Introduction

In the vast world of combinatorics and string theory, one intriguing problem involves creating a list of 10-digit strings where every 4-digit substring is unique. This problem not only highlights the complexities of permutation and combination but also shows how constraints can shape the solution space.

Understanding the Problem

To form a 10-digit string where every 4-digit substring is unique, one must ensure that no repeated sequence of four consecutive numbers appears within the string. This means that as you slide a window of four digits across the string, each snapshot of four should not have been seen before at any prior position.

Example

Consider the string "1234567890". As we slide the 4-digit window across this string, the substrings are:

1. "1234"
2. "2345"
3. "3456"
4. "4567"
5. "5678"
6. "6789"
7. "7890"

Each of these substrings is unique. Therefore, "1234567890" is a valid 10-digit string under the specified condition.

Technical Explanation

Constraints and Considerations

Given the problem's constraints, we must think about:

• Permutations and Combinations: There are $10^4 = 10,000$ possible 4-digit sequences using digits 0-9.
• Overlap and Continuity: Each 4-digit substring overlaps with the previous by three digits, except for the first occurance.
• Uniqueness: Ensuring that as each new digit is appended to the sequence, the resultant 4-digit sequence forms a previously unused combination.

For a string of length $n$, there are $n - 3$ possible 4-digit substrings. For a 10-digit string, this results in 7 substrings, each needing to be unique.

Algorithm

To generate such strings systematically:

1. Initialize a set to store 4-digit substrings for uniqueness checking.
2. Iterate through potential strings, verifying that each resultant 4-digit substring is not in the set.
3. Append the 4-digit substring to the set once confirmed unique and continue if the whole 10-digit sequence can be created.

Pseudocode Example

• The possible combinations for a valid string can be constrained using permutation theory.
• Since digits repeat and permutations are drawn from a set, direct calculation of possibilities can be tricky.
• The computational complexity for generating sequences can grow if we consider longer strings.
• A theoretical limit is imposed by the number of digits despite the large permutation potential (10! sequences for unique digits).