Some Four 4s solutions

1. 1 =
4 · 4
4 · 4
=
4
4
+ 4 − 4
2 =
4
4
+
4
4
= 4 −
√
4 + 4 − 4
3 =
4 + 4 + 4
4
=
√
4 +
√
4 −
4
4
4 = 4(4 − 4) + 4 = 4 + 4 −
√
4 −
√
4
5 = 44−4
+ 4 =
√
4 +
√
4 +
4
4
6 =
4 + 4
4
+ 4 = 4 +
√
4 + 4 − 4
7 = 4 + 4 −
4
4
=
4! +
√
4 +
√
4
4
8 = 4 − 4 + 4 + 4
9 =
4
4
+ 4 + 4
10 =
4 · 4 + 4!
4
= 4 + 4 + 4 −
√
4
11 =
4 + 4!
4
+ 4 =
4
.4
+
4
4
=
44
√
4 +
√
4
12 = 4
(
4 −
4
4
)
= 4 + 4 +
√
4 +
√
4
13 =
4 + 4! + 4!
4
=
4!
√
4
+
4
4
14 =
4!
4
+ 4 + 4 = 4 + 4 + 4 +
√
4
15 = 4 · 4 −
4
4
=
4
.4
+
√
4
.4
=
4!
(
√
4 +
√
4).4
16 = (−4)(−4) + 4 − 4 =
4!
4
+ .4
.4
17 = 4 · 4 +
4
4
18 =
4 · 4! − 4!
4
= 4 · 4 + 4 −
√
4
19 = 4! −
(
4 +
4
4
)
20 =
(
4
4
+ 4
)
· 4 = 4! − 4 + 4 − 4
21 = 4! +
4
4
− 4
22 = 4! −
4 + 4
4
= 4! −
√
4 + 4 − 4
23 = 4! − 44−4
= 4! −
√
4 +
4
4
24 = 4 · 4 + 4 + 4 = 4! + 4 −
(√
4 +
√
4
)
25 = 4! + 44−4
= 4! +
√
4 −
4
4
26 = 4! +
4!
4
− 4 = 4! +
√
4 + 4 − 4
27 = 4! +
√
4 +
4
4
=
4! ·
√
4
4 · .4
28 = 4(4 + 4) − 4 = 4! + 4 + 4 − 4
29 = 4! +
4! − 4
4
= 4! + 4 +
4
4
30 =
4 · 4! + 4!
4
= 4! +
√
4 +
√
4 +
√
4
31 =
4 + 4!
4
+ 4! = 4! +
√
4 +
√
4
.4
=
4!√
4
+ .4
.4
32 =
44
4 + 4
= 4 · 4 + 4 · 4
33 =
4 − .4
.4
+ 4! = 4! + 4 +
√
4
.4
34 =
4!
4
+ 4 + 4! = 4! +
√
4 +
√
4
.4
35 =
4.4
.4
+ 4! = 44 −
4
.4
36 = 44 − 4 − 4 = (4 +
√
4)(4 +
√
4)
37 =
4
.4
+ 4 + 4! =
4!
√
.4
+
4
4
38 = 44 −
4!
4
= 44 − 4 −
√
4 =
4
.4
+ 4 + 4!
39 =
4 · 4 − .4
.4
= 44 −
√
4
.4

2. 40 =
44
4
− 4! = 44 −
√
4 −
√
4
41 =
4 · 4 + .4
.4
=
4! +
√
4
.4
− 4!
42 = 4! + 4! −
4!
4
=
44
√
.4
− 4!
43 = 44 −
4
4
44 = 44.4 − .4 = 44 + 4 − 4
45 =
(
4!
4
)
! ÷ (4 · 4) = 44 +
4
4
46 =
(4! − 4)
.4
− 4 = 44 + 4 −
√
4
47 = 4! ·
√
4 −
4
4
48 = 4(4 · 4 − 4) = 44 +
√
4
√
4
49 = 4! + 4! +
4
4
= 44 +
√
4
.4
=
4! − 4 − .4
.4
50 =
4 · 4 + 4
.4
= 44 + 4 +
√
4
51 =
4! + 4
.4
√
.4
52 = 44 + 4 + 4
53 = 44 +
4
.4
=
(
√
4 +
√
4)! − .4
.4
54 = 44 +
4
.4
=
4! · 4
4 · .4
=
(
4
.4
)√
4
·
√
.4
55 =
4! − 4 +
√
4
.4
=
44
√
4 · .4
56 = (4! +
√
4 +
√
4) ·
√
4
57 =
4! −
√
4
.4
+
√
4 =
(
4
.4
)√
4
− 4!
58 =
(
4! +
√
4
.4
)
√
4 =
4!
.4
+
√
4
√
4
59 =
4! −
√
4
.4
+ 4 =
4! + .4
.4
−
√
4
60 = 4 · 4 · 4 − 4 =
4!
.4
+
4
√
.4
61 =
4! +
√
4
.4
− 4
62 = 4 · 4 · 4 −
√
4 =
4! + .4 + .4
.4
63 =
4! +
√
4
.4
−
√
4 =
4! + .4
.4
+
√
4
64 = 44−4
4 = 4 · 4 ·
√
4 ·
√
4
65 =
4! + 4 −
√
4
.4
=
4! + .4
.4
+ 4
66 = 4 · 4 · 4 +
√
4 =
44 ·
√
.4
.4
67 =
4! +
√
4
.4
+
√
4
68 = 4 · 4 · 4 + 4 = 4
√ 4
.4 + 4
69 =
4! +
√
4
.4
+ 4 =
4! + 4 − .4
.4
70 = 44 + 4! +
√
4 =
44
√
.4
+ 4
71 =
4! + 4! −
√
.4
√
.4
=
4! + 4 + .4
.4
72 = 44 + 4! + 4 =
4! · 4
√
4 ·
√
.4
73 =
4! + 4! +
√
.4
√
.4
74 =
4! ·
√
4
√
.4
+
√
4 =
4! − 4
.4
+ 4!
75 =
44
.4
− 4!
76 =
4! ·
√
4
√
.4
+ 4
77 =
(
4
.4
)√
4
− 4
78 =
4!
.4
+ (
√
4 +
√
4)!
79 =
4! −
√
4
.4
+ 4! =
(
4
.4
)√
4
−
√
4

3. 80 = (44 − 4) ·
√
4
81 =
4
.4
·
4
.4
82 =
4!
.4
+ 4! + 4
83 =
(
4
.4
)√
4
+
√
4 =
4! − .4
.4
+ 4!
84 = 44 ·
√
4 − 4
85 =
(
4
.4
)√
4
+ 4 =
4! + .4
.4
+ 24 =
4! + 4
.4
.4
86 = 44 ·
√
4 −
√
4
87 = 4! · 4 −
4
.4
88 = 4! · 4 − 4 − 4 = 44 + 44
89 =
4! +
√
4
.4
+ 4! =
(
4!
√
.4
− .4
)
÷ .4
90 = 44 ·
√
4 +
√
4 =
4!
.4
+
4!
√
.4
91 = 4! · 4 −
√
4
.4
=
(
4!
√
.4
+ .4
)
÷ .4
92 = 44 ·
√
4 + 4 = 4! · 4 −
√
4 −
√
4
93 = 4! · 4 −
√
4
√
.4
94 =
4! + 4
.4
+ 4! = 4! · 4 − 4 +
√
4
95 =
44
.4
− 4 = 4! · 4 −
4
4
96 = 4! · 4 + 4 − 4
97 =
44
.4
−
√
4 = 4! · 4 +
4
4
98 = 4! · 4 + 4 −
√
4
99 = 4! · 4 +
√
4
√
.4
=
4! + 4! − 4
.4
100 =
44
.44
=
(
4! +
4
4
)
4
101 =
44
.4
+
√
4 = 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 =
44 − .4
.4
110 = (4! + 4) · 4 −
√
4
111 =
44 + .4
.4
=
444
4
112 = (4! +
√
4 ·
√
4) · 4
113 =
114 = (4! + 4) · 4 +
√
4
115 =
4! + 4! −
√
4
.4
116 =
4! ·
√
4
.4
− 4
117 = (4! +
√
4)
√
4
.4
=
4! ·
√
4 + 4
.4
118 =
4! ·
√
4
.4
−
√
4
119 =
4! + 4! − .4
.4

4. 120 =
4! · 4
√
4 · .4
121 =
4! + 4! + .4
.4
122 = 4! · 4 + 4! +
√
4
123 =
44
.4
+ 4! =
4! · 4
√
.4
− 4!
124 = 4! · 4 + 4! + 4
125 =
(√
4
.4
)
! +
√
4
.4
126 =
4!
.4 · .4
− 4!
127 =
128 = (4! + 4 + 4) · 4
129 =
(√
4
.4
)
! +
4
.4
130 = (4! +
√
4) ·
√
4
.4
131 =
4!
.4 · .4
− 4
132 =
44 ·
√
4
√
.4
133 =
4!
.4 · .4
−
√
4
134 =
(
4!
.4
− .4
)
÷ .4
135 =
(
√
4 +
√
4)!
.4 · .4
136 =
(
4!
.4
+ .4
)
÷ .4
137 =
4!
.4 · .4
+
√
4
138 =
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 =
148 =
4!
.4 · .4
−
√
4
149 =
(
4!
.4
− .4
)
÷ .4
150 =
(
√
4 +
√
4)!
.4 · .4
151 =
(
4!
.4
+ .4
)
÷ .4
152 =
4!
.4 · .4
+
√
4
153 =
154 =
4!
.4 · .4
+ 4
155 =
156 =
4!
.4
+ 4! · 4 =
(
4!
4
)
! ÷ 4 − 4!
157 =
158 =
159 =
4!
.4 · .4
+ 4!
160 =
(
4!
4
)
! ·
(√
.4 − .4
)

5. 160 =
(
4!
4
)
! ·
(√
.4 − .4
)
161 =
162 =
163 =
164 =
165 =
166 =
167 =
168 =
169 =
170 =
171 =
172 =
173 =
174 =
175 =
176 =
177 =
178 =
179 =
180 =
181 =
182 =
183 =
184 =
185 =
186 =
187 =
188 =
189 =
190 =
191 =
192 =
193 =
194 =
195 =
196 =
197 =
198 =
199 =
200 =

6. Numbers Made With One 4
2
5
= .4
4
9
= .4
2
3
=
√
.4
2 =
√
4
4 = 4
24 = 4!
Numbers Made With Two 4’s
0 = 4 − 4
3
5
=
.4
√
.4
9
10
=
.4
.4
1 =
4
4
2 = 4 −
√
4
3 =
√
4
√
.4
4 =
√
4 +
√
4
4.5 =
√
4
.4
5 =
√
4
.4
6 = 4 +
√
4 =
4!
4
=
4
√
.4
8 = 4 + 4
9 =
4
.4
10 =
4
.4
12 =
4!
√
4
16 = 4 · 4
20 = 4! − 4
22 = 4! −
√
4
24 = (
√
4 +
√
4)!
26 = 4! +
√
4
28 = 4! + 4
32 =
.4
√
4
36 =
4!
√
.4
44 = 44
48 = 4! + 4!
54 =
4!
.4
60 =
4!
.4
96 = 4! · 4
120 =
(√
4
.4
)
!
256 = 44
576 = 4! · 4!
720 = (4 +
√
4)!

7. The Year 2010 Puzzle
0 = 20
− 10
1 = 2 + 0 − 1 + 0
2 = 2 + 0 − 1 · 0 = 20 ÷ 10
3 = 2 + 0 + 10
4 = 2 + 0! + 10
5 = 2 + 0! + 1 + 0!
6 = (2 + 0 + 1 + 0)!
7 = (2 + 0 + 1)! + 0!
8 = 2(0!+1+0!)
9 = (2 + 0!)(1+0!)
10 = 2 · 0 + 10 = 20 − 10
11 =
√
[(2 + 0!)! − 1]! + 0!
12 = (2 + 0!)! · (1 + 0!)
13 = (.2 + 0!) ÷ .1 + 0!
14 =
15 =
16 =
17 =
18 = 20 − (1 + 0!)
19 = 20 − 10
20 = 20 − 1 · 0
21 = 20 + 10
22 = 20 + 1 + 0!
23 = (2 + 0! + 1)! − 0!
24 = (2 + 0! + 1)! + 0
25 = (2 + 0! + 1)! + 0!
26 =
27 =
28 =
29 =
30 = 20 + 10
31 =
32 =
33 =
34 =
35 =
36 =
37 =
38 =
39 =
40 =
41 =
42 =
43 =
44 =
45 =
46 =
47 =
48 =
49 =
50 =
72 = [(2 + 0!)!]! ÷ 10

8. The Year 2011 Puzzle
0 = 20
− 11
1 = 2 + 0 − 1 · 1
2 = 2 + 0 · 1 · 1
3 = 2 + 0 + 11
4 = 2 + 0 + 1 + 1
5 = 2 + 0! + 1 + 1
6 = (2 + 0 + 1)! · 1 = (.2 − 0) ÷ (.1 ·
√
.1 )
7 = (2 + 0 + 1)! + 1
8 = (2 + 0!)! + 1 + 1
9 = (2 + 0!)(1+1)
10 = −20
+ 11
11 = 2 · 0 + 11
12 = 20
+ 11 = [(.2 + 0!) ÷ .1] · 1
13 = 2 + 0 + 11
14 = 2 + 0! + 11
15 =
16 =
17 = (2 + 0!)! + 11
18 = 20 − 1 − 1 = (.2 + 0) ÷ (.1 · .1 )
19 = 20 − 11
20 = 20 − 1 + 1
21 = 20 + 11
= (2 + 0 + .1) ÷ .1
22 = 20 + 1 + 1
23 = (2 + 0! + 1)! − 1
24 = (2 + 0! + 1)! · 1
25 = (2 + 0! + 1)! + 1
26 =
27 = (20
) ÷ (.1 ·
√
.1 )
28 =
29 =
30 = (20
) ÷ (.1 ·
√
.1 )
31 = 20 + 11
32 =
33 = (.2 + 0!) ÷ (.1 ·
√
.1 )
34 =
35 =
36 =
37 =
38 =
39 =
40 =
41 =
42 =
43 =
44 =
45 =
46 =
47 =
48 =
49 =
50 =

9. 54 = (2 − 0) ÷ (.1 ·
√
.1 )
60 = (2 + 0) ÷ (.1 ·
√
.1 )
81 = (2 + 0!)! ÷ (.1 ·
√
.1 )
90 = (2 + 0!)! ÷ (.1 ·
√
.1 )
108 = (.2 + 0!) ÷ (.1 · .1 )
Numbers Made With One 0, 1, or 2
0 = 0
1
10
= .1
1
9
= .1
1
5
= .2
2
9
= .2
1
3
=
√
.1
1 = 1 = 0!
2 = 2
Numbers Made With Two 0’s, 1’s, or 2’s
3 = 2 + 0!
6 = (2 + 0!)! = (2 + 1)!
10 = 10
11 = 11
20 = 20 = 2 ÷ .1
21 = 21