According to [url="http://ourworld.compuserve.com/homepages/DavidandPenny/200Up.htm"]http://ourworld.compuserve.com/homepages/DavidandPenny/200Up.htm[/url] the remaining 4-4's-representations up to 160 must have at least one squaring operation.

83 = (4! -.4)/.4 + 4!


Added
162 = 4!/(.4)sq + 4sq - 4
170 = 4!/(.4)sq + 4sq + 4
178 = 4!/(.4)sq + 4! + 4
182 = 4!/(.4)sq + 4sq + 4sq
186 = 4!/(.4)sq
194 = 4!/(.4)sq + 44

83 = (4!-.4)/.4+4!

165 = (4!-.4[sup]sq[/sup])/(.4[sup]sq[/sup])+4[sup]sq[/sup]

167 = (4![sup]sq[/sup]-4)/4+4!
168 = 4![sup]sq[/sup]/4+4!
169 = (4![sup]sq[/sup]+4)/4+4!

173 = (4!-.4[sup]sq[/sup])/(.4[sup]sq[/sup])+4!

I thought someone earlier posted: 192 = 4!*4+4!*4

And we've been using: 576 = 4!sq

174 = (4!)/(.4sq)+4!
175 = (4!+.4sq)/(.4sq)+4!

161= 159 + 2, 164 = 4 * 41
 

161 = (4!sq + 4)/4 + 4sq
164 = 4*(4sq + .4)/.4

Solutions with fewer 4's than on the last summary:

14 = 4/.4+4
14 = 4! -4/.4

22 = 4sq+4!/4

30 = 4!+4!/4

63 = (4[sup]sqsq[/sup]-4)/4

65 = (4[sup]sqsq[/sup]+4)/4

76 = (4/.4)[sup]sq[/sup]-4!

124 = (4/.4)[sup]sq[/sup]+4!

126 = 4!/(.4[sup]sq[/sup])-4!

143 = (4![sup]sq[/sup]-4)/4

146 = 4!/(.4[sup]sq[/sup])-4

and a typo correction:

144 = 4!sq/4 + 4/4 should have been
14[b]5[/b] = 4!sq/4 + 4/4

Earlier, while under the influence of beer :-) , rogue posted several solutions in which 4[sup]sqsq[/sup] was erroneously evaluated to equal 64 instead of 256.

[quote="rogue"]50 = 4sqsq - 4/.4 - 4
53 = 4sqsq - 44/4
54 = 4sqsq - 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
87 = 4sqsq + 4! - 4/4
88 = 4sqsq + 4!
89 = 4sqsq + 4! + 4/4
98 = 4sqsq + 4! + 4/.4
[/quote]

All that work by rogue can be salvaged by simply adding 192 to the left side of each of those equations:

242 = 4sqsq-4/.4-4
245 = 4sqsq-44/4
246 = 4sqsq-4/.4
247 = 4sqsq-(4-4/4)sq
249 = 4sqsq-(4!+4)/4
254 = 4sqsq-(4+4)/4
256 = 4sqsq
259 = 4sqsq+4-4/4
261 = 4sqsq+4+4/4
263 = 4sqsq+(4!+4)/4
265 = 4sqsq+(4-4/4)sq
266 = 4sqsq+4/.4
267 = 4sqsq+44/4
270 = 4sqsq+4/.4+4
271 = 4sqsq+4sq-4/4
273 = 4sqsq+4sq+4/4
279 = 4sqsq+4!-4/4
280 = 4sqsq+4!
281 = 4sqsq+4!+4/4
290 = 4sqsq+4!+4/.4

[QUOTE=Wacky][b]These still need a solution[/b]
78
93
107
131
157

161-255[/QUOTE]

(0.4[sup]sq[/sup]+(4-.4)[sup]sq[/sup])/0.4[sup]sq[/sup]=82
4+(4[sup]sq[/sup]-.4[sup]sq[/sup])/.4[sup]sq[/sup]=103
4[sup]sq[/sup]+(4[sup]sq[/sup]-.4[sup]sq[/sup])/.4[sup]sq[/sup]=115
4!+(4[sup]sq[/sup]-.4[sup]sq[/sup])/.4[sup]sq[/sup]=123

And, impressingly enough:

[QUOTE=S80780]According to [url="http://ourworld.compuserve.com/homepages/DavidandPenny/200Up.htm"]http://ourworld.compuserve.com/homepages/DavidandPenny/200Up.htm[/url] the remaining 4-4's-representations up to 160 must have at least one squaring operation.[/QUOTE]

4*(.4+4!)+.4=98 :showoff:

... and fetofs's solutions lead to:

99 = (4[sup]sq[/sup]-.4[sup]sq[/sup])/.4[sup]sq[/sup]

101 = (4[sup]sq[/sup]+.4[sup]sq[/sup])/.4[sup]sq[/sup]

... which lead to:

157 = 4[sup]sqsq[/sup]-(4[sup]sq[/sup]-.4[sup]sq[/sup])/.4[sup]sq[/sup]

[QUOTE=fetofs][QUOTE=Wacky]
93
107
131

163
166
171
172

176
(and more)[/QUOTE][/QUOTE]

78=4[sup]sq[sup]sq[/sup][/sup]*.4-(.4+4!)

Working on 93.

:banana:

Well, I guess it's time to post solutions for a few you hadn't gotten earlier (I don't know if you've solved them yet:

78 = 4!*4!sq/4sqsq+4!
82 = (4sqsq-4!)/4+4!
83 = (4sq-.4sq)/.4sq-4sq
98 = (4sq-.4sq-.4sq)/.4sq
103= (4sq-.4sq)/.4sq+4
107= 4sqsq-(4!-.4sq)/.4sq
115= (4sq-.4sq)/.4sq+4sq
123= (4sq-.4sq)/.4sq+4!
131= 4sqsq-(4sq+4)/.4sq
157= 4sqsq-(4sq-.4sq)/.4sq

93 is the only one under 160 I haven't found. Between 160 and 300, I couldn't find the following 25 (yet):

163
177
181
183
185
187
189
202
203
205
206
211
213
214
217
218
223
237
243
269
283
287
294
295
298

Also, I found a copy of the problem that's apparently from an earlier edition of my book on [url=http://www.geocities.com/TimesSquare/Arcade/7810/44sfaq.htm]this page[/url]. It's almost the same as mine, but his is copyright 1976 and mine is 1980. Basically, that sentence allowing 4! and repeating decimals is gone in mine, and the ! key is allowed generally.