12 = 4sq-4
40 = 4sq/.4

Thru 75, I'm having some trouble computing the following:

38, 42, 55, 57, 58, 62, 67 and 75

[QUOTE=grandpascorpion]Thru 75, I'm having some trouble computing the following:

38, 42, 55, 57, 58, 62, 67 and 75[/QUOTE]

38 = 4! + 4/.4 + 4

42 = 4! + 4! - 4!/4

58 = 4! + 4! + 4/.4

I wonder how difficult it would be to write a program to evaluate all possible expressions (given the limitations) and output the simplest solution for each, i.e. the one that uses the fewest 4's.

Spoilers at: [url]http://www.cut-the-knot.org/arithmetic/funny/4_4.shtml[/url]

Ah but many of those solutions use the square root sign (which is verboten under Ken's rules).

Here are some others without resorting to the link I produced above. If they are the same, it is coincidental. There are multiple solutions to many of these. Many of the ones below follow easily from other solutions.

59 = 4!/.4 - 4/4
61 = 4!/.4 + 4/4
66 = 4!/.4 + 4!/4
68 = 4!/.4 + 4 + 4
68 = 4! + 4! + 4! - 4
70 = (4 + 4)!/(4! * 4!)
76 = 4! + 4! + 4! + 4
82 = 4!/.4 + 4! - 4
86 = 4!/.4 + 4!
90 = 4!/.4 + 4! + 4
92 = 4! * 4 - 4
95 = 4! * 4 - 4/4
97 = 4! * 4 + 4/4
101 = (4/.4)sq + 4/4
104 = (4/.4)sq + 4
108 = (4/.4)sq + 4 + 4
116 = (4+4/4)! - 4
117 = (44/4)sq - 4
124 = (4+4/4)! + 4
125 = (44/4)sq + 4
128 = 4!sq/4 - 4sq
132 = 4!sq/4 - 4sq + 4
136 = 4!sq/4 - 4 - 4
140 = 4!sq/4 - 4
143 = 4!sq/4 - 4/4
144 = 4!sq/4 + 4/4

Here is the current list to save everyone from scrolling. It contains duplicates as those might provide ideas for expanding the table.

0 = 4-4
1 = 4/4
2 = (4+4)/4
3 = (4+4+4)/4
4 = 4
5 = 4 + 4/4
6 = 4! / 4
7 = (4!+4)/4
7 = 4 + 4 - 4/4
8 = 4 + 4
9 = 4 + 4 + 4/4
10 = 4 / .4
11 = 44 / 4
12 = 4 + 4 + 4
12 = 4sq-4
13 = 4sq -4 + 4/4
14 = 4sq - (4+4)/4
15 = 4sq - 4/4
16 = 4sq
17 = 4sq + 4/4
18 = 4/.4 + 4 + 4
19 = 4! - 4 - 4/4
20 = 4! - 4
21 = 4! -4 + 4/4
22 = 4! - (4+4)/4
23 = 4! - 4/4
24 = 4!
25 = 4! + 4/4
26 = 4sq + 4/.4
27 = 4! + 4 - 4/4
28 = 4sq + 4sq - 4
29 = 4! + 4 + 4/4
30 = (4+4/4)!/4
31 = 4sq + 4sq - 4/4
32 = 4sq + 4sq
33 = 4sq + 4sq + 4/4
34 = 4! + 4/.4
35 = 4! + 44/4
36 = 4!sq / 4sq
37 = 4!sq / 4sq + 4/4
38 = 4! + 4/.4 + 4
39 = 4! + 4sq - 4/4
40 = 4! + 4sq
40 = 4sq/.4
41 = 4! + 4sq + 4/4
42 = 4! + 4! - 4!/4
43 = 44 - 4/4
44 = 44
45 = 44 + 4/4
47 = 4! + 4! - 4/4
48 = 4! + 4!
49 = 4! + 4! + 4/4
52 = 4! + 4! + 4
56 = 4*4sq-4-4
58 = 4! + 4! + 4/.4
59 = 4!/.4 - 4/4
60 = 4!/.4
60 = 4*4sq-4
61 = 4!/.4 + 4/4
63 = 4*4sq-4/4
64 = 4sq * 4
65 = 4*4sq+4/4
66 = 4!/.4 + 4!/4
68 = 4! + 4! + 4! - 4
68 = 4!/.4 + 4 + 4
70 = (4 + 4)!/(4! * 4!)
72 = 4! * 4 - 4!
76 = 4! + 4! + 4! + 4
80 = 4sq * 4 + 4sq
81 = ((4-4/4)sq)sq
82 = 4!/.4 + 4! - 4
86 = 4!/.4 + 4!
90 = 4!/.4 + 4! + 4
92 = 4! * 4 - 4
95 = (4! * 4sq-4)/4
95 = 4! * 4 - 4/4
96 = 4! * 4
96 = 4! * 4sq/4
97 = (4! * 4sq+4)/4
97 = 4! * 4 + 4/4
100 = ( 4 / .4 )sq
101 = (4/.4)sq + 4/4
104 = (4/.4)sq + 4
108 = (4/.4)sq + 4 + 4
109 = (44 -.4)/.4
110 = 44 / .4
111 = 44.4 / .4
116 = (4+4/4)! - 4
117 = (44/4)sq - 4
120 = (4+4/4)!
121 = (44/4)sq
124 = (4+4/4)! + 4
125 = (44/4)sq + 4
128 = 4!sq/4 - 4sq
128 = 4sq * 4 + 4sq * 4
132 = 4!sq/4 - 4sq + 4
136 = 4!sq/4 - 4 - 4
140 = 4!sq/4 - 4
143 = 4!sq/4 - 4/4
144 = 4!sq / 4
144 = 4!sq/4 + 4/4
256 = 4sq * 4sq

And some more:

3 = 4 - 4/4
50 = 4sqsq - 4/.4 - 4
53 = 4sqsq - 44/4
54 = 4sqsq - 4/.4
54 = 4!/.4 - 4!/4
54 = 4! + 4! + 4!/4
55 = 4sqsq - (4 - 4/4)sq
57 = 4sqsq - (4!+4)/4
62 = 4sqsq - (4+4)/4
64 = 4sqsq
67 = 4sqsq + 4 - 4/4
69 = 4sqsq + 4 + 4/4
71 = 4sqsq + (4!+4)/4
73 = 4sqsq + (4 - 4/4)sq
74 = 4sqsq + 4/.4
75 = 4sqsq + 44/4
78 = 4sqsq + 4/.4 + 4
79 = 4sqsq + 4sq - 4/4
81 = 4sqsq + 4sq + 4/4
84 = 4sq * 4 + 4sq + 4
87 = 4sqsq + 4! - 4/4
89 = 4sqsq + 4! + 4/4
94 = (4/.4)sq - 4!/4
99 = (4/.4)sq - 4/4
102 = 4! * 4 + 4!/4
106 = (4/.4)sq + 4!/4
112 = 4! * 4 + 4sq
114 = 44/.4 + 4

S'more
 

46 = (4sq+4)/.4-4
50 = (4sq+4)/.4
51 = (4sq+4+.4)/.4
53 = ((4!+4)/4)sq+4
54 = (4sq+4)/.4+4
69 = (4!+4-.4)/.4
71 = (4!+4+.4)/.4
74 = (4!+4)/.4+4
84 = 4*(4sq+4)+4
94 = ( 4 / .4 )sq-4!/4
99 = ( 4 / .4 )sq-4/4
106 = ( 4 / .4 )sq+4!/4
112 = 4*(4!+4)

edit : sorry, there's redundancies

How did I miss this one:

88 = 4sqsq + 4!

and

77 = (4 - 4/4)sqsq - 4
85 = (4! + 4/.4)/.4 (this one is a cheat from the link I provided)
85 = (4 - 4/4)sqsq + 4 (this one is mine)
98 = 4sqsq + 4! + 4/.4
105 = (4 - 4/4)sqsq + 4!

That leaves 83, 91, 93, and 107 without solutions under 110.

0 = 4-4
1 = 4/4
2 = (4+4)/4
3 = 4-4/4
4 = 4
5 = 4+4/4
6 = 4+(4+4)/4 = 4/.4-4 = 4!/4
7 = 4+4-4/4 = (4!+4)/4
8 = 4+4
9 = (4-.4)/.4
10 = 4/.4
11 = 44/4
12 = 4^2-4
13 = 4^2-4+4/4 = (4!+4!+4)/4
14 = 4^2-(4+4)/4
15 = 4^2-4/4
16 = 4^2
17 = 4^2+4/4
18 = 4^2+(4+4)/4
19 = (4+4-.4)/.4
20 = (4+4)/.4
21 = (4+4.4)/.4
22 = 4!-(4+4)/4
23 = 4!-4/4
24 = 4!
25 = 4!+4/4
26 = 4!+(4+4)/4
27 = 4!+4-4/4
28 = 4!+4
29 = 4!+4+4/4
30 = (4+4/4)!/4 = (4+4+4)/.4
31 = 4^2+4^2-4/4
32 = 4^2+4^2
33 = 4^2+4^2+4/4
34 = 4! + 4/.4
35 = 4! + 44/4
36 = 4*(4-.4)/.4 = (4!/4)^2
37 = (4!/4)^2 + 4/4
38 = (4^2-.4-.4)/.4
39 = (4^2-.4)/.4
40 = 4^2/.4
41 = (4^2+.4)/.4
42 = (4^2+.4+.4)/.4
43 = 44-4/4
44 = 44
45 = 44+4/4

47 = 4!+4!-4/4
48 = ((4!+4)^2-4^2)/4^2 = 4!*(4+4)/4 = 4*(4^2-4) = 44+4 = 4!+4!
49 = (4+4-4/4)^2 = 4!+4!+4/4 = ((4!+4)/4)^2
50 = ((4!+4)^2+4^2)/4^2 = (4!-4)/.4
51 = (4!-4+.4)/.4
52 = 44+4+4 = 4!+4!+4
53 = ((4!+4)/4)^2+4

55 = (4!-.4)/.4-4
56 = 4!+4!+4+4 = 4!/.4-4
57 = (4!+.4)/.4-4
58 = (4!-.4-.4)/.4
59 = (4!-.4)/.4
60 = 4!/.4
61 = (4!+.4)/.4
62 = (4!+.4+.4)/.4
63 = 4*4^2-4/4
64 = 4*4^2 = (4+4)^2
65 = 4*4^2+4/4 = ((4!+4)/4)^2+4^2

68 = 4!+44
69 = (4!+4-.4)/.4
70 = (4!+4)/.4
71 = (4!+4.4)/.4
72 = 4!+4!+4!

74 = (4!+4)/.4+4

76 = 4!+4!+4!+4

80 = (4!+4+4)/.4 = (4^2-4)^2-(4+4)^2 = 4*(4!-4)
81 = ((4-.4)/.4)^2

84 = 4*(4!-4)+4 = 4!/.4+4!
85 = ((4-.4)/4)^2+4 = (4!+.4)/.4+4!

88 = 4!+4!+4!+4^2

92 = (4/.4)^2-4-4 = 4*4!-4

95 = 4!*4-4/4
96 = 4!*4
97 = 4!*4+4/4

99 = (4/.4)^2-4/4
100 = 4*4!+4 = (4/.4)^2
101 = (4/.4)^2+4/4

104 = (4/.4)^2+4
105 = (44/4)^2-4^2
106 = 44/.4-4

108 = (4/.4)^2+4+4
109 = (44-.4)/.4
110 = 44/.4
111 = 444/4
112 = (4^2-4)^2-4^2-4^2

114 = 44/.4+4

116 = (4/.4)^2+4^2
117 = (44/4)^2-4

119 = (4!+4!-.4)/.4
120 = (4+4/4)! = (4!+4!)/.4
121 = (44/4)^2

124 = (4^2-4)^2-4^2-4
125 = (44/4)^2+4 = (4!-4)/(.4)^2

128 = (4^2-4)^2-4^2

132 = (4/.4)^2+4^2+4^2 = (4^2-4)^2-4^2+4

136 = (4^2-4)^2-4-4
137 = (44/4)^2+4^2
138 = (4!^2-4!)/4

140 = (4^2-4)^2-4

142 = (4!^2-4-4)/4
143 = (4^2-4)^2-4/4 = (4!^2-4)/4
144 = (4^2-4)^2 = 4!^2/4
145 = (4^2-4)^2+4/4 = (4!^2+4)/4
146 = (4!^2+4+4)/4

148 = (4^2-4)^2+4
149 = (4!-(.4)^2)/(.4)^2
150 = 4!/(.4)^2
151 = (4!+(.4)^2)/(.4)^2
152 = (4^2-4)^2+4+4 = 4*44-4!
153 = 4+(4!-(.4)^2)/(.4)^2
154 = 4+4!/(.4)^2
155 = 4+(4!+(.4)^2)/(.4)^2

160 = (4^2-4)^2+4^2