Unlock an answering machine using minimum number of digits
$begingroup$
You know these answering machines with remote inquiry facility, where you can call the answering machine and enter a 4 digit passcode into your telephone keypad, so you can listen to your messages from anywhere you like.
Many of these machines are quite dumb in the way you can feed the codes. They let you in if they encounter the right combinations of digits, regardless what you sent before.
Example: code is 1234
if you feed the machine 1234, you're in
if you feed the machine 01234, you're in
if you feed the machine 0121234 you're in
It just waits for the right 4 digits to come in, regardless of what happened before.
In order to hack the machine, could try all numbers from 0000 to 9999, sending 40000 sounds across the wire. But since you are a smart hacker, you see that there's room for optimization.
What is the shortest series of digits you have to send to the answering machine in order to break the code in any case?
P.S: In the comments section, Hugh has mentioned links to other questions asking the exact same question (but with different wording) . All answers to those questions use graph theory and de Bruijn sequence.
I would like an answer to this question which does not require any knowledge of graph theory or de Bruijn sequence.
logical-deduction combinatorics
edited 49 mins ago
Rubio♦
28.8k565178
asked 2 hours ago
Hemant AgarwalHemant Agarwal
12
New contributor
Hemant Agarwal is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
You know these answering machines with remote inquiry facility, where you can call the answering machine and enter a 4 digit passcode into your telephone keypad, so you can listen to your messages from anywhere you like.
Many of these machines are quite dumb in the way you can feed the codes. They let you in if they encounter the right combinations of digits, regardless what you sent before.
Example: code is 1234
if you feed the machine 1234, you're in
if you feed the machine 01234, you're in
if you feed the machine 0121234 you're in
It just waits for the right 4 digits to come in, regardless of what happened before.
In order to hack the machine, could try all numbers from 0000 to 9999, sending 40000 sounds across the wire. But since you are a smart hacker, you see that there's room for optimization.
What is the shortest series of digits you have to send to the answering machine in order to break the code in any case?
P.S: In the comments section, Hugh has mentioned links to other questions asking the exact same question (but with different wording) . All answers to those questions use graph theory and de Bruijn sequence.
I would like an answer to this question which does not require any knowledge of graph theory or de Bruijn sequence.
logical-deduction combinatorics
edited 49 mins ago
Rubio♦
28.8k565178
asked 2 hours ago
Hemant AgarwalHemant Agarwal
12
New contributor
Hemant Agarwal is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
Already answered on Math.SE — math.stackexchange.com/questions/2131588/…
$endgroup$
– Hugh
2 hours ago
$begingroup$
...a-a-a-a-a-a-and also on Puzzling.SE as "one beer too many". I've marked this question as a duplicate.
$endgroup$
– Hugh
1 hour ago
$begingroup$
Reopened as non-duplicate following OP's request for a solution not requiring graph theory of use of the de Bruijn sequence. This excludes the existing answers both here and on the previously asked questions; we usually don't like edits to questions that invalidate previously valid answers, but in this case @Hugh's answer was mostly a rehash of those answers (though with the string explicitly spelled out), so we haven't lost effort in solving this. I'm not sure there's a good way to solve this without using the tools just excluded, so ... good luck solvers! :)
$endgroup$
– Rubio♦
42 mins ago
add a comment |
$begingroup$
You know these answering machines with remote inquiry facility, where you can call the answering machine and enter a 4 digit passcode into your telephone keypad, so you can listen to your messages from anywhere you like.
Many of these machines are quite dumb in the way you can feed the codes. They let you in if they encounter the right combinations of digits, regardless what you sent before.
Example: code is 1234
if you feed the machine 1234, you're in
if you feed the machine 01234, you're in
if you feed the machine 0121234 you're in
It just waits for the right 4 digits to come in, regardless of what happened before.
In order to hack the machine, could try all numbers from 0000 to 9999, sending 40000 sounds across the wire. But since you are a smart hacker, you see that there's room for optimization.
What is the shortest series of digits you have to send to the answering machine in order to break the code in any case?
P.S: In the comments section, Hugh has mentioned links to other questions asking the exact same question (but with different wording) . All answers to those questions use graph theory and de Bruijn sequence.
I would like an answer to this question which does not require any knowledge of graph theory or de Bruijn sequence.
logical-deduction combinatorics
edited 49 mins ago
Rubio♦
28.8k565178
asked 2 hours ago
Hemant AgarwalHemant Agarwal
12
New contributor
Hemant Agarwal is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
You know these answering machines with remote inquiry facility, where you can call the answering machine and enter a 4 digit passcode into your telephone keypad, so you can listen to your messages from anywhere you like.
Many of these machines are quite dumb in the way you can feed the codes. They let you in if they encounter the right combinations of digits, regardless what you sent before.
Example: code is 1234
if you feed the machine 1234, you're in
if you feed the machine 01234, you're in
if you feed the machine 0121234 you're in
It just waits for the right 4 digits to come in, regardless of what happened before.
In order to hack the machine, could try all numbers from 0000 to 9999, sending 40000 sounds across the wire. But since you are a smart hacker, you see that there's room for optimization.
What is the shortest series of digits you have to send to the answering machine in order to break the code in any case?
P.S: In the comments section, Hugh has mentioned links to other questions asking the exact same question (but with different wording) . All answers to those questions use graph theory and de Bruijn sequence.
I would like an answer to this question which does not require any knowledge of graph theory or de Bruijn sequence.
logical-deduction combinatorics
logical-deduction combinatorics
edited 49 mins ago
Rubio♦
28.8k565178
asked 2 hours ago
Hemant AgarwalHemant Agarwal
12
New contributor
Hemant Agarwal is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 49 mins ago
Rubio♦
28.8k565178
asked 2 hours ago
Hemant AgarwalHemant Agarwal
12
New contributor
Hemant Agarwal is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 49 mins ago
Rubio♦
28.8k565178
edited 49 mins ago
Rubio♦
28.8k565178
edited 49 mins ago
Rubio♦
28.8k565178
28.8k565178
asked 2 hours ago
Hemant AgarwalHemant Agarwal
12
New contributor
Hemant Agarwal is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 hours ago
Hemant AgarwalHemant Agarwal
12
asked 2 hours ago
Hemant AgarwalHemant Agarwal
12
12
New contributor
Hemant Agarwal is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Hemant Agarwal is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Hemant Agarwal is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$begingroup$
Already answered on Math.SE — math.stackexchange.com/questions/2131588/…
$endgroup$
– Hugh
2 hours ago
$begingroup$
...a-a-a-a-a-a-and also on Puzzling.SE as "one beer too many". I've marked this question as a duplicate.
$endgroup$
– Hugh
1 hour ago
$begingroup$
Reopened as non-duplicate following OP's request for a solution not requiring graph theory of use of the de Bruijn sequence. This excludes the existing answers both here and on the previously asked questions; we usually don't like edits to questions that invalidate previously valid answers, but in this case @Hugh's answer was mostly a rehash of those answers (though with the string explicitly spelled out), so we haven't lost effort in solving this. I'm not sure there's a good way to solve this without using the tools just excluded, so ... good luck solvers! :)
$endgroup$
– Rubio♦
42 mins ago
add a comment |
$begingroup$
Already answered on Math.SE — math.stackexchange.com/questions/2131588/…
$endgroup$
– Hugh
2 hours ago
$begingroup$
...a-a-a-a-a-a-and also on Puzzling.SE as "one beer too many". I've marked this question as a duplicate.
$endgroup$
– Hugh
1 hour ago
$begingroup$
Reopened as non-duplicate following OP's request for a solution not requiring graph theory of use of the de Bruijn sequence. This excludes the existing answers both here and on the previously asked questions; we usually don't like edits to questions that invalidate previously valid answers, but in this case @Hugh's answer was mostly a rehash of those answers (though with the string explicitly spelled out), so we haven't lost effort in solving this. I'm not sure there's a good way to solve this without using the tools just excluded, so ... good luck solvers! :)
$endgroup$
– Rubio♦
42 mins ago
$begingroup$
Already answered on Math.SE — math.stackexchange.com/questions/2131588/…
$endgroup$
– Hugh
2 hours ago
$begingroup$
Already answered on Math.SE — math.stackexchange.com/questions/2131588/…
$endgroup$
– Hugh
2 hours ago
$begingroup$
...a-a-a-a-a-a-and also on Puzzling.SE as "one beer too many". I've marked this question as a duplicate.
$endgroup$
– Hugh
1 hour ago
$begingroup$
...a-a-a-a-a-a-and also on Puzzling.SE as "one beer too many". I've marked this question as a duplicate.
$endgroup$
– Hugh
1 hour ago
$begingroup$
Reopened as non-duplicate following OP's request for a solution not requiring graph theory of use of the de Bruijn sequence. This excludes the existing answers both here and on the previously asked questions; we usually don't like edits to questions that invalidate previously valid answers, but in this case @Hugh's answer was mostly a rehash of those answers (though with the string explicitly spelled out), so we haven't lost effort in solving this. I'm not sure there's a good way to solve this without using the tools just excluded, so ... good luck solvers! :)
$endgroup$
– Rubio♦
42 mins ago
$begingroup$
Reopened as non-duplicate following OP's request for a solution not requiring graph theory of use of the de Bruijn sequence. This excludes the existing answers both here and on the previously asked questions; we usually don't like edits to questions that invalidate previously valid answers, but in this case @Hugh's answer was mostly a rehash of those answers (though with the string explicitly spelled out), so we haven't lost effort in solving this. I'm not sure there's a good way to solve this without using the tools just excluded, so ... good luck solvers! :)
$endgroup$
– Rubio♦
42 mins ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
It sounds like you're looking for the order-$4$ de Bruijn sequence with alphabet $A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}$. In that case—the shortest string is $10003$ digits long.
00001000200030004000500060007000800090011001200130014001500160017001800190021002200230024002500260027002800290031003200330034003500360037003800390041004200430044004500460047004800490051005200530054005500560057005800590061006200630064006500660067006800690071007200730074007500760077007800790081008200830084008500860087008800890091009200930094009500960097009800990101020103010401050106010701080109011101120113011401150116011701180119012101220123012401250126012701280129013101320133013401350136013701380139014101420143014401450146014701480149015101520153015401550156015701580159016101620163016401650166016701680169017101720173017401750176017701780179018101820183018401850186018701880189019101920193019401950196019701980199020203020402050206020702080209021102120213021402150216021702180219022102220223022402250226022702280229023102320233023402350236023702380239024102420243024402450246024702480249025102520253025402550256025702580259026102620263026402650266026702680269027102720273027402750276027702780279028102820283028402850286028702880289029102920293029402950296029702980299030304030503060307030803090311031203130314031503160317031803190321032203230324032503260327032803290331033203330334033503360337033803390341034203430344034503460347034803490351035203530354035503560357035803590361036203630364036503660367036803690371037203730374037503760377037803790381038203830384038503860387038803890391039203930394039503960397039803990404050406040704080409041104120413041404150416041704180419042104220423042404250426042704280429043104320433043404350436043704380439044104420443044404450446044704480449045104520453045404550456045704580459046104620463046404650466046704680469047104720473047404750476047704780479048104820483048404850486048704880489049104920493049404950496049704980499050506050705080509051105120513051405150516051705180519052105220523052405250526052705280529053105320533053405350536053705380539054105420543054405450546054705480549055105520553055405550556055705580559056105620563056405650566056705680569057105720573057405750576057705780579058105820583058405850586058705880589059105920593059405950596059705980599060607060806090611061206130614061506160617061806190621062206230624062506260627062806290631063206330634063506360637063806390641064206430644064506460647064806490651065206530654065506560657065806590661066206630664066506660667066806690671067206730674067506760677067806790681068206830684068506860687068806890691069206930694069506960697069806990707080709071107120713071407150716071707180719072107220723072407250726072707280729073107320733073407350736073707380739074107420743074407450746074707480749075107520753075407550756075707580759076107620763076407650766076707680769077107720773077407750776077707780779078107820783078407850786078707880789079107920793079407950796079707980799080809081108120813081408150816081708180819082108220823082408250826082708280829083108320833083408350836083708380839084108420843084408450846084708480849085108520853085408550856085708580859086108620863086408650866086708680869087108720873087408750876087708780879088108820883088408850886088708880889089108920893089408950896089708980899090911091209130914091509160917091809190921092209230924092509260927092809290931093209330934093509360937093809390941094209430944094509460947094809490951095209530954095509560957095809590961096209630964096509660967096809690971097209730974097509760977097809790981098209830984098509860987098809890991099209930994099509960997099809991111211131114111511161117111811191122112311241125112611271128112911321133113411351136113711381139114211431144114511461147114811491152115311541155115611571158115911621163116411651166116711681169117211731174117511761177117811791182118311841185118611871188118911921193119411951196119711981199121213121412151216121712181219122212231224122512261227122812291232123312341235123612371238123912421243124412451246124712481249125212531254125512561257125812591262126312641265126612671268126912721273127412751276127712781279128212831284128512861287128812891292129312941295129612971298129913131413151316131713181319132213231324132513261327132813291332133313341335133613371338133913421343134413451346134713481349135213531354135513561357135813591362136313641365136613671368136913721373137413751376137713781379138213831384138513861387138813891392139313941395139613971398139914141514161417141814191422142314241425142614271428142914321433143414351436143714381439144214431444144514461447144814491452145314541455145614571458145914621463146414651466146714681469147214731474147514761477147814791482148314841485148614871488148914921493149414951496149714981499151516151715181519152215231524152515261527152815291532153315341535153615371538153915421543154415451546154715481549155215531554155515561557155815591562156315641565156615671568156915721573157415751576157715781579158215831584158515861587158815891592159315941595159615971598159916161716181619162216231624162516261627162816291632163316341635163616371638163916421643164416451646164716481649165216531654165516561657165816591662166316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000
This was shamelessly taken from the following source.
answered 2 hours ago
HughHugh
2,0231923
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$begingroup$
It sounds like you're looking for the order-$4$ de Bruijn sequence with alphabet $A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}$. In that case—the shortest string is $10003$ digits long.
00001000200030004000500060007000800090011001200130014001500160017001800190021002200230024002500260027002800290031003200330034003500360037003800390041004200430044004500460047004800490051005200530054005500560057005800590061006200630064006500660067006800690071007200730074007500760077007800790081008200830084008500860087008800890091009200930094009500960097009800990101020103010401050106010701080109011101120113011401150116011701180119012101220123012401250126012701280129013101320133013401350136013701380139014101420143014401450146014701480149015101520153015401550156015701580159016101620163016401650166016701680169017101720173017401750176017701780179018101820183018401850186018701880189019101920193019401950196019701980199020203020402050206020702080209021102120213021402150216021702180219022102220223022402250226022702280229023102320233023402350236023702380239024102420243024402450246024702480249025102520253025402550256025702580259026102620263026402650266026702680269027102720273027402750276027702780279028102820283028402850286028702880289029102920293029402950296029702980299030304030503060307030803090311031203130314031503160317031803190321032203230324032503260327032803290331033203330334033503360337033803390341034203430344034503460347034803490351035203530354035503560357035803590361036203630364036503660367036803690371037203730374037503760377037803790381038203830384038503860387038803890391039203930394039503960397039803990404050406040704080409041104120413041404150416041704180419042104220423042404250426042704280429043104320433043404350436043704380439044104420443044404450446044704480449045104520453045404550456045704580459046104620463046404650466046704680469047104720473047404750476047704780479048104820483048404850486048704880489049104920493049404950496049704980499050506050705080509051105120513051405150516051705180519052105220523052405250526052705280529053105320533053405350536053705380539054105420543054405450546054705480549055105520553055405550556055705580559056105620563056405650566056705680569057105720573057405750576057705780579058105820583058405850586058705880589059105920593059405950596059705980599060607060806090611061206130614061506160617061806190621062206230624062506260627062806290631063206330634063506360637063806390641064206430644064506460647064806490651065206530654065506560657065806590661066206630664066506660667066806690671067206730674067506760677067806790681068206830684068506860687068806890691069206930694069506960697069806990707080709071107120713071407150716071707180719072107220723072407250726072707280729073107320733073407350736073707380739074107420743074407450746074707480749075107520753075407550756075707580759076107620763076407650766076707680769077107720773077407750776077707780779078107820783078407850786078707880789079107920793079407950796079707980799080809081108120813081408150816081708180819082108220823082408250826082708280829083108320833083408350836083708380839084108420843084408450846084708480849085108520853085408550856085708580859086108620863086408650866086708680869087108720873087408750876087708780879088108820883088408850886088708880889089108920893089408950896089708980899090911091209130914091509160917091809190921092209230924092509260927092809290931093209330934093509360937093809390941094209430944094509460947094809490951095209530954095509560957095809590961096209630964096509660967096809690971097209730974097509760977097809790981098209830984098509860987098809890991099209930994099509960997099809991111211131114111511161117111811191122112311241125112611271128112911321133113411351136113711381139114211431144114511461147114811491152115311541155115611571158115911621163116411651166116711681169117211731174117511761177117811791182118311841185118611871188118911921193119411951196119711981199121213121412151216121712181219122212231224122512261227122812291232123312341235123612371238123912421243124412451246124712481249125212531254125512561257125812591262126312641265126612671268126912721273127412751276127712781279128212831284128512861287128812891292129312941295129612971298129913131413151316131713181319132213231324132513261327132813291332133313341335133613371338133913421343134413451346134713481349135213531354135513561357135813591362136313641365136613671368136913721373137413751376137713781379138213831384138513861387138813891392139313941395139613971398139914141514161417141814191422142314241425142614271428142914321433143414351436143714381439144214431444144514461447144814491452145314541455145614571458145914621463146414651466146714681469147214731474147514761477147814791482148314841485148614871488148914921493149414951496149714981499151516151715181519152215231524152515261527152815291532153315341535153615371538153915421543154415451546154715481549155215531554155515561557155815591562156315641565156615671568156915721573157415751576157715781579158215831584158515861587158815891592159315941595159615971598159916161716181619162216231624162516261627162816291632163316341635163616371638163916421643164416451646164716481649165216531654165516561657165816591662166316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000
This was shamelessly taken from the following source.
answered 2 hours ago
HughHugh
2,0231923
$endgroup$
add a comment |
$begingroup$
It sounds like you're looking for the order-$4$ de Bruijn sequence with alphabet $A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}$. In that case—the shortest string is $10003$ digits long.
00001000200030004000500060007000800090011001200130014001500160017001800190021002200230024002500260027002800290031003200330034003500360037003800390041004200430044004500460047004800490051005200530054005500560057005800590061006200630064006500660067006800690071007200730074007500760077007800790081008200830084008500860087008800890091009200930094009500960097009800990101020103010401050106010701080109011101120113011401150116011701180119012101220123012401250126012701280129013101320133013401350136013701380139014101420143014401450146014701480149015101520153015401550156015701580159016101620163016401650166016701680169017101720173017401750176017701780179018101820183018401850186018701880189019101920193019401950196019701980199020203020402050206020702080209021102120213021402150216021702180219022102220223022402250226022702280229023102320233023402350236023702380239024102420243024402450246024702480249025102520253025402550256025702580259026102620263026402650266026702680269027102720273027402750276027702780279028102820283028402850286028702880289029102920293029402950296029702980299030304030503060307030803090311031203130314031503160317031803190321032203230324032503260327032803290331033203330334033503360337033803390341034203430344034503460347034803490351035203530354035503560357035803590361036203630364036503660367036803690371037203730374037503760377037803790381038203830384038503860387038803890391039203930394039503960397039803990404050406040704080409041104120413041404150416041704180419042104220423042404250426042704280429043104320433043404350436043704380439044104420443044404450446044704480449045104520453045404550456045704580459046104620463046404650466046704680469047104720473047404750476047704780479048104820483048404850486048704880489049104920493049404950496049704980499050506050705080509051105120513051405150516051705180519052105220523052405250526052705280529053105320533053405350536053705380539054105420543054405450546054705480549055105520553055405550556055705580559056105620563056405650566056705680569057105720573057405750576057705780579058105820583058405850586058705880589059105920593059405950596059705980599060607060806090611061206130614061506160617061806190621062206230624062506260627062806290631063206330634063506360637063806390641064206430644064506460647064806490651065206530654065506560657065806590661066206630664066506660667066806690671067206730674067506760677067806790681068206830684068506860687068806890691069206930694069506960697069806990707080709071107120713071407150716071707180719072107220723072407250726072707280729073107320733073407350736073707380739074107420743074407450746074707480749075107520753075407550756075707580759076107620763076407650766076707680769077107720773077407750776077707780779078107820783078407850786078707880789079107920793079407950796079707980799080809081108120813081408150816081708180819082108220823082408250826082708280829083108320833083408350836083708380839084108420843084408450846084708480849085108520853085408550856085708580859086108620863086408650866086708680869087108720873087408750876087708780879088108820883088408850886088708880889089108920893089408950896089708980899090911091209130914091509160917091809190921092209230924092509260927092809290931093209330934093509360937093809390941094209430944094509460947094809490951095209530954095509560957095809590961096209630964096509660967096809690971097209730974097509760977097809790981098209830984098509860987098809890991099209930994099509960997099809991111211131114111511161117111811191122112311241125112611271128112911321133113411351136113711381139114211431144114511461147114811491152115311541155115611571158115911621163116411651166116711681169117211731174117511761177117811791182118311841185118611871188118911921193119411951196119711981199121213121412151216121712181219122212231224122512261227122812291232123312341235123612371238123912421243124412451246124712481249125212531254125512561257125812591262126312641265126612671268126912721273127412751276127712781279128212831284128512861287128812891292129312941295129612971298129913131413151316131713181319132213231324132513261327132813291332133313341335133613371338133913421343134413451346134713481349135213531354135513561357135813591362136313641365136613671368136913721373137413751376137713781379138213831384138513861387138813891392139313941395139613971398139914141514161417141814191422142314241425142614271428142914321433143414351436143714381439144214431444144514461447144814491452145314541455145614571458145914621463146414651466146714681469147214731474147514761477147814791482148314841485148614871488148914921493149414951496149714981499151516151715181519152215231524152515261527152815291532153315341535153615371538153915421543154415451546154715481549155215531554155515561557155815591562156315641565156615671568156915721573157415751576157715781579158215831584158515861587158815891592159315941595159615971598159916161716181619162216231624162516261627162816291632163316341635163616371638163916421643164416451646164716481649165216531654165516561657165816591662166316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000
This was shamelessly taken from the following source.
answered 2 hours ago
HughHugh
2,0231923
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It sounds like you're looking for the order-$4$ de Bruijn sequence with alphabet $A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}$. In that case—the shortest string is $10003$ digits long.
00001000200030004000500060007000800090011001200130014001500160017001800190021002200230024002500260027002800290031003200330034003500360037003800390041004200430044004500460047004800490051005200530054005500560057005800590061006200630064006500660067006800690071007200730074007500760077007800790081008200830084008500860087008800890091009200930094009500960097009800990101020103010401050106010701080109011101120113011401150116011701180119012101220123012401250126012701280129013101320133013401350136013701380139014101420143014401450146014701480149015101520153015401550156015701580159016101620163016401650166016701680169017101720173017401750176017701780179018101820183018401850186018701880189019101920193019401950196019701980199020203020402050206020702080209021102120213021402150216021702180219022102220223022402250226022702280229023102320233023402350236023702380239024102420243024402450246024702480249025102520253025402550256025702580259026102620263026402650266026702680269027102720273027402750276027702780279028102820283028402850286028702880289029102920293029402950296029702980299030304030503060307030803090311031203130314031503160317031803190321032203230324032503260327032803290331033203330334033503360337033803390341034203430344034503460347034803490351035203530354035503560357035803590361036203630364036503660367036803690371037203730374037503760377037803790381038203830384038503860387038803890391039203930394039503960397039803990404050406040704080409041104120413041404150416041704180419042104220423042404250426042704280429043104320433043404350436043704380439044104420443044404450446044704480449045104520453045404550456045704580459046104620463046404650466046704680469047104720473047404750476047704780479048104820483048404850486048704880489049104920493049404950496049704980499050506050705080509051105120513051405150516051705180519052105220523052405250526052705280529053105320533053405350536053705380539054105420543054405450546054705480549055105520553055405550556055705580559056105620563056405650566056705680569057105720573057405750576057705780579058105820583058405850586058705880589059105920593059405950596059705980599060607060806090611061206130614061506160617061806190621062206230624062506260627062806290631063206330634063506360637063806390641064206430644064506460647064806490651065206530654065506560657065806590661066206630664066506660667066806690671067206730674067506760677067806790681068206830684068506860687068806890691069206930694069506960697069806990707080709071107120713071407150716071707180719072107220723072407250726072707280729073107320733073407350736073707380739074107420743074407450746074707480749075107520753075407550756075707580759076107620763076407650766076707680769077107720773077407750776077707780779078107820783078407850786078707880789079107920793079407950796079707980799080809081108120813081408150816081708180819082108220823082408250826082708280829083108320833083408350836083708380839084108420843084408450846084708480849085108520853085408550856085708580859086108620863086408650866086708680869087108720873087408750876087708780879088108820883088408850886088708880889089108920893089408950896089708980899090911091209130914091509160917091809190921092209230924092509260927092809290931093209330934093509360937093809390941094209430944094509460947094809490951095209530954095509560957095809590961096209630964096509660967096809690971097209730974097509760977097809790981098209830984098509860987098809890991099209930994099509960997099809991111211131114111511161117111811191122112311241125112611271128112911321133113411351136113711381139114211431144114511461147114811491152115311541155115611571158115911621163116411651166116711681169117211731174117511761177117811791182118311841185118611871188118911921193119411951196119711981199121213121412151216121712181219122212231224122512261227122812291232123312341235123612371238123912421243124412451246124712481249125212531254125512561257125812591262126312641265126612671268126912721273127412751276127712781279128212831284128512861287128812891292129312941295129612971298129913131413151316131713181319132213231324132513261327132813291332133313341335133613371338133913421343134413451346134713481349135213531354135513561357135813591362136313641365136613671368136913721373137413751376137713781379138213831384138513861387138813891392139313941395139613971398139914141514161417141814191422142314241425142614271428142914321433143414351436143714381439144214431444144514461447144814491452145314541455145614571458145914621463146414651466146714681469147214731474147514761477147814791482148314841485148614871488148914921493149414951496149714981499151516151715181519152215231524152515261527152815291532153315341535153615371538153915421543154415451546154715481549155215531554155515561557155815591562156315641565156615671568156915721573157415751576157715781579158215831584158515861587158815891592159315941595159615971598159916161716181619162216231624162516261627162816291632163316341635163616371638163916421643164416451646164716481649165216531654165516561657165816591662166316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000
This was shamelessly taken from the following source.
answered 2 hours ago
HughHugh
2,0231923
$endgroup$
It sounds like you're looking for the order-$4$ de Bruijn sequence with alphabet $A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}$. In that case—the shortest string is $10003$ digits long.
00001000200030004000500060007000800090011001200130014001500160017001800190021002200230024002500260027002800290031003200330034003500360037003800390041004200430044004500460047004800490051005200530054005500560057005800590061006200630064006500660067006800690071007200730074007500760077007800790081008200830084008500860087008800890091009200930094009500960097009800990101020103010401050106010701080109011101120113011401150116011701180119012101220123012401250126012701280129013101320133013401350136013701380139014101420143014401450146014701480149015101520153015401550156015701580159016101620163016401650166016701680169017101720173017401750176017701780179018101820183018401850186018701880189019101920193019401950196019701980199020203020402050206020702080209021102120213021402150216021702180219022102220223022402250226022702280229023102320233023402350236023702380239024102420243024402450246024702480249025102520253025402550256025702580259026102620263026402650266026702680269027102720273027402750276027702780279028102820283028402850286028702880289029102920293029402950296029702980299030304030503060307030803090311031203130314031503160317031803190321032203230324032503260327032803290331033203330334033503360337033803390341034203430344034503460347034803490351035203530354035503560357035803590361036203630364036503660367036803690371037203730374037503760377037803790381038203830384038503860387038803890391039203930394039503960397039803990404050406040704080409041104120413041404150416041704180419042104220423042404250426042704280429043104320433043404350436043704380439044104420443044404450446044704480449045104520453045404550456045704580459046104620463046404650466046704680469047104720473047404750476047704780479048104820483048404850486048704880489049104920493049404950496049704980499050506050705080509051105120513051405150516051705180519052105220523052405250526052705280529053105320533053405350536053705380539054105420543054405450546054705480549055105520553055405550556055705580559056105620563056405650566056705680569057105720573057405750576057705780579058105820583058405850586058705880589059105920593059405950596059705980599060607060806090611061206130614061506160617061806190621062206230624062506260627062806290631063206330634063506360637063806390641064206430644064506460647064806490651065206530654065506560657065806590661066206630664066506660667066806690671067206730674067506760677067806790681068206830684068506860687068806890691069206930694069506960697069806990707080709071107120713071407150716071707180719072107220723072407250726072707280729073107320733073407350736073707380739074107420743074407450746074707480749075107520753075407550756075707580759076107620763076407650766076707680769077107720773077407750776077707780779078107820783078407850786078707880789079107920793079407950796079707980799080809081108120813081408150816081708180819082108220823082408250826082708280829083108320833083408350836083708380839084108420843084408450846084708480849085108520853085408550856085708580859086108620863086408650866086708680869087108720873087408750876087708780879088108820883088408850886088708880889089108920893089408950896089708980899090911091209130914091509160917091809190921092209230924092509260927092809290931093209330934093509360937093809390941094209430944094509460947094809490951095209530954095509560957095809590961096209630964096509660967096809690971097209730974097509760977097809790981098209830984098509860987098809890991099209930994099509960997099809991111211131114111511161117111811191122112311241125112611271128112911321133113411351136113711381139114211431144114511461147114811491152115311541155115611571158115911621163116411651166116711681169117211731174117511761177117811791182118311841185118611871188118911921193119411951196119711981199121213121412151216121712181219122212231224122512261227122812291232123312341235123612371238123912421243124412451246124712481249125212531254125512561257125812591262126312641265126612671268126912721273127412751276127712781279128212831284128512861287128812891292129312941295129612971298129913131413151316131713181319132213231324132513261327132813291332133313341335133613371338133913421343134413451346134713481349135213531354135513561357135813591362136313641365136613671368136913721373137413751376137713781379138213831384138513861387138813891392139313941395139613971398139914141514161417141814191422142314241425142614271428142914321433143414351436143714381439144214431444144514461447144814491452145314541455145614571458145914621463146414651466146714681469147214731474147514761477147814791482148314841485148614871488148914921493149414951496149714981499151516151715181519152215231524152515261527152815291532153315341535153615371538153915421543154415451546154715481549155215531554155515561557155815591562156315641565156615671568156915721573157415751576157715781579158215831584158515861587158815891592159315941595159615971598159916161716181619162216231624162516261627162816291632163316341635163616371638163916421643164416451646164716481649165216531654165516561657165816591662166316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000
This was shamelessly taken from the following source.
answered 2 hours ago
HughHugh
2,0231923
edited 1 hour ago
answered 2 hours ago
HughHugh
2,0231923
answered 2 hours ago
HughHugh
2,0231923
answered 2 hours ago
HughHugh
2,0231923
2,0231923
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$begingroup$
Already answered on Math.SE — math.stackexchange.com/questions/2131588/…
$endgroup$
– Hugh
2 hours ago
$begingroup$
...a-a-a-a-a-a-and also on Puzzling.SE as "one beer too many". I've marked this question as a duplicate.
$endgroup$
– Hugh
1 hour ago
$begingroup$
Reopened as non-duplicate following OP's request for a solution not requiring graph theory of use of the de Bruijn sequence. This excludes the existing answers both here and on the previously asked questions; we usually don't like edits to questions that invalidate previously valid answers, but in this case @Hugh's answer was mostly a rehash of those answers (though with the string explicitly spelled out), so we haven't lost effort in solving this. I'm not sure there's a good way to solve this without using the tools just excluded, so ... good luck solvers! :)
$endgroup$
– Rubio♦
42 mins ago