Using floor, ceiling, square root, and factorial functions to get integers

Solution 1:

I've numerically verified that this is possible for all numbers up to 208. Here's a heuristic argument why we can expect it to be possible for all numbers:

After the last factorial, the remaining operations are floors, ceilings and square roots. The natural number $n$ is reachable from a number $m$ using these operations iff $(n-1)^{2^k}<m<(n+1)^{2^k}$ for some $k\in\mathbb N_0$: If this condition is fulfilled, we can draw $k$ square roots and then take either the floor or the ceiling to reach $n$; if it isn't fulfilled, then taking the floor or ceiling in between drawing square roots won't help, since it will never cause the value to cross the boundaries $(n\pm1)^{2^k}$. (Note that the expressions for $n$ up to $11$ reflect this; the floor or ceiling operations are never directly followed by a square root.)

Taking logarithms, we have that $n$ is reachable from $m$ iff $2^k\ln(n-1)<\ln m<2^k\ln(n+1)$. If we consider the logarithm of $m$ to be "randomly" distributed, we can ask for the "probability" of a factorial fulfilling this condition. The length of the admissible interval is $2^k(\ln(n+1)-\ln(n-1))$, and this has to be put in relation to the distance between successive intervals, which is $2^{k+1}\ln n-2^k\ln n=2^k\ln n$. (This is the distance to the next higher interval; the distance to the next lower interval is $2^{k-1}\ln n$, but the difference isn't relevant in what follows.)

Thus, the proportion of values fulfilling the condition is $2^k(\ln(n+1)-\ln(n-1))/(2^k\ln n)=(\ln(n+1)-\ln(n-1))/\ln n$. The exact value isn't important; the key point is that this is independent of $k$ and thus doesn't decrease as $m$ increases. Thus, for each $m$, the "probability" of $m$ fulfilling the condition is roughly the same, as long as it makes sense to regard the logarithm of $m$ as uniformly distributed. But we know that infinitely many values of $m$ are reachable, namely at the very least the ones we get by repeatedly taking factorials, starting with $4$, and it's plausible that there is no systematic relationship between these values and the above intervals, so that the logarithms of these values of $m$ can be regarded as uniformly distributed. Since for any given $n$ each of these values of $m$ has the same finite probability of $n$ being reachable from it, we can expect to eventually find some $m$ from which $n$ is reachable.