Problem

Source: ELMO SL 2018 C2

Tags: combinatorics



We say that a positive integer $n$ is $m$-expressible if it is possible to get $n$ from some $m$ digits and the six operations $+,-,\times,\div$, exponentiation $^\wedge$, and concatenation $\oplus$. For example, $5625$ is $3$-expressible (in two ways): both $5\oplus (5^\wedge 4)$ and $(7\oplus 5)^\wedge 2$ yield $5625$. Does there exist a positive integer $N$ such that all positive integers with $N$ digits are $(N-1)$-expressible? Proposed by Krit Boonsiriseth