Puzzle No. 122

Try to find mathematical formulae to compute the intergers from 1 up to 100 by using the digit 4 four times - and any mathematical operations you want (but: see below)!

Examples:

  1 = 4/4 * 4/4    =  1 * 1
  2 = 4/4 + 4/4    =  1 + 1
  3 = 4 - 4^(4-4)  =  4 - 1
  4 = 4! + 4! - 44 = 24 + 24 - 44
...

After some consideration, I had to restrict this puzzle a little more:

The reason is that allowing all the three operators

• sqrt(n)
• n!
• [n]

one can construct (almost?) every integer using only one 4.

Examples:

• 1 = [sqrt(sqrt(4))]

• 2 = sqrt(4)

• 3 = [sqrt[sqrt[sqrt(sqrt(sqrt(sqrt(4!))))!]!]]

• 4 = 4 :-))

• 5 = [sqrt(sqrt(sqrt(sqrt(4!))))!

• 6 = [sqrt[sqrt[sqrt(sqrt(sqrt(sqrt(4!))))!]!]]!

• 7 = ...

Therefore the mathematical operations to solve this puzzle are restricted to +, -, *, /, !,^ and sqrt.
Similarly, notation tricks like .4 for 0.4 and dot(.4) (.4 with a dot above 4, meaning 0.4444...) are not valid. Let's see if a solution for n = 1,...,100 is still possible!!

Source: Gordon Lee (gerontius@geocities.com)