"Clean-er" list so far
 

0 = 4 + 4 - 4 - 4
1 = (4+4)^(4-4) = (4!-4)^(4-4)
2 = 4/4 + 4/4
3 = 4-4^(4-4)
4 = 4*4^(4-4)
5 = 4+4^(4-4)
6 = 4 + ((4 + 4) / 4)
7 = 4 + 4 - (4 / 4)
8 = 4 * 4 - 4 - 4
9 = 4 + 4 + (4 / 4)
10 = 4*4-4-√4
11 = 44/(√4+√4)
12 = 4+4+√(4*4)
13 = 4!-44/4
14 = 4/.4 + √(4*4)
15 = 44/4 + 4 = 4*4 -4/4
16 = 4*4*4/4
17 = 4*4 + 4/4
18 = 44 - 4! - √4
19 = 4! - 4 - 4/4
20 = (44 - 4) / √4
21 = 4! - 4 + 4/4
22 = 44/(4/√4) = 4!-4/(4-√4)
23 = 4!-4^(4-4)
24 = 4!-4+(√4+√4)
25 = 4!+4^(4-4)
26 = 4!+√4 +4-4
27 = 4!+√4 +4/4
28 = 4!+4 +4-4
29 = 4!+4 +4/4
30 = 4!+√4+√4+√4
31 = 4! + 4 / .4~ - √4 = 4! + √4 + √4/.4
32 = 4 ^ √(√4 + √4) * √4
33 = 4! + (√4 + √4) / .4~ = 4! + 4 + √4/.4
34 = 4! + 4 + 4 + √4
35 = (4 * 4 - √4) / .4
36 = ((√4/.4)! + 4!) / 4
37 = (4!+.4)/.4 - 4! = 4! + (4! + √4) / √4 = 4! + 4 + 4 / .4~
38 = 4! + 4 + 4 / .4 = 44 - 4 - √4
39 = 44 - √4 / .4
40 = 44 - √4 - √4
41 = (4*4+.4)/.4
42 = 44 - 4/√4
43 = 44 - 4/4
44 = 44 +4 -4
45 = 44 + 4 / 4
46 = 44 + 4 / √4
47 = 4! + 4! - 4 / 4
48 = 4! + 4! + 4 - 4 = 44 + √4 + √4
49 = (√4/.4 + √4) ^ √4 = 4! + 4! + 4/4
50 = (√4 * 4) / (.4^√4) = √4 * (4! + 4/4)
51 = (4! - √4) /.4 - 4
52 = ((√4 + √4)! + √4) * √4 = (4! + √4) * √4
53 = 44 + 4/.4~ = (4! - √4) /.4 -√4
54 = 4! * √4 + 4 + √4
55 = (44 / √4) /.4
56 = (4 / .4 + 4) * 4
57 = (4! - √4) /.4 +√4
58 = 4! * √4 + 4/.4
59 = (4! -√4) / .4 + 4
60 = 4 * 4 * 4 - 4 = 4^4 / 4 - 4
61 = ((√4 / .4)! + √4) / √4
62 = ((√4 / .4)! + 4) / √4
63 = (4! + √4) /.4 - √4
64 = 4*4*√4*√4
65 = 4!/.4 + √4/.4
66 = 4 * 4 * 4 + √4
67 = (4! + √4) / .4 + √4
68 = 4 * 4 * 4 + 4
69 = (4! + √4) / .4 + 4
70 = 4! + 4! + 4! - √4
71 = (4! + 4 + .4) / .4
72 = 44 + 4! + 4
73 = ( √4 *4! + √(.4~) ) / √(.4~) = 4/.4~ + √(√(√4))^(4!)
74 = 4/.4 + √(√(√4)^(4!))
75 = (4! + 4 + √4) / .4
76 = √(√(√(4^(4!)))) + 4! / √4
77 = ( √4 / √(.4~))^4 - 4 = (4! + 4!/4) / .4 + √4
78 = (4! / .4~) + (√4 + √4)!
79 = (√4 / √(.4~))^4 - √4
80 = (4 / .4) * (4 + 4)
81 = (4 - 4/4 ) ^ 4
82 = (4! / .4~) + 4! + 4
83 = (√4 / √(.4~))^4 + √4
84 = 44*√4 - 4
85 = (4/.4~)^√4+4
86 = 44*√4 - √4
87 = 4! * 4 - 4 / .4~
88 = 4! * 4 - 4 - 4
89 = (4! / √(.4~) - .4) / .4
90 = 4! * 4 - 4 - √4
91 = (4! / √(.4~) + .4) / .4 = 4! * 4 - √4 / .4
92 = 4*4!-√4-√4
93 = 4! * 4 - √4 / √(.4~)
94 = 4! * 4 - 4 / √4
95 = 4! * 4 - 4 / 4
96 = 4! * 4 - 4 + 4
97 = 4! * 4 + 4 / 4
98 = 4! * 4 + 4 / √4
99 = 44 / √(.4~ * .4~)
100 = (4! + 4/4) * 4 = 4/.4 * 4/.4
101 = 4! * 4 + √4 / .4
102 = 4! * 4 + 4 + √4
103 = 44 / .4~ + 4
104 = 4! * 4 + 4 + 4
105 = 4*4! + 4/.4~
106 = 4! * 4 + 4/.4
107 = (4! * √4 - .4~) / .4~
108 = 4! *4 + 4! / √4
109 = (4! *√4 + .4~) / .4~
110 = (√4 / .4)! - 4 / .4
111 = 444/4
112 = 44/.4 + √4
113 =
114 = (√4 / .4)! - 4 - √4
115 = (√4 / .4)! - (√4 /.4)
116 = (√4 / .4)! - √4 - √4
117 = (√4 / .4)! - √(4/.4~)
118 = (√4 / .4)! - 4 + √4
119 = (√4 / .4)! - 4 / 4
120 = (√4 / .4)! + 4 - 4
121 = (√4 / .4)! + 4 / 4
122 = (√4 / .4)! + 4/ √4
123 = (√4 / .4)! + √(4 /.4~)
124 = (√4 / .4)! + √4 +√4
125 = (√4 / .4)! + √4 / .4
126 = √(√(√(4^4!))) * √4 - √4
127 = √(√(√((√4 / .4)^4!))) + √4
128 = √(√(√(4^4!))) * (4 - √4)
129 = √(√(√((√4 / .4)^4!))) + 4
130 = (√4 / .4)! + 4/.4
131 = (4! / .4) / .4~ - 4
132 = (√4 / .4)! + 4!/√4
133 = (4! / .4) / .4~ -√4
134 = (4! /.4 - .4~) / .4 = 44 / .4 + 4!
135 = ((√4 * √4)! / .4) /.4~
136 = (4! /.4 + .4~) / .4 = (√4/.4)! + 4^√(4)
137 = (4! /.4~) / .4 + √4
138 = (4! * 4 - 4) / √(.4~) = (4! ^ √4 - 4!) / 4
139 = (4! / .4~) / .4 + 4
140 = (4! / √4) ^ (√4) - 4
141 = (4! * 4 - √4)/ √(.4~)
142 = 4! * 4! /4 - √4
143 = (4! *4! -4) /4
144 = 4! *4! /√4 /√4
145 = (4! ^ √4 + 4) / 4
146 = (4! / √4) ^ √4 + √4
147 = (4! * 4 + √4) / √.4~
148 = (4! / √4) ^ √4 + 4
149 = √( √( √( (√4/.4) ^ 4! ) ) ) + 4! = (4!/.4-.4)/.4
150 = (4! * 4! + 4!) / 4
151 = (4! / .4 + .4)/.4
152 = (4! / .4) / .4 + √4
153 = (√(√(√(4^4!))) + 4) / .4~
154 = (4! / .4) / .4 + 4
155 = ((4! / .4) + √4)/ .4
156 = √(√(√(4^4!))) / .4 - 4
157 =
158 = √(√(√(4^4!))) / .4 - √4
159 = (√(√(√(4^4!)))-.4) / .4
160 = 4 * 4 * 4/.4 = 4^4 - 4!*4
161 = (√(√(√(4^4!)))+.4) / .4
162 = (4 / .4~) ^ √4 * √4
163 =

================================
113, 157 and 163 are sticklers so far....P.S. I too am stumped on them.

I've added a few more building blocks
[spoiler]
576 = √(4!^4)

55 = (4! + .4~) / .4~
61 = (4! + .4) / .4
1296 = 4!^√4 / (√.4~)
1440 = 4!^√4 / .4
[/spoiler]

E-X-T-R-E-M-E four fours
 

See this link: [url]http://www.dwheeler.com/fourfours/[/url]

He allows every operations under the sun and has answers for 0 to 40,000

[QUOTE=petrw1;98985]See this link: [url]http://www.dwheeler.com/fourfours/[/url]

He allows every operations under the sun and has answers for 0 to 40,000[/QUOTE]

Not EVERY operation under the sun, for example logarithms are NOT allowed - for good reason:

see this link: [url]http://www.jimloy.com/puzz/4-4s.htm[/url]

[quote]Surprisingly, if logarithms are allowed, it turns out that any positive integer can be expressed with three fours:

n=-ln[ln[sqrt(sqrt(sqrt...(sqr(4))...))]/ln(4)]/ln(4)

where the number of square roots is twice n. I haven't checked this out to see if it is true.
[/quote]

Edit:

Fo(u)r our game I would suggest the following expansion, that our game does not get stuck:

Allow gamma and % if and only if an other solution can not be found. If somebody finds a gamma- and %-free solution later, this new solution should replace the gamma'd or %'ed solution in our list.

...fo(u)r the sake of progress I'm game with allowing % and gamma.

And I like your recommendation that the preference is to NOT use them and solutions without them will replace solutions with them.

...I guess you could say we are letting % and gamma into our village but restricting them to live on the North side of the tracks (in my city you don't go there after dark.)

Whenever I publish the complete list I will hilight these solutions as "second class". :grin:

Ah, using one of the new operators there's a sneaky way of creating 1 with only one 4. For the earlier link, that can be used for a solution to 163 ...

Ah, using one of the new operators there's a sneaky way of creating 1 with only one 4. From the earlier link, that can be used for a solution to 163 ...

@Grandpascorpion: You mean 1 = gamma(√4)?

163 = [b]([/b]gamma(gamma(4))% + √(√(√(4^4!)))[b])[/b] / .4

Edit: without gamma, but with %, found with the program [URL="http://ourworld.compuserve.com/homepages/DavidandPenny/Amamas.htm"]200up[/URL]:

163 = ((4 + 4!)% + .4~) / .4~%

Yep, but you didn't need it apparently :)

164 = √(√(√(4^4!)))) / .4 + 4
165 = (√4+ √(√(√(4^4!))))) / .4
166 = √(√(√(4^4!)))) / .4 + gamma(4)

I confess I did not find this myself - never thought of using 6.25 = .4 ^ (-√4):

gamma-free 166:

166 = 4! * (.4 ^ (-√4) + √.4~)

Nobody for 167?

167 = gamma(gamma(4)) + 4! * √4 - gamma(√4) (anyone to find a gamma-free solution?)
168 = ((√4 / .4~) + 4) * 4!
169 = ((4 / .4~) + 4) ^ √4
170 = (√(√(√(4^4!))) + 4) / .4