Four Fours

\begin{matrix} 0 = & 44 - 44 & 1 = & \frac{44}{44} & 2 = & 4 \frac{4}{4 + 4} & 3 = & \frac{4 + 4 + 4}{4} \\ 4 = & 4 (4 - 4) + 4 & 5 = & \frac{4 \times 4 + 4}{4} & 6 = & \sqrt{44 - 4 - 4} & 7 = & \frac{44}{4} - 4 \\ 8 = & 4 \frac{4}{4} + 4 & 9 = & 4 + \frac{4}{4} + 4 & 10 = & \frac{44}{4.4} & 11 = & \frac{44}{\sqrt{4} \sqrt{4}} \\ 12 = & \frac{44 + 4}{4} & 13 = & 4! - \frac{44}{4} & 14 = & \sqrt{4} + 4 + 4 + 4 & 15 = & \frac{44}{4} + 4 \\ 16 = & 4 \frac{4}{4} 4 & 17 = & 4 \times 4 + \frac{4}{4} & 18 = & 4 \times 4.4 + .4 & 19 = & 4! - \frac{4}{4} - 4 \\ 20 = & 4 \frac{4}{.4 \sqrt{4}} & 21 = & \frac{4.4 + 4}{.4} & 22 = & \frac{44 \sqrt{4}}{4} & 23 = & \frac{\sqrt{4} + 44}{\sqrt{4}} \\ 24 = & 4 + 4 \times 4 + 4 & 25 = & \frac{4}{4 % \sqrt{4} \sqrt{4}} & 26 = & \frac{44}{\sqrt{4}} + 4 & 27 = & 4 + 4! - \frac{4}{4} \\ 28 = & 44 - 4 \times 4 & 29 = & \frac{4}{4 \times 4 %} + 4 & 30 = & \frac{4 + 4 + 4}{.4} & 31 = & \frac{4! + 4}{4} + 4! \\ 32 = & 4 \frac{4}{\sqrt{4}} 4 & 33 = & \frac{44}{\sqrt{4} \sqrt{\bar{.4}}} & 34 = & 44 - \frac{4}{.4} & 35 = & \frac{44}{4} + 4! \\ 36 = & 44 - 4 - 4 & 37 = & \frac{4}{\bar{.4}} + 4! + 4 & 38 = & 44 - \sqrt{4} - 4 & 39 = & 44 - \frac{\sqrt{4}}{.4} \\ 40 = & 44 - \sqrt{4} \sqrt{4} & 41 = & \frac{4 \times 4 + .4}{.4} & 42 = & \sqrt{4} + 44 - 4 & 43 = & 44 - \frac{4}{4} \\ 44 = & 44 \frac{4}{4} & 45 = & \frac{4}{4} + 44 & 46 = & 4 + 44 - \sqrt{4} & 47 = & 4! \sqrt{4} - \frac{4}{4} \\ 48 = & 4 (4 + 4 + 4) & 49 = & \frac{\sqrt{4}}{.4} + 44 & 50 = & \sqrt{4} + 44 + 4 & 51 = & \frac{\sqrt{4}}{4 %} + \frac{4}{4} \\ 52 = & 44 + 4 + 4 & 53 = & \frac{4}{\bar{.4}} + 44 & 54 = & \frac{4}{.4} + 44 & 55 = & \frac{44}{.4 \sqrt{4}} \\ 56 = & \frac{4}{4 %} - 44 & 57 = & \frac{4! + .4}{.4} - 4 & 58 = & \frac{4^{4} - 4!}{4} & 59 = & \frac{4!}{.4} - \frac{4}{4} \\ 60 = & 4 \times 4 + 44 & 61 = & \frac{4!}{.4} + \frac{4}{4} & 62 = & 4 \times 4 \times 4 - \sqrt{4} & 63 = & \frac{4^{4} - 4}{4} \\ 64 = & 4^{4 - \frac{4}{4}} & 65 = & \frac{4^{4} + 4}{4} & 66 = & 4! + 44 - \sqrt{4} & 67 = & \frac{4! + 4}{\bar{.4}} + 4 \\ 68 = & 4 \times 4 \times 4 + 4 & 69 = & \frac{4 + 4! - .4}{.4} & 70 = & \sqrt{4} + 44 + 4! & 71 = & \frac{4.4 + 4!}{.4} \\ 72 = & 4! + 44 + 4 & 73 = & \frac{\sqrt{4} - 4 %}{4 %} + 4! & 74 = & \frac{4! + 4}{.4} + 4 & 75 = & \frac{4 - \frac{4}{4}}{4 %} \\ 76 = & 4 (4! - 4) - 4 & 77 = & {\sqrt{(\frac{4}{\bar{.4}})}}^{4} - 4 & 78 = & 4 (4! - 4) - \sqrt{4} & 79 = & \frac{\sqrt{4}}{4 % \sqrt{\bar{.4}}} + 4 \\ 80 = & 4 \frac{4 + 4}{.4} & 81 = & {(4 - \frac{4}{4})}^{4} & 82 = & 4 (4! - 4) + \sqrt{4} & 83 = & \frac{4! - .4}{.4} + 4! \\ 84 = & 44 \sqrt{4} - 4 & 85 = & \frac{\frac{4}{.4} + 4!}{.4} & 86 = & 44 \sqrt{4} - \sqrt{4} & 87 = & 4 \times 4! - \frac{4}{\bar{.4}} \\ 88 = & 44 + 44 & 89 = & \frac{4 - .44}{4 %} & 90 = & 44 \sqrt{4} + \sqrt{4} & 91 = & \frac{.4 % + 4}{4.4 %} \\ 92 = & 44 \sqrt{4} + 4 & 93 = & 4 \times 4! - \sqrt{(\frac{4}{\bar{.4}})} & 94 = & \frac{\sqrt{4}}{4 %} + 44 & 95 = & \frac{44}{\bar{.4}} - 4 \\ 96 = & 4 \frac{4}{4} 4! & 97 = & 4 \times 4! + \frac{4}{4} & 98 = & \frac{4!}{\bar{.4}} + 44 & 99 = & \frac{4}{4 %} - \frac{4}{4} \\ 100 = & \frac{44}{.44} & 101 = & \frac{4}{4 %} + \frac{4}{4} & 102 = & .4 \times 4^{4} - .4 & 103 = & \frac{44}{\bar{.4}} + 4 \\ 104 = & .4 (4^{4} + 4) & 105 = & \frac{44 - \sqrt{4}}{.4} & 106 = & \frac{44}{.4} - 4 & 107 = & \frac{\sqrt{4 %} + 4}{4 %} + \sqrt{4} \\ 108 = & \frac{44 + 4}{\bar{.4}} & 109 = & \frac{44 - .4}{.4} & 110 = & \frac{44}{\sqrt{4} \sqrt{4 %}} & 111 = & \frac{444}{4} \\ 112 = & 44 \sqrt{4} + 4! & 113 = & \frac{\frac{\sqrt{4}}{\bar{.4} %} + \sqrt{4}}{4} & 114 = & \frac{44}{.4} + 4 & 115 = & \frac{\sqrt{4} + 44}{.4} \\ 116 = & 4 (4! + 4) + 4 & 117 = & \frac{4! \sqrt{4} + 4}{\bar{.4}} & 118 = & \sqrt{4} \frac{4! - .4}{.4} & 119 = & \frac{4! \sqrt{4} - .4}{.4} \\ 120 = & \frac{44 + 4}{.4} & 121 = & {(\frac{44}{4})}^{\sqrt{4}} & 122 = & \sqrt{4} \frac{4! + .4}{.4} & 123 = & \frac{44}{\bar{.4}} + 4! \\ 124 = & \frac{4^{4}}{\sqrt{4}} - 4 & 125 = & \frac{\frac{4}{4} + 4}{4 %} & 126 = & \frac{4^{4} - 4}{\sqrt{4}} & 127 = & \frac{4^{4} - \sqrt{4}}{\sqrt{4}} \\ 128 = & 4 \times 4 (4 + 4) & 129 = & \frac{4^{4} + \sqrt{4}}{\sqrt{4}} & 130 = & \frac{4^{4} + 4}{\sqrt{4}} & 131 = & \frac{4!}{.4 \times \bar{.4}} - 4 \\ 132 = & 44 \sqrt{(\frac{4}{\bar{.4}})} & 133 = & \frac{4!}{.4 \times \bar{.4}} - \sqrt{4} & 134 = & \frac{44}{.4} + 4! & 135 = & \frac{\frac{\sqrt{4}}{4 %} + 4}{.4} \\ 136 = & \sqrt{4} (4! + 44) & 137 = & \frac{4!}{.4 \times \bar{.4}} + \sqrt{4} & 138 = & \frac{\sqrt{{4!}^{4}} - 4!}{4} & 139 = & \frac{4!}{.4 \times \bar{.4}} + 4 \\ 140 = & 4 \times 4! + 44 & 141 = & \frac{4 - 4! %}{4 % \sqrt{\bar{.4}}} & 142 = & 4! \frac{4!}{4} - \sqrt{4} & 143 = & \frac{\sqrt{{4!}^{4}} - 4}{4} \\ 144 = & \frac{4}{4 %} + 44 & 145 = & \frac{\sqrt{{4!}^{4}} + 4}{4} & 146 = & 4! \frac{4!}{4} + \sqrt{4} & 147 = & \frac{4 \times 4! + \sqrt{4}}{\sqrt{\bar{.4}}} \\ 148 = & 4! \frac{4!}{4} + 4 & 149 = & \frac{\frac{4!}{4 %} - 4}{4} & 150 = & \frac{\frac{4}{.4} - 4}{4 %} & 151 = & \frac{\frac{4!}{4 %} + 4}{4} \\ 152 = & 4 \times 44 - 4! & 153 = & \frac{4! + 44}{\bar{.4}} & 154 = & \frac{4!}{4 \times 4 %} + 4 & 155 = & \frac{\frac{4!}{.4} + \sqrt{4}}{.4} \\ 156 = & 4^{4} - \frac{4}{4 %} & 157 = & \frac{\sqrt{\bar{.4}} + 4 %}{\bar{.4} %} - \sqrt{4} & 158 = & \frac{\sqrt{\bar{.4}}}{\bar{.4} %} + 4 + 4 & 159 = & \frac{4!}{.4 \times \bar{.4}} + 4! \\ 160 = & 4 (44 - 4) & 161 = & \frac{\sqrt{\sqrt{\sqrt{4^{4!}}}} + .4}{.4} & 162 = & 4! \frac{4!}{4 - \bar{.4}} & 163 = & \frac{\sqrt{\bar{.4}} + 4 %}{\bar{.4} %} + 4 \\ 164 = & \frac{\sqrt{4}}{.4}! + 44 & 165 = & \frac{44}{.4 \sqrt{\bar{.4}}} & 166 = & 4^{4} - \frac{.4}{\bar{.4} %} & 167 = & \frac{4 % % \sqrt{4} + .4}{4! % %} \\ 168 = & 4 (44 - \sqrt{4}) & 169 = & {(\frac{4}{\bar{.4}} + 4)}^{\sqrt{4}} & 170 = & \frac{4! + 44}{.4} & 171 = & \frac{\frac{4}{4 %} - 4!}{\bar{.4}} \\ 172 = & 4 \times 44 - 4 & 173 = & \frac{4 - \frac{4!}{\bar{.4}} %}{\sqrt{4} %} & 174 = & 4 \times 44 - \sqrt{4} & 175 = & \frac{4! + 4}{4 \times 4 %} \\ 176 = & 44 \sqrt{4} \sqrt{4} & 177 = & \frac{4! - .4}{\sqrt{4} \sqrt{\bar{.4} %}} & 178 = & 4 \times 44 + \sqrt{4} & 179 = & \frac{(\sqrt{4} + 4)! - 4}{4} \\ 180 = & 4 \times 44 + 4 & 181 = & \frac{(\sqrt{4} + 4)! + 4}{4} & 182 = & \frac{4 - .4}{\sqrt{4} %} + \sqrt{4} & 183 = & \frac{4! + .4}{\sqrt{4} \sqrt{\bar{.4} %}} \\ 184 = & 4 (\sqrt{4} + 44) & 185 = & \frac{44 + .4}{4! %} & 186 = & \frac{.4}{\bar{.4} %} + 4 \times 4! & 187 = & \frac{4 - (4! + \sqrt{4}) %}{\sqrt{4} %} \\ 188 = & 4 \times 4! \sqrt{4} - 4 & 189 = & \frac{.44 + .4}{\bar{.4} %} & 190 = & \frac{4 + 4 - .4}{4 %} & 191 = & \frac{4}{\sqrt{4} %} - \frac{4}{\bar{.4}} \\ 192 = & 4 (44 + 4) & 193 = & \frac{4 - \sqrt{(\sqrt{4} - 4 %) %}}{\sqrt{4} %} & 194 = & \sqrt{44^{4} %} + .4 & 195 = & \frac{4 - \frac{.4}{4}}{\sqrt{4} %} \\ 196 = & 4 \times 4! \sqrt{4} + 4 & 197 = & \frac{4 - \frac{4! %}{4}}{\sqrt{4} %} & 198 = & \frac{44 \sqrt{4}}{\bar{.4}} & 199 = & \frac{4}{\sqrt{4} %} - \frac{4}{4} \\ 200 = & 4 \times 44 + 4! & 201 = & \frac{4}{\sqrt{4} %} + \frac{4}{4} & 202 = & \frac{4 + 4}{4 %} + \sqrt{4} & 203 = & \frac{4 - \sqrt{4} %}{\sqrt{4} %} + 4 \\ 204 = & \frac{4 + 4}{4 %} + 4 & 205 = & \frac{\frac{.4}{4} + 4}{\sqrt{4} %} & 206 = & 4^{4} - \frac{\sqrt{4}}{4 %} & 207 = & \frac{4 \times 4! - 4}{\bar{.4}} \\ 208 = & 4 (4! \sqrt{4} + 4) & 209 = & \frac{4}{\sqrt{4} %} + \frac{4}{\bar{.4}} & 210 = & \frac{4.4 + 4}{4 %} & 211 = & 4^{4} - \frac{\sqrt{4 %}}{\bar{.4} %} \\ 212 = & 4^{4} - 44 & 213 = & \frac{\frac{\sqrt{4}}{\bar{.4} %} - 4!}{\sqrt{4}} & 214 = & 4 \frac{4!}{\bar{.4}} - \sqrt{4} & 215 = & \frac{\frac{4}{\bar{.4}} - .4}{4 %} \\ 216 = & \frac{44}{\sqrt{4 %}} - 4 & 217 = & \frac{4 \times 4! + \bar{.4}}{\bar{.4}} & 218 = & \frac{44 - .4}{\sqrt{4 %}} & 219 = & \frac{44 - \sqrt{4 %}}{\sqrt{4 %}} \\ 220 = & \frac{44 \sqrt{4}}{.4} & 221 = & \frac{4}{4 \times \bar{.4} %} - 4 & 222 = & \frac{444}{\sqrt{4}} & 223 = & \frac{4}{4 \times \bar{.4} %} - \sqrt{4} \\ 224 = & \frac{44}{\sqrt{4 %}} + 4 & 225 = & \frac{4 - .4}{4 \times .4 %} & 226 = & \frac{4}{4 \times .4 %} - 4! & 227 = & \frac{4}{4 \times \bar{.4} %} + \sqrt{4} \\ 228 = & 4^{4} - 4! - 4 & 229 = & \frac{4}{4 \times \bar{.4} %} + 4 & 230 = & \frac{4 \times 4! - 4}{.4} & 231 = & \frac{\frac{4}{\bar{.4} %} + 4!}{4} \\ 232 = & \sqrt{4^{4 + 4}} - 4! & 233 = & \frac{4 + \sqrt{\bar{.4}} - \sqrt{\bar{.4}} %}{\sqrt{4} %} & 234 = & \sqrt{4} + 4^{4} - 4! & 235 = & \frac{4 \times 4! - \sqrt{4}}{.4} \\ 236 = & 4! \frac{4}{.4} - 4 & 237 = & \frac{4 \times 4 - \sqrt{4 %}}{\sqrt{\bar{.4} %}} & 238 = & 4! \frac{4}{.4} - \sqrt{4} & 239 = & \frac{4 \times 4! - .4}{.4} \\ 240 = & 4^{4} - 4 \times 4 & 241 = & \frac{4 \times 4! + .4}{.4} & 242 = & 4! \frac{4}{.4} + \sqrt{4} & 243 = & 4! \sqrt{(\frac{4}{{\bar{.4}}^{4}})} \\ 244 = & 4! \frac{4}{.4} + 4 & 245 = & \frac{4 \times 4! + \sqrt{4}}{.4} & 246 = & 4^{4} - \frac{4}{.4} & 247 = & 4^{4} - \frac{4}{\bar{.4}} \\ 248 = & 4^{4} - 4 - 4 & 249 = & \frac{\frac{4}{.4 %} - 4}{4} & 250 = & 4^{4} - \sqrt{4} - 4 & 251 = & 4^{4} - \frac{\sqrt{4}}{.4} \\ 252 = & 4^{4} - \sqrt{4} \sqrt{4} & 253 = & 4^{4} - \sqrt{(\frac{4}{\bar{.4}})} & 254 = & \sqrt{4} + 4^{4} - 4 & 255 = & 4^{4} - \frac{4}{4} \end{matrix}