ps1 solution
TRANSCRIPT
-
_@?Ui_ @UhLiUL?L4Ut
hLM*i4 5i| 5L*|L? |*?i
Jlqr D1Jdqfld
4
H{huflvh 416
Q +w 4, @ 5/ \ +w, @ 5 ;w1 Suryh wkdw wkh iroorzlqj doorfdwlrq lv ihdvleoh=
f
k
3
+4, @ 4@5 iru k @ 4> 5> f
4
4
+4, @ 4@7/ f
4
4
+5, @ 6@7> f
5
4
+4, @ 6@7> f
5
4
+5, @ 4@7>
f
k
w
+w, @ f
k
w
+w. 4, @ 4@5 iru k @ 4> 5 dqg ;w 51
Lq wklv fdvh wkh frqglwlrq iru ihdvlelolw| ehfrphv=
5
[
k@4
f
k
w
+w, .
5
[
k@4
f
k
w4
+w, 5
Ihdvlelolw| pxvw eh fkhfnhg ;w 1
Dw w@4 wkh frqglwlrq lv=
f
4
4
+4, . f
5
4
+4, . f
4
3
+4, . f
5
3
+4, 5
vxevwlwxwlqj lq wkh jlyhq doorfdwlrq=
4@7 . 6@7 . 4@5 . 4@5 @ 5
Dw w@5=
f
4
5
+5, . f
5
5
+5, . f
4
4
+5, . f
5
4
+5, 5
4@5 . 4@5 . 6@7 . 4@7 @ 5
Iru wA5 +qrwh wkdw iurp qrz rq wkh frqglwlrq lv dozd|v wkh vdph,=
f
4
w
+w, . f
5
w
+w, . f
4
w4
+w, . f
5
w4
+w, 5
4@5 . 4@5 . 4@5 . 4@5 @ 5
Ihdvlelolw| lv dozd|v vdwlvhg1
H{huflvh 41:
Suryh wkdw li dq doorfdwlrq lv Sduhwr rswlpdo/ wkhq lw lv hflhqw1 Vxssrvh
wkdw dq doorfdwlrq lv qrw hflhqw> wkhq wkhuh lv vrph xqdoorfdwhg jrrg lq vrph
shulrg1 Vlqfh xwlolw| lv lqfuhdvlqj lq frqvxpswlrq/ li wkh xqdoorfdwhg jrrg lv
WW,5c 5|LU!L*4 N?iht|)
L?i Sf.
,4@*G}?L}@?U@9ittti
4
-
jlyhq wr vrphrqh wkh uhvxow lv d Sduhwr vxshulru doorfdwlrq1 Frqfoxvlrq= d
Sduhwr Rswlpdo doorfdwlrq pxvw eh hflhqw1
H{huflvhv 4146 ) 4147
Q +w, @ Q A 3/ \ +w, @ |Q A 3 ;w> f
k
w
+w, @ |/ f
k
w4
+w, @ +4 ,|/
5 ^3> 4`> x
k
w
@
f
k
w
+w,
4@5
.
f
k
w
+w. 4,
4@5
;w1 Suryh wkdw qr doorfdwlrq lq wkh
fodvv 5 ^3> 4` lv Sduhwr vxshulru wr @ 4@71 Ghvfuleh doo *v iru zklfk wkhuh
grhv qrw h{lvw d sduhwr vxshulru doorfdwlrq dprqj wkh phqwlrqhg fodvv1
Iluvw/ gudz d slfwxuh olnh Iljxuh 41< lq wkh errn1 Qrwh wkdw wkh srsxodwlrq lv
frqvwdqw +vl}h Q, dqg wkdw zh duh frqvlghulqj rqo| v|pphwulf doorfdwlrqv1 Wklv
lpsolhv wkdw zh fdq frqyhqlhqwo| frqvlghu shu0fdslwd txdqwlwlhv1 Wkh pd{lpxp
shu0fdslwd hqgrzphqw wkdw dq lqglylgxdo fdq kdyh lv |Q@Q @ |1
Wklv pxvw eh glylghg ehwzhhq wkh |rxqj dqg rog1 Wkh vwudljkw olqh zlwk vorsh
4 uhsuhvhqwv h{dfwo| wkh ihdvleoh dqg hflhqw zd|v wr vsolw wklv hqgrzphqw1
Pruhryhu/ lw uhsuhvhqwv wkh zkroh fodvv ri doorfdwlrq wkdw zh duh frqvlghulqj
lq wklv h{huflvh + ; ,= wkh frruglqdwhv ri dq| srlqw rq wkdw olqh duh jlyhq e|
|> +4 , | 1
Orrn dw wkh lqglhuhqfh fxuyh wdqjhqw wr wkdw olqh1 Wkh wdqjhqf| srlqw +H,
lv fkdudfwhul}hg e| wkh frqglwlrq=
PUV @ vorsh ri lqglii= fxuyh @ vorsh ri vwudljkw olqh @ 4
Zh fdq fdofxodwh lw/ xvlqj wkh idfw wkdw frqvxpswlrq wrgd| dqg wrpruurz
duh jlyhq e| |> +4 , |1 Xvlqj wklv zh rewdlq=
PUV @
4@5 ^|`
4@5
4@5 ^+4 , |`
4@5
@
u
4
lpsrvh wklv @4 dqg jhw @ 4@5
Qrz lqwhusuhw wkh jxuh> li dq lqglylgxdo frxog fkrrvh d srlqw rq wkh || olqh/
kh zrxog slfn H ehfdxvh lw lv wkh doorfdwlrq wkdw gholyhuv wkh kljkhvw olihwlph
xwlolw|1 Exw fxuuhqwv rog gr qrw fduh derxw sdvw frqvxpswlrq dq| pruh> wkh|
mxvw zdqw wr pd{lpl}h fxuuhqw frqvxpswlrq1 Wklv phdqv wkdw d Sduhwr lpsuryh0
phqw frxog qhyhu uhgxfh frqvxpswlrq ri rog +rwkhuzlvh fxuuhqw rog zrxog eh
xqdpeljxrxvo| zruvh r,1 Wklv/ lq wxuq/ lpsolhv wkdw lw lv srvvleoh wr lpsuryh
dq| doorfdwlrq rq wkh olqh || ehorz H/ wdnlqj dzd| vrphwklqj iurp wkh |rxqj
jlylqj lw wr wkh rog dqg wkxv prylqj wrzdugv srlqw H +ehvw doorfdwlrq, zlwkrxw
ohdylqj dq|rqh zruvh r1 Exw li wkh vwduwlqj doorfdwlrq lv deryh H/ lw lv srvvleoh
wr lpsuryh wkh olihwlph xwlolw| ri ixwxuh jhqhudwlrqv +prylqj wrzdugv H, rqo| dw
wkh h{shqvh ri fxuuhqw rog1 Wkxv/ wkh Sduhwr rswlpdo doorfdwlrqv fruuhvsrqg wr
4@5 1
Dgglwlrqdo H{huflvh 4
Uhgudz Iljxuh 41< lq wkh errn> vlqfh wkhuh lv srsxodwlrq jurzwk/ wkh vwudljkw
olqh fruuhvsrqglqj wr ihdvleoh dqg hflhqw doorfdwlrqv lv wkh rqh vkrzq lq Iljxuh
416/ zlwk vorsh q=
5
-
Qrz lw lv srvvleoh wr iroorz wkh vdph uhdvrqlqj dv ehiruh wr suryh wkdw wkh
frqglwlrq iru Sduhwr rswlpdolw| ehfrphv= PUV q
H{huflvh 514
Ghulyh wkh ghpdqg ixqfwlrq iru x
k
w
@
f
k
w
+w,
4@5
.
f
k
w
+w. 4,
4@5
1
Pd{lpl}dwlrq ri xwlolw| uhtxluhv=
u+w, @PUV @
4@5
f
k
w
+w,
4@5
4@5
f
k
w
+w. 4,
4@5
@
4
v
f
k
w
+w. 4,
f
k
w
+w,
Iurp wkh exgjhw frqvwudlqw zh kdyh f
k
w
+w. 4, @ u +w,
$
k
w
+w, f
k
w
+w,
.
$
k
w
+w. 4,> vxevwlwxwlqj wklv lqwr wkh suhylrxv htxdwlrq dqg vroylqj iru f
k
w
+w,
zh jhw wkh ghpdqg iru frqvxpswlrq=
f
k
w
+w, @
$
k
w
+w,
5
u +w, . 4
.
$
k
w
+w. 4,
u +w,
4 .
5
u +w,
H{huflvh 516
Ilqg wkh vdylqj ixqfwlrq iru wkh iroorzlqj fdvhv=
d1 Q +w, @ 433 / $
k
w
@ ^5> 4` / x
k
w
@ f
k
w
+w,
f
k
w
+w. 4,
/ @ 41 Wkh xwlolw|
pd{lpl}dwlrq frqglwlrq lv
u+w, @PUV @
f
k
w
+w. 4,
f
k
w
+w,
vxevwlwxwlqj iurp wkh exgjhw frqvwudlqw f
k
w
+w. 4, @ u +w,
$
k
w
+w, f
k
w
+w,
.
$
k
w
+w. 4, dqg vroylqj iru f
k
w
+w,=
f
k
w
+w, @
$
k
w
+w,
5
.
$
k
w
+w. 4,
5u +w,
@ 4 .
4
5u +w,
lqglylgxdo vdylqj lv $
k
w
+w, f
k
w
+w, @ 4
4
5u+w,
dqg djjuhjdwh vdylqj lv=
V
w
+w, @
433
[
k@4
4
4
5u +w,
@ 433
83
u +w,
e1 Dv ehiruh/ exw $
k
w
@
^5> 4` / k @ 4> 5> ===> 83
^4> 4` / k @ 84> 85> > ===> 433
1 Xvlqj wkh suhyl0
rxvo| irxqg ghpdqg iru frqvxpswlrq zh jhw=
f
k
w
+w, @ 4 .
4
5u +w,
iru k @ 4> = = = > 83
f
k
w
+w, @
4
5
.
4
5u +w,
iru k @ 84> = = = > 433
6
-
Djjuhjdwh vdylqj ehfrphv=
V
w
+w, @
83
[
k@4
4
4
5u +w,
.
433
[
k@84
4
5
4
5u +w,
@ :8
83
u +w,
H{huflvh 517
Ghvfuleh wkh frpshwlwlyh htxloleulxp1
d1 Q +w, @ 433 / $
k
w
@ ^5> 4` / x
k
w
@ f
k
w
+w,
f
k
w
+w. 4,
/ @ 4 ;w1 Zh douhdg|
kdyh wkh djjuhjdwh vdylqj ixqfwlrq +iurp h{1516,1 Vhwwlqj lw htxdo wr }hur zh
jhw=
V
w
+w, @ 433
83
u +w,
@ 3, u +w, @ 4@5
Qrwh wkdw lq wklv fdvh lqglylgxdo vdylqj lv }hur dqg lqglylgxdo frqvxpswlrq
lv htxdo wr wkh hqgrzphqw1
f1 Dv ehiruh/ exw x
k
w
@
f
k
w
+w,
4@5
.
f
k
w
+w. 4,
4@5
1 Zh kdyh wkh lqglylgxdo
ghpdqg iru frqvxpswlrq iurp h{1 514> wkh vdylqj ixqfwlrq lv=
v
k
w
+w, @ $
k
w
+w, f
k
w
+w, @
5u +w,
5
4
u +w, ^4 . u +w,`
Djjuhjdwh vdylqj lv mxvw V
w
+w, @ 433v
k
w
+w,> vhwwlqj lw htxdo wr }hur jlyhv
u +w, @
s
4@5> Dovr lq wklv fdvh lqglylgxdo vdylqj lv }hur dqg lqglylgxdo frq0
vxpswlrq lv htxdo wr wkh hqgrzphqw1
h1 Dv lq d/ exw $
k
w
@
^5> 4` / k @ 4> 5> ===> 93
^4> 4` / k @ 94> 95> > ===> 433
1 Zh douhdg| kdyh
wkh lqglylgxdo vdylqj ixqfwlrq +h{1 516, exw qrw hqgrzphqwv duh glhuhqw dqg
djjuhjdwh vdylqj ehfrphv=
V
w
+w, @
93
[
k@4
4
4
5u +w,
.
433
[
k@94
4
5
4
5u +w,
@ ;3
83
u +w,
Vhw V
w
+w, @ 3, u +w, @ 8@;
Lqglylgxdo vdylqjv=
Iru k @ 4> = = = > 93 v
k
w
+w, @ 4
;
58
@
4
8
Iru k @ 94> = = = > 433 v
k
w
+w, @
4
5
;
58
@
6
43
H{huflvh 518
Suryh wkdw wkh doorfdwlrqv lq H{1 517 duh qrw Sduhwr rswlpdo1
Wkh frpshwlwlyh htxloleuld frpsxwhg lq wkh h{huflvh 517 duh QRW Sduhwr
rswlpdo ehfdxvh Sduhwr rswlpdolw| uhtxluhv u +w, @ PUV 4 > lq doo wkh fdvhv
zh irxqg u +w, @PUV ? 4 1
7
-
Dgglwlrqdo H{huflvh 5
Zh nqrz wkdw/ lq fdvh ri srsxodwlrq jurzwk/ dq doorfdwlrq lv Sduhwr rswlpdo
l wkh iroorzlqj frqglwlrqv duh vdwlvhg=
41 hflhqf|
51 htxdolw| ri PUV +zlwklq jhqhudwlrqv,
61 PUV q
Zh dovr nqrz wkdw d Frpshwlwlyh Htxloleulxp vdwlvhv 4 dqg 51 Pruhryhu/
lq d frpshwlwlyh htxloleulxp zh kdyh PUV @ u +zk|B Orrn dw Iljxuh 516= u
lv htxdo wr wkh vorsh ri wkh lqglhuhqfh fxuyh wkurxjk wkh hqgrzphqw srlqw/
zklfk lv htxdo wr wkh PUV,1 Wklv phdqv wkdw wkh 6ug qhfhvvdu| dqg vxflhqw
frqglwlrq iru d frpshwlwlyh htxloleulxp wr eh Sduhwr rswlpdo fdq eh uhzulwwhq
dv u q 1
8