I and a mate of mine were trying to push the three 3s challenge (https://mindyourdecisions.com/blog/2018/09/03/can-you-solve-the-3s-challenge/) to its limit by incorporating other common mathematical functions. These functions included exp(), floor(), sign(), ceiling(), factorial(), and EulersTotient (phi()). We quickly came to realize that by chaining the phi function and the factorial function you are almost able to hit any number by starting with only three 3s.
For example, 66 = phi(phi(3!!)) + sign(3) + sign(3)
My question is can anyone come up with a counter-example of a number not able to be written in this way? After much conversation, we believe that you can just basically factorial if your number is too low and phi if your number is too high.
It's just a question for fun but thank you!