Perhaps we should define f(n) as the highest factorial which must be taken to go from n to n via those operations, or infinity if it cannot be done. So your example shows that f(7) <= 71, since you take 71! and no higher factorials. Then you could look at the relative maxima of f(n) to see if good heuristics can be developed. Computationally, I think this problem is equivalent to one with operations n!, floor(sqrt(n)), and ceil(sqrt(n)) and this might be the best way to implement the function practically. Charles Greathouse Analyst/Programmer Case Western Reserve University On Sun, Nov 10, 2013 at 11:59 AM, Victor Miller <victorsmiller@gmail.com>wrote:

I just saw the following problem on the POW (problem of the week) list run by Stan Wagon. This looks like an interesting computing challenge:

Problem 1171 Four Unary Functions

Call n, a positive integer, "good" if one can start with n and use the operations factorial, square root, floor and ceiling to obtain n, using at least one of the first two operations. For example, 7 is good because Ceiling[Sqrt[7!]]! after extracting 7 square roots gives 6.3, and so a final ceiling gives 7 (see http://mathforum.org/wagon/fall13/p1171.jpg).

Show that n is good for all n <= K, where K is as large as you can get.

Note: Here factorial can operate only on integers. So while the example above, using the gamma function, works using Sqrt[7!]!instead of Ceiling[Sqrt[7!]]!, that is not legal. The assertion that all positive integers are good is a conjecture of Don Knuth and Richard Hess.

Note that the actual representations are not sought, just the proof that the representation exists. It has been conjectured by D. Knuth and R. Hess that all positive integers are good.

Source: Crux Math, M506, 38:8 Oct 2012, 310-313 _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun