Sequence containing every four-digit combination

[Mike314]

What is the shortest numerical string containing every possible combination of four digits? (0000 - 9999 inclusive)
(To clarify, the string 123456 would contain the combinations 1234, 2345 and 3456.)

In theory, the shortest possible sequence should be 10,003 digits long. But does such a sequence actually exist? Is there a way to prove or disprove it (without writing the whole thing out)?
If it doesn't exist, is there a way to calculate the minimum possible length of our ideal sequence?

Apologies if this is in the wrong forum. Please let me know if I should post elsewhere.

 

[Steven G]

What have you tried? Where are you stuck? Have you tried to solve this same problem with 2 digits? Or do you just want the answer?

 

[Mike314]

More after the answer. It's academic curiosity, rather than a homework problem.

Context: my flat door has a four-digit code, and I found out recently that there's more than one code that works (and still work if part of a longer string).
Which led me to wondering what the best way to find all possible entry codes was, and that led me here.

 

[Steven G]

Sorry, but we do not offers solutions on this forum. Rather we give leading hints. Why not try the problem with 2 digits. It will only take ,maybe ten minutes.

 

[Agent Smith]

There seems to be [imath]10^4[/imath] numbers you have to count. The upper bound would be [imath]4 \times 10^4[/imath] digits then (4 digits for each permutation).

 

[Agent Smith]

The lower bound [imath]y = x - 3[/imath]