Problem Detail: A safe is protected by a four-digit $(0-9)$ combination. The safe only considers the last four digits entered when deciding whether an input matches the passcode. For instance, if I enter the stream $012345$, I am trying each of the combinations $0123$, $1234$, and $2345$. Clearly, a 40000-length string $000000010002…9999$ is guaranteed to crack the safe. Can we try each of the 10000 combinations using a shorter string? What’s the shortest string we can devise to try every combination?
Asked By : jonshao
Answered By : David Richerby
The answer is to use a de Bruijn sequence, as discussed in response to this question on CS Theory. This gives a sequence of length $10^4=10,000$. However, the sequence is cyclic, in the sense that if you wrote it on a paper tape and joined the ends together to form a loop, only then would it contain every possible 4-digit sequence and some of those sequences would cross the join. So, for a linear sequence, you need to repeat the first three items at the end, giving $10,003$, improving on the obvious solution of length $40,000$. (Thanks to PålGD for pointing out the issue with cyclicity.)
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