Your goal is to use the ten digits (including zero) to get as close to 1 as you can without actually getting 1.

You may use +, -, *, and /. You may combine two numbers like 4 and 7 to get 47. You may use fraction bars. You may use the numbers as powers. You may use decimals. You may use square roots, but you may only use three of them (Otherwise, you could have a near-infinite number of square roots, and the result would approach 1). You may use as many other roots as you wish, as long as you're counting the digit (i.e. if you use a cube root, that counts as your use of the digit 3). You may put a bar over part of a decimal to make it repeating. You may use a factorial, but only one.

You may not round. You may not use floor or ceiling functions. You may not use any digit more than once. You must use each digit. You may not use e, pi, etc.

Examples:

72   3   4   0
-- - - + - - - = 1.188888... (off by .1888...)
81   6   5   9

would not be as good as

57.3*4096
--------- = 0.8953125 (off by .1046875)
  (2^18)

Joe's answer is off by about 10^-(10^8.9)
My first solution is in this ballpark

Penny's is much closer.  It is off by about
10^-(10^(10^(10^(10^(10^(5.3)

Old Original Oscar!'s is much much closer.  The difference doesn't look much smaller but it is about
10^-(10^(10^(10^(10^(10^(10^(10^(8.1)

I didn't bother finding my improvement or Steve Herman's as .2^n converges to zero much faster than the nth root of .9
Maybe someone else would care to.

Posted by Jer on 2006-12-19 12:31:05