There is a way to express 64 with only two fours and no symbols beyond +, -, *, /, ^, √, !, and parenthesis, although some may be used more than once. It isn't too hard. Can you find it?

It is asserted in a reliable source that 64 can also be expressed with a single 4 using 57 square root signs, nine factorials (no double or higher factorials), and 18 sets of parentheses. I can't figure it out. Can you?

(In reply to re: Reliable source by Jer)

The online version of the articles published for Mathematics Magazine do not (currently) go further back than 1974, and of those that were published on or after 1974, only a summary of the article is given.  Luckily, the article was re-published in 1974, Volume 47, Number 1, Pages 308-310, and so this excerpt of the summary can be viewed:

"All integers from 1 to 200 can be generated by starting with the number 4 and repeatedly taking the factorial or the square root or the greatest integer of the current result. The same is probably true of all positive integers, although it may be unprovable."

As can be noted, D. E. Knuth used the floor function to arrive at some of his integers.  Thus, I suspect the same is true for the arrival at number 64.   As such, the post by Demau Senae entitled "Second Part" is most probably the same solution given by D. E. Knuth.

The floor function's use of brackets, [ x ], is an older notation of the floor function's symbols, |_ x _|. (The floor function's notation is not the combined pipe symbol and underscore, as shown here, but a single character for the left- and single character for the right-handed notation symbols which are not included in this font.) 

Edited on April 10, 2006, 10:11 pm

Posted by Dej Mar on 2006-04-10 17:59:02