When I'm a passenger in a car and don't have anything to do, I try to turn the strings of numbers on license plates into equations (I call this solving the license plate).

For example, $$4161 \ \Longrightarrow \ 4 + 1= 6 - 1$$ $$333 \ \Longrightarrow \ 3 = 3 = 3$$

However, if I only allow addition and subtraction there are some license plates I can't solve. $$119$$ is one such example.

I was wondering if there is a string length long enough so that every license plate of that length can be solved.

More precisely, is there an $$n$$ so that for all $$a_1, ..., a_n$$ with $$a_i \in \{1, ..., 9\}$$ can we find a sequence of operations (only allowing addition, subtraction, and equality) that can solve the license plate problem?

Note that I excluded $$0$$ from the problem because any license plate that goes $$00...001$$ is unsolvable.

I have no idea how to approach this problem so any help would be greatly appreciated.

• 1 = 1^9. (If that is allowed.) – marty cohen Jul 19 at 19:26

No. The string $$122222\cdots 2$$ is never able to be "solved" (using only addition, subtraction, and equals signs) for any length $$n$$ as any positioning of addition, subtraction, and equals signs will always leave the expression to the far left as an odd number while any other expression in the equality chain is always going to be an even number.
• Good point! What if we also included multiplication? That eliminates the $1222...2$ counterexample. I wonder if there are any obvious "unsolvable" strings with addition, subtraction, multiplication, and equality. – Connor Goldstick Jul 19 at 19:40