What is the minimum number of digits required to make the numbers 1-20?

Question

What is the minimum number of digits required to make the numbers $1-20$?

Any $n$ consecutive numbers can be counted as a $n$-digit consecutive number. So, $$12\\34$$ has the numbers $$1, 2, 3, 4, 12, 13, 21, 24, 31, 34, 42, 43$$ but not the numbers $$123, 14$$ because the digits $1, 2, 3$ are not in a straight line, and $1, 4$ are not consecutive. They have to be touching vertically or horizontally.

Note: your answer should be in a grid form, like above, but the answer doesn't have to be in a complete rectangle, so like $$\space\space23\\45\space\space\\678$$ would still be a valid (although incorrect) answer.

EDIT:

Changed the question from $1-100$ to $1-20$ because it's going to take forever to find the $1-100$ question :)

Answer 1

It looks to me as if this is one optimal solution:

0 2 9 3 1 1 5 7 1 8 4 6

Why I think it's an optimal solution:

First of all, it's easy to check that it's a solution. Now, since one 1 is next to at most 4 other digits we need to have at least three of them. And we need one of every other digit; so at least 3+9=12 digits. Above, we have exactly 12 digits.