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 Reliable source by Jer)

I guess I should reveal my reliable source. 

The book "536 Puzzle & Curious Problems" by H. E. Dudneny (1967) Edited by Martin Gardner contains the two fours version.

In the answers there is a note by Gardner mentioning D. E. Knuth and his article "Representing Numbers Using Only One 4," Mathematics Magazine, Vol. 37, November-December, 1964, pp. 308-310.

I cannot find the acticle itself, though.  But on page 15 of Professor Knuth's vita http://www-cs-faculty.stanford.edu/~knuth/vita.pdf
it is listed.

Posted by Jer on 2006-04-10 15:59:32