Let us consider formation of a given positive integer N by the following rules
- You may use only one digit, which one is your choice and you may use it multiple times, though the fewer times the better
- You may use the following arithmetic operations: exponentiation, multiplication, division, addition, subtraction and unary minus; the order is your choice, in other words you are allowed parentheses but only for grouping (no binomial coefficients and other shenanigans)
- You may use concatenation but only of the raw digit you chose, for example if you chose 7 you can make 777 but you can't make 17 = 7/7 concat 7 ; and no decimal point
The task is usually to find an expression for N by the rules above involving the minimum number of digits.
Now, given rule 3 asking for a number like N=3333 or N=111 would be rather boring, or wouldn't it?
The question I'm asking is:
Which is the smallest number whose decimal representation xx...xx is formed from a single digit x but has an optimal formation by the above rules that is not based on x but on a different digit.
Same question but additional requirement optimal formation must have fewer digits than decimal representation.
Example: 99999999999 (11 9's) can be written (11-1)^11-1 which is only 6 1s. This is not the smallest example, though.
Note 1: this is brute-forceable with a computer, but if someone can come up with a paper and pencil solution that would be preferred.
Note 2: please let me emphasize the absence of the lateral-thinking tag.