The **answer** to the ten-digit problem is:

6210001000

True enough, but is there a logical or algebraic approach to the solution?

You can arrive at the solution through logic.

Consider that you have 10 digits. 9 of them zeros.

9,000,000,000 However, in this case you must have one - 9.

9,000,000,001 However, here you only have 8 zeros, and one 8.

8,000,000,010 However, you also have one 1 (which would leave you with 7 zeros).

7,100,000,100 However, in this case, you have two - ones.

7,200,000,100 However, you now have one two, and thus 6 remaining zeros.

6,210,001,000 Which is your answer.

Now, the question would be whether this answer is unique.

I believe that it is. Now, proving that it is unique is more difficult, but one seems to get nowhere by adding in a three.

Algebra is difficult because it doesn't deal with whole numbers as smoothly.