(c) + s(e) + s(d) + s(f) = (s(c) + s(d)) + (s(e) + s(f)) = s(10c + d) + s(10e + f) = s(a) + s(b)
Ȥʤꡢ
s(a + b) < s(a) + s(b)
d + f ˷η꤬ΤǡΩޤ
ʾꡢa k+1 οb m ʲοǡޤ
ˡͿ줿 a, b ˤĤоΤʤΤǡa k ʲοb m+1 οξϡa b ȤؤOKǤ
ʾǾϽǤ

5)ξϤꡢ˴ؤŪǼˡȤޤ
a, b 1ο0 9ξˤϡºݤˤäƤߤСޤΩޤ
a k ʲοb m ʲοǡΩȤޤ
ββǡa k+1 οȤƤߤޤ
a = 10c + dc k οd 1 ο
b = 10e + fe m-1 οf 1 ο
Ƚ񤱤Τǡ
s(a * b) = s((10c + d) * (10e + f)) = s(100 * ce + 10 * (de + cf) + df)
4)
<= s(100 * ce) + s(10 * (de + cf)) + s(df)
2)
= s(ce) + s(de + cf) + s(df)
Ƥӡ4)
<= s(ce) + s(de) + s(cf) + s(df)
ǡc k οd 1 οe m-1 οf 1 οʤΤǡǼˡβ꤬Ȥơ
<= s(c) * s(e) + s(d) * s(e) + s(c) * s(f) + s(d) * s(f)
= (s(c) + s(d)) * (s(e) + s(f))
3)Ȥä
= (s(10c + d)) * (s(10e + f))
= s(a) * s(b)
ϡޤa * b η׻ǰǤη꤬äˤΩʤΤǡ
ɤϡa * b ˷η꤬ʤǤ
ʾꡢa k+1 οb m ʲοǡޤ
ˡͿ줿 a, b ˤĤоΤʤΤǡa k ʲοb m+1 οξϡa b ȤؤOKǤ
ʾǾϽǤ

ơϫޤޤФ⤦Ǥ
s(n) = 1000, s(n^2) = 1000^2
äΤǡ
s(n^2) = (s(n))^2
Ĥޤꡢ椬ΩǤǡn^2 η׻Ƿη꤬äƤϤޤ
ºݤη׻ͤƤߤȡ4 ʾοɤηˤˤϡ餫 4^2 = 16 Ƿ夬Τǡ٤Ƥο 3 ʲǤ
ξ硢n k Ȥȡ
0 < s(n) <= 3k
Ȥʤꡢk >= 1000/3 > 333 ǡn 334 ʾοǤ
k οǡŪˡο 0 ǤʤĿϡ
334 ġ3 333 Ĥ 1 1 ġ 3 332 Ĥ 2 2

1000 ġ1 1000
ˤʤޤ
n^2 η׻ͤƤߤȡ
n = 10^(k-1) * a(k-1) + 10^(k-2) * a(k-2) + ... + 10^2 * a(2) + 10^1 * a(1) + 10^0 * a(0)
= [i=0,k-1]{10^i * a(i)}
Ȥơ
n^2 = ([i=0,k-1]{10^i * a(i)})^2
= [0<=i,j<=k-1]{10^(i+j) * a(i) * a(j)}
= [m=0,2k-2]{10^m * [max(0,m-(k-1))<=i<=min(k-1,m)]{a(i) * a(m-i)}}
η꤬äƤϤʤΤǡm = 0, ..., 2k-2 Фơ
0 <= [max(0,m-(k-1))<=i<=min(k-1,m)]{a(i) * a(m-i)} <= 9
äˡm = k-1 ξ硢
0 <= [0<=i<=k-1]{a(i) * a(k-1-i)} <= 9
0 <= a(i) <= 3 θȡ嵭 k ¤Τ 0 ǤʤΤϡ⡹ 9 ǡʤȤ k-9 0 ǤʤФʤޤ
a(i) ˴ؤƹͤȡa(i) * a(k-1-i) ϡi <-> k-1-i ˴ؤоΤʤΤǡʤȤ (k-9)/2 or (k-8)/2 Ĥ 0 ǤʤФʤޤ
ǡ (k+9)/2 or (k+8)/2 Ĥ 0 ǤϤޤ

(餬Ǥ(k+9)/2 1000 ĶʤΤǡפȤΤϡŪ˺򤬤ޤ)
äơۤɤ 0 ǤʤθĿ顢(k+9)/2 1000 ĶʤΤǡ
k/2 < (k+9)/2 <= 1000
줫顢
333 < k < 2000
Ǥ
(ޤǤꡣ)

ΰǡ
0 < s(n) < 10^k
0 < s(n^2) = (s(n))^2 < 10^(2k)
ʤΤǡ
0 < s(n^2) = 1000^2 = 1000000 <= 9 * 2k = 18k
k >= 1000000/18 = 500000/9 > 55555

(ꡣɡk > 55555 ¸ߤ뤫ɤʬʤޤޤǤ)
̷Ǥ
äơs(n) = 1000, s(n^2) = 1000^2 Ȥʤ褦 n ¸ߤޤ
(ޤǤꡣ)

Ǹɾ԰¤Ǥ
ˤ衢⤦ޤǤǤ͡
ͥνȡ 810ڡ 20:11:24 MAIL:uchi@sco.bekkoame.ne.jp 28052
uchinyan
#28049
ˡѤǡ
줰⤴̵ʤʤ褦ˡ
ͥνȡ 89ʿ 23:15:02 MAIL:uchi@sco.bekkoame.ne.jp 28053
fumio
ޥ뤵󡢤ˤƤ͡
⤬ʬμ̿콨ʥ硼ˤ˻Ƥȸä
襤Ȥ櫓狼ʻäƤޤ
89ʿ 23:56:14 28054
⡼ޥ
#28052
uchinyanء
Ĥʤ顢Ѥʻ׹ͤϤȴǤ
¤Ū˲ǽʤǤ衪줬
s(n)=10,s(n^2)=10^2 ΤΤޤ
줬С n^2 ˤʤΤǤ뤳ȤƱʬϤǤ

¿ʬuchinyanǺޤƤ餫ʤʤǤ礦ͤ
ߤСʤʤŤʤäΤäƻפäƤޤ(^^;
810ڡ 0:09:47 28055
fumio
ޥ뤵󡢤ˤƤ͡
⤬ʬμ̿콨ʥ硼ˤ˻Ƥȸä
襤Ȥ櫓狼ʻäƤޤ
810ڡ 0:11:00 28056

ˤʤäƲ
810ڡ 0:51:06 28057
ޥ
ޤΤա꤬Ȥޤm(__)m

ʤ󤫡īϡ֤ХפäƴäǤȥ饯ˤʤäƤơϼ֤ҤƤƤޤʤ櫓ǡޤ֤פǤǡŻˤ褦ʤȻפäƤꤷޤ꤬Ȥޤm(__)m
PowerBook G4 810ڡ 1:31:24 MAIL:masaru-y@sansu.org HomePage:28058
Melvy
#28055
ġ
ʷϵǤ#28044 ιͤǡͤᤷޤз夬ɤΤǤϤʤȡ
äȤ⡤ֳ֤ (10101ĤΤ褦) ͤƤޤȷɽŤʤäƤޤΤǡ
θĿ򲼤 0 ġ1 ġ3 ġ7 ġĤ䤷Ƥ(ΤȤΤη 2^{s(n)-1} ˤʤ)
n^2 η 0 1 2 ǹ뤳Ȥˤʤ褦ʵޤ
9 ޤǻȤäƤ褱СäȽ̤뤫⤷ޤ󤬡

ޤäƤʤΤǡ꤬ޤ餴Ŧ
810ڡ 1:51:52 HomePage:رMelvy28059
⡼ޥ
#28059
Melvyء
ȤǤäȳμ¤¸ߤޤ
ҥȤϡ(x+y)^2
810ڡ 7:51:43 28060
uchinyan
#28055
ޤä
#28052ϡȾ k ɾŪ˸꤬褦Ǥ
k < 2000 ϡľȹ鷺餫˸Ǥ
ƹͤޤ
ͥνȡ 810ڡ 8:16:58 MAIL:uchi@sco.bekkoame.ne.jp 28061
uchinyan
#28055

n = 1111111110000000001, 10101010101010101001, ... ?
s(n) = 10, s(n^2) = 100
ͥνȡ 810ڡ 10:55:00 MAIL:uchi@sco.bekkoame.ne.jp 28062
⡼ޥ
#28062
uchinyanء

n = 1111111110000000001, 10101010101010101001, ... ?
s(n) = 10, s(n^2) = 100


Ǥޤ͡
n=111111111*10^9+1
ʤ顢
n^2=(111111111*10^9+1)^2=111111111^2*10^18+2*111111111*10^9+1
ʤΤǡ
s((111111111*10^9+1)^2)=s(111111111^2)+2*s(111111111)+1=9^2+2*9+1
=(9+1)^2=10^2
򷫤֤СǤǽˤʤȤ櫓Ǥ͡
n θȤƤϤޤ¾ˤ뤫⤷ޤ󤬡

ͧͤβϡ

¸ߤ

S(n)ϼnγƷο¤Ȥ롣

nkοǡS(n)=mȤ롣⤷S(n^2)=m^2Ǥ

N=10^(k+1)*n+1ȤS(N)=m+1S(N^2)=(m+1)^2

Ȥʤ롣

n1=1ȤǼŪniޤޤäȤn=ni

ȤƾΤ褦Nn(i+1)Ȥ

ȡS(ni)=iS(ni^2)=i^2Ȥʤ뤫n1000

Ǥ롣

Ǥ

ǽϡȡȥˤʤ褦ʵơ(^^;
810ڡ 11:18:52 28063
uchinyan
#28063
ʤۤɤʤ

ΰ֤԰ϡݤǡǤˤʤΤǤϡȻפäƤޤäȤȻפޤ
ѤиĤȤ֤ʤȿʤǤ

ͧͤβϡǤåʤ顢⤦äɤޤƤ餤ޤ
äȽ񤤤Ƥ뤱ɡ䤬Ȼפ (^^;s ˤʤäƤޤ͡

ȤǡȹͤФƤͧͤäơɤʤǤ
̣š ^^
ͥνȡ 810ڡ 11:41:45 MAIL:uchi@sco.bekkoame.ne.jp 28064
⡼ޥ
#28064
ͧͤǤ
Ŀ;ߤ뤫ʤ(^^)
ػͧͤǤ錄ؤ¿ʬդʡ
̣ǡɤⵤʬžˤɤߤäƤ褦ǡ
ܤ򶵤Ƥޤ󡣡(^^)
ߤ˥᡼νФäƤޤ
ΥȤϤᤤʥΤ餻ƤäƤ⡢ɤ⹥Ǥʤ褦ʡۤǤ
˾ˤϪƤäƤΤϤΤΰǡΤȤΤäƤޤ
810ڡ 14:02:17 28065
uchinyan
#28065
ͧͤǤ
Ŀ;ߤ뤫ʤ(^^)
Ǥ̣ϤʤǤ
ꥸʥʤΤʤǤʤȤСʤʤȡפäΤǡ

#28063⤦٤äɤߤޤ
1111111110000000001 ˶ᤫäǤ͡θΤ⤦ޤǤϤޤǤ
ͧͤβϡǤȤ褦ʤǤä (^^;
ʤͧͤβǡn1 1 ʳˤ⡢㤨С10, 100, 1000, ... ʤɤǤ⤤
1111111110000000001 Τ褦 n9 = 111111111 ȯƤ⤤Τǡ
n ̵¤ˤޤ͡
ơ#28059MelvyήΥץǤǽʤ褦Ǥ
MelvyΤϡͧͤΤ򡢾岼ηҤä֤褦ʲˤʤäƤ褦Ǥ
ˡݥȤϡn^2 η׻Ƿη꤬ʤȤʤΤǡʳμβ⤢뤫⤷ޤ
ͥνȡ 810ڡ 21:39:58 MAIL:uchi@sco.bekkoame.ne.jp 28066
⡼ޥ
#28066
n1 1 ʳˤ⡢㤨С10, 100, 1000, ... ʤɤǤ⤤
1111111110000000001 Τ褦 n9 = 111111111 ȯƤ⤤Τǡ
n ̵¤ˤޤ͡

錄⤽פޤ־ʤΤϤǤ͡
ɤȤŪʿ⤫Фʤơ10^2 ΤΤ줿ȤˡʤۤɤáäƤ錄ⴶޤʬȤɤƻפĤʤääƤäȲäꡣ(^^)

ݥȤϡn^2 η׻Ƿη꤬ʤȤʤΤǡʳμβ⤢뤫⤷ޤ

Ǥ͡x 10^(k+1) k+1 οηʾˤʤ褦ˤƤз夬Ϥʤ櫓顣¸ߤФΤǰĤǤʬ䤹Τ뤳ȤФ櫓ǡˡǺΤִñäƤȤʤǤ礦͡
811ʶ 8:46:26 28067
uchinyan
#28067
錄⤽פޤ־ʤΤϤǤ͡
֤פؤˤޤˡȤ̣Ǥϡ0 ǾʤΤǡʡȻפޤ
ɤǶϡĤ⡩ʤΤǡְäƤ뤫⤷ޤ (^^;
˷Ȥ̣Ǥϡ2 ȯơ
2, 22, 2202, 22020002 = 2202 * 10^4 + 2, 22020002 * 10^8 + 2, ...
s(n) 2, 4, 6, 8, ... ȤʤΤǡ1000 ®Ť⸺Ǥ礦
n ǻȤϡ0 3 3 1 ĤʤΤǡ򤦤ޤѤƤС򸺤餹פϤǤ뤫⤷ޤ
ϡMelvy#28059ǻŦʤäƤޤ͡

¤ϡMelvyϡ䤬ۤʤȤͤƤ֤ˡäȴʷDz򤫤Ƥ櫓ǡ
餿ơʤƤޤ (^^;

äɲá
ä긫ƨƤޤ#28063ͧͤβˡǡ
N=10^(k+1)*n+1ȤS(N)=m+1S(N^2)=(m+1)^2
ϡ2n ηηθơ10^(k+1) ȤƤΤǤ礦¸ߤˤϽʬ
εǼˡβǡs(n^2) = (s(n))^2 ȤƤΤǡn^2 η׻˷η꤬äƤϤʤ餺
äơ2n ηϤʤ
N = 10^k * n + 1 Ȥ S(N) = m+1S(N^2) = (m+1)^2
ǽʬǤ͡򸺤餻ޤ
MelvyβˡϡǤͥƤ褦Ǥ

βҤ轵οޤǤߵ٤ߤǤϤɤǤ礦
äȤ⡢Ǻ˻ä
ϽмҤʤΤǡꥢ륿໲äϤޤǤˤʤʤ
ˤʤäƤޤ͡ʤ
ͥνȡ 811ʶ 17:42:20 MAIL:uchi@sco.bekkoame.ne.jp 28068
⡼ޥ
#26068
ˤǤ꤯s(n)=1000,s(n^2)=1000^2 Ǥ뤫ɤ꤫ǤϤʤǤ͡

N=10^(k+1)*n+1ȤS(N)=m+1S(N^2)=(m+1)^2
ϡ2n ηηθơ10^(k+1) ȤƤΤǤ礦¸ߤˤϽʬ
εǼˡβǡs(n^2) = (s(n))^2 ȤƤΤǡn^2 η׻˷η꤬äƤϤʤ餺
äơ2n ηϤʤ
N = 10^k * n + 1 Ȥ S(N) = m+1S(N^2) = (m+1)^2
ǽʬǤ͡򸺤餻ޤ

ϾʤȤ1ϤޤäǤ⡢ο5099Ȥ1夬뤫顢10^(k+1) ꥹϾʤΤʤäƻפΤǤ(^^;

811ʶ 18:55:57 28069
uchinyan
#28069
䤿ˤǤ꤯s(n)=1000,s(n^2)=1000^2 Ǥ뤫ɤ꤫ǤϤʤǤ͡
꤫Ǥʤ褦ˡۤƤ뤫Ǥ
դ˸ȡ10^k ܡ n ǻȤ 0 3 3 1 Ĥ2 3 ϰˤϻȤʤ ꥮθ³Τ褦Ǥ
10^(k+1) ܤƤгΤ˥ꥹϸޤs(n^2) = (s(n))^2 褦˽缡ۤƤΤǡn^2 η׻˷ηϤꤨ
䤳ϾʤȤ1ϤޤäǤ⡢ο5099Ȥ1夬뤫顢
ȤȤϡιˡ餷ơꤨʤȻפޤ
ºݡn k Ȥ n -> 10^k * n + 1 ΤȤ
n, n^2
1, 1
11, 121
1101, 1212201 = 121 22 01
11010001, 121220122020001 = 1212201 2202 0001
1101000100000001, 1212201220200012202000200000001 = 121220122020001 22020002 00000001
11010001000000010000000000000001, 1212201220200012202000200000001 2202000200000002 0000000000000001
...
Ǥ
ʤߤˡMelvy#28059ͿƤ빽ˡϡ岼η夬Ҥä֤äˤʤޤ
10^k ܤμˡȸŪˤƱȻפޤ
ͥνȡ 811ʶ 23:36:00 MAIL:uchi@sco.bekkoame.ne.jp 28070
⡼ޥ
#28070
2 ϤޤäϲǽȤʬޤ(^^)
1Ϥޤä硢ɤοǽο1ʳˤϤʤäƤΤ餫ʤΤʤȤС2ˤʤ뤳ȤС4ˤ⡢8ˤʤꤽ顢8λϼϰ夬ޤ͡
餫ʤ褦˻פΤǤñˤޤǤ礦
Ȥ⡢夬ꤷäơο0ʤΤ
ŪʤȤʬʤʤäƤʤ(^^;
812ڡ 8:58:29 28071
uchinyan
#28071
󡢺äʤˡμºݤη׻βͤС餫ȻפǤ
ͥνȡ 812ڡ 15:22:54 MAIL:uchi@sco.bekkoame.ne.jp 28072
⡼ޥ
#28072
ˡ顢ǽο1Ǥ͡
ʤ顢n*10^k+1 ʤ(^^;

ޤOrZ
813 13:34:33 28073