mersenneforum.org - Four Fours

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[QUOTE=Wacky;98601]But that will be impossible. There is no way to create a square without consuming an additional 4. And we cannot generate 64 without using at least one (actually more) 4. Therefore the 7^2 + 64 approach will not work.[/QUOTE]

Of course not ... that was just my feeble attempt at sarcastic humor.

Getting back on track ...

[quote=Andi47;98521]162 = (4 / .4~) ^ √4 * √4[/quote]163 = [I]another damn prime[/I]

Anyone?

- - -

And we don't have 157 within the OP's rules yet.

"Clean-er" list so far

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

171 =
172 = 44 * 4 - 4
173 =
174 = 44 * 4 - √4

Yes, I took the two easy ones ... I'm still trying the other two

[quote=petrw1;100109]171 =
172 = 44 * 4 - 4
173 =
174 = 44 * 4 - √4

Yes, I took the two easy ones ... I'm still trying the other two[/quote]Back in post #10 of this thread, Wacky proposed an "interesting" rule we've followed since then:
[quote=Wacky;97581]And just to keep it "interesting", no skipping around. :)
You have to wait to post "7" until someone posts a solution for 6", etc.
Posting "better" solutions for smaller numbers is fine.[/quote]So, we need 171 before proceeding further. :-)

I did not find anything without gamma...

171 = gamma(gamma(4)) / √.4~ - 4/.4~

173 = gamma(gamma(4)) + (4! - .4~)/.4~

Edit: going further:

175 = ((4 + 4!) / .4) / .4
176 = 44 * (√4 + √4)

We move on after 177:

177 = ((√4 / .4)! - √4) / √.4~
178 = (4 * 44) + √4
179 = ((4 + √4)! - 4) / 4
180 = 4 + (4 * 44)

Some better solutions (note that I consider too many square roots to be ugly):

[QUOTE]
164 = √(√(√(4^4!)))) / .4 + 4
165 = (√4+ √(√(√(4^4!))))) / .4
170 = (√(√(√(4^4!))) + 4) / .4[/QUOTE]

164 = (√4 / .4)! + 44
165 = 44 / (.4 * √.4~)
170 = (4! + 44) / .4

I suggest you allow two additional functions, sum and subfacorial.

S4 = 4+3+2+1= 10 (I do not have a sigma on my computer)
!4= 9

These are sometimes accepable in this game.

181 = ((√4 / .4)! + √.4~)/√.4~

181 = (4 + (4 + sqrt(4))!) / 4
182 = ((4 + sqrt(4))! / 4) + sqrt(4)
183 = (sqrt(4) + (sqrt(4) / .4)!) / sqrt(.4~)
184 = 4 * (sqrt(4) + 44)

Is 185 possible?

[QUOTE=fetofs;101718]

Is 185 possible?[/QUOTE]

I didn't think 181 or 183 was possible ... but you proved me wrong.

So I won't try to guess if 185 is possible.

I did not find anything without gamma.

185 = gamma(gamma(4)) + √(√(√(4^4!))) + gamma(√4))

[QUOTE=PrimeCrazzy;101700]I suggest you allow two additional functions, sum and subfacorial.

S4 = 4+3+2+1= 10 (I do not have a sigma on my computer)
!4= 9

These are sometimes accepable in this game.[/QUOTE]

You can get the gamma function for positive integers quite easily:

gamma(n) = (n-1)!

examples:

gamma(√4) = 1! = 1
gamma(4) = 3! = 6
gamma(gamma(4)) = 5! = 120

You can also get 185 by using the sum and subfactorial functions ad follows:

(sum(sum4))*sqrt (4 subfactorial)+sum4+sum4

[QUOTE=PrimeCrazzy;101818]You can also get 185 by using the sum and subfactorial functions ad follows:

(sum(sum4))*sqrt (4 subfactorial)+sum4+sum4[/QUOTE]

Most of us are not interested in your "change of the rules". If you are allowed to change the rules to suit yourself, then any value becomes trivial.

Please stick to the established restrictions.

186 = ((4 / .4)! + 4) / √.4~

[quote=PrimeCrazzy;101700]I suggest you allow two additional functions, sum and subfacorial.

S4 = 4+3+2+1= 10 (I do not have a sigma on my computer)
!4= 9

These are sometimes accepable in this game.[/quote]PrimeCrazzy,

How about starting a new thread, titled something like "PrimeCrazzy's Four Fours", where you lay out the set of rules [u]you[/u] particularly like? Then those of us interested in that set can start building 1, 2, 3, ... :smile:

[quote=Andi47;101839]186 = ((4 / .4)! + 4) / √.4~[/quote]I think you meant:

186 = ((√4 / .4)! + 4) / √.4~

187 = ((gamma(4))! + 4 + 4!) / 4
188 = (4 + 4) * 4! - 4
189 = (4! + 4! / .4) / .4~
190 = (4 + 4) * 4! - √4

... and now one of those exciting prime decades!

191 = ((gamma(4))! + 44) / 4
192 = (44 + 4) * 4

[QUOTE=cheesehead;101945]I think you meant:

186 = ((√4 / .4)! + 4) / √.4~[/QUOTE]

Ooops, catch that loose square root! :wink:

193 = (4 + 4) * 4! + gamma(√4)
194 = (4 + 4) * 4! + √4
195 = (4! + (4! / .4~)) / √.4~
196 = (4! / √4 + √4) ^ √4

[quote=Andi47;101963]Ooops, catch that loose square root! :wink:

< snip >

195 = (4! + (4! / .4~)) / √.4~[/quote]... but not by snagging it into another formula like a fishing hook with line dangling!

195 = (4! + (4! / .4~)) / .4

[QUOTE=cheesehead;101969]... but not by snagging it into another formula like a fishing hook with line dangling!

195 = (4! + (4! / .4~)) / .4[/QUOTE]

*grrrr* I should double- and triplecheck my formulas... :blush:

P.S.: the [URL="http://www.dwheeler.com/fourfours/"]EXTREME four fours[/URL] page lists 197 with a complexity of 6, i.e. using the "sq" function and not having found a formula with % and/or gamma. But it IS possible to do 197 using %.

Indeed.

197 = (4 - (4 + √4)%) / √4%

198 = (44 * √4) / .4~
199 = (4 + 4) / 4% - gamma(√4)
200 = 44 * 4 + 4!
201 = (4 + 4) / 4% + gamma(√4)

202 = ( ( 4 + 4 ) / 4% ) + sqrt(4)
202 = 4 ^ 4 - ( 4! / .4~ )

203 = gamma(gamma(4)) / √.4~ + 4! - gamma(√4)
204 = (√4 / .4)! / √.4~ + 4!
205 = gamma(gamma(4)) / √.4~ + gamma(√4)
206 = gamma(gamma(4)) / √.4~ + √4

[QUOTE=Andi47;102007]203 = gamma(gamma(4)) / √.4~ + 4! - gamma(√4)
204 = (√4 / .4)! / √.4~ + 4!
205 = gamma(gamma(4)) / √.4~ + gamma(√4)
206 = gamma(gamma(4)) / √.4~ + √4[/QUOTE]

205 and 206 are presumably missing '+4!'

207 = (4! * gamma(4) - gamma(4)) / √.4~
208 = 4 ^ 4 - 4! - 4!

*grrrrrr*

catching two loose 4!'s:

205 = gamma(gamma(4)) / √.4~ + 4! + gamma(4)
206 = gamma(gamma(4)) / √.4~ + 4! + √4

moving on:
209 = ((gamma(4) + gamma(√(4)))! - 4!) / 4!
210 = (4! + 4! / .4) / .4
211 = (gamma(4 + 4) + 4!) / 4!
212 = 4 ^ 4 - 44

[QUOTE=cheesehead;101950]... and now one of those exciting prime decades!

191 = ((gamma(4))! + 44) / 4
192 = (44 + 4) * 4[/QUOTE]

May I suggest you start a new thread "Cheesehead prime four" with your rules. Pleasde note that GAMMA IS NOT IN THE ORIGINAL RULES!

[QUOTE=Wacky;101833]Most of us are not interested in your "change of the rules". If you are allowed to change the rules to suit yourself, then any value becomes trivial.

Please stick to the established restrictions.[/QUOTE]

Does the useof gamma and % make all numbers trivial?

[QUOTE=PrimeCrazzy;102075]May I suggest you start a new thread "Cheesehead prime four" with your rules. Pleasde note that GAMMA IS NOT IN THE ORIGINAL RULES![/QUOTE]
No, it isn't; but surely the fact that gamma (and per cent) has been used by several others as well as Cheesehead suggests that there may have been an amendment to the rules. If you go and read post 104 in this thread, the person who started this thread clearly says;
[QUOTE=petrw1;99011]...fo(u)r the sake of progress I'm game with allowing % and gamma. And I like [Andi47's] recommendation that the preference is to NOT use them and solutions without them will replace solutions with them.[/QUOTE]
So, in order that the continuation of consecutive numbers being solved does not get bogged down, the person who set up the thread has decided to allow gamma and per cent, although gamma-less and per-cent-less solutions found for lower numbers will supercede the earlier solutions.
The key difference is that gamma and per cent were deemed acceptable only when a point was reached where a number was apparently unobtainable using the initial functions. Sum and subfactorial, on the other hand, were put forward when not strictly necessary by someone who had clearly not bothered to read the thread properly. Looked at in this strictly logical manner, I think it's understandable why Cheesehead may have been a little impatient with you.
Now, if you'd like to join in on this puzzle [1] using the rules so far established I'm sure no-one would object. If you would like to start a new thread using sum and subfactorial, but neither gamma nor per cent, then by all means do so - although, I recommend choosing a different set of numbers as well otherwise people may be discouraged from taking part in something which is very similar to an existing active thread.
Either way, calm down, take a deep breath, and relax.

[1] Is this thread a puzzle or a problem? Is there a technical difference between the two?

[quote=PrimeCrazzy;102075]May I suggest you start a new thread "Cheesehead prime four" with your rules. Pleasde note that GAMMA IS NOT IN THE ORIGINAL RULES![/quote]I wanted to facilitate progress of the thread to higher numbers, without using any operator (such as %)[sup]*[/sup] to which I object more strongly than to the gamma. I didn't propose, introduce, or approve the gamma function here. I had previously stated my preference not to extend the allowed operators, and didn't want to repeat that. Also, I admired the simplicity of the 192 solution, and wanted to be the one to post it.

In view of my earlier objections, [I]it would have been a good idea for me to have stated my motivations[/I] in order to ward off misunderstanding. I apologize for omitting that.

I was not irritated by your suggestions to add sum and subfactorial. My post #124 suggestion about starting a new thread was punctuated with a :smile: to signal that it was intended as a friendly, not hostile, suggestion about how to achieve what you wanted without further modifying the present thread. I wrote "... those of us interested in that set ..." because I was (and still am) genuinely interested in participating in such a thread, but I didn't state my interest explicitly. I wouldn't object to using either the sum or subfactorial if the thread originator specified them. I'd simply avoid participating if I objected to the originator's set of rules in that context.

- - - - -

[sup]*[/sup] I previously objected to % and decimal point equally. I've changed my opinion since then, to accept decimal point (implied division by 10, if followed by a single-digit number) more readily than percent (implied division by 10[sup]2[/sup]).

213 = (gamma(gamma(4)) + 4! - √4) / √.4~
214 = 4 * 4! / .4~ - √4
215 = 4 * 4! / .4~ - gamma(√4)
216 = 4! * (4 + 4) + 4!

[QUOTE=DJones;102163]
215 = 4 * 4! / .4~ - gamma(√4)[/QUOTE]

215 = ((4 * 4!) - .4~) / .4~

217 = (4 * 4! + .4~) / .4~
218 = 4 * 4! / .4~ + √4
219 = (4! * gamma(4) + √4) / √.4~
220 = 4 * 4! / .4~ + 4

[quote=DJones;102163]213 = (gamma(gamma(4)) + 4! - √4) / √.4~[/quote]With single gamma:

213 = (4! * gamma(4) - √4) / √.4~

[quote=cheesehead;102185]218 = 4 * 4! / .4~ + √4
< snip >
220 = 4 * 4! / .4~ + 4[/quote]Lazy me. :blush: There are solutions using more elementary (to me, anyway ~) operations.

[QUOTE=cheesehead;102185]With single gamma:
213 = (4! * gamma(4) - √4) / √.4~[/QUOTE]

Ah, [i]that's[/i] how I got to 144 the first time. Must write things down in future, rather than relying on memory. Of course, I'll probably forget to do that.

220 = 44 * √4 / .4

221 = (gamma(4) / .4) ^ √4 - 4
222 = 444 / √4
223 = (gamma(4) / .4) ^ √4 - √4
224 = (4 + 4) * (4 + 4!)

225 = ((4 + √4) / .4) ^ √4

226 is stumping me at the moment

226 = (gamma(4) / .4)[sup]√4[/sup] + gamma(√4)
227 = (gamma(4) / .4)[sup]√4[/sup] + √4
228 = 4[sup]4[/sup] - 4! - 4
229 = (gamma(4) / .4)[sup]√4[/sup] + 4

With only one gamma:

226 = 4[sup]4[/sup] - 4! - gamma(4)

230 = 4 ^ 4 - 4! - √4
231 = 4 ^ 4 - 4! - gamma(√4)
232 = (4 * 4) ^ √4 - 4!
233 = 4 ^ 4 - 4! + gamma(√4)

I think I'll give up here for now, and see if 'better' (i.e. fewer / no gammas and/or fewer / no %s) solutions are do-able for the smaller numbers.

EDIT:
Actually, cheeky question, could petrw1 do another invaluable 'solutions so far' list, please?

234 = (4 * (√4 + 4!)) / .4~
or 234 = 4[sup]4[/sup] - 4! + √4

235 = (4 * 4! - √4) / .4
236 = 4[sup]4[/sup] - 4! + 4
237 = gamma(gamma(4)) * √4 - √4 / √(.4~)

[quote=cheesehead;102510]234 = 4 ^ 4 - 4! + √4
235 = (4! * 4 - √4) / .4
236 = 4 ^ 4 - 4! + 4[/quote]
Is there an echo in here? :crank:

If you look really hard, there are erasure marks between posts #151 and #152, timestamped a few minutes before this post. :-}

- - -

238 = 4! * 4 / .4 - √4
239 = (4! * 4 - .4) / .4
240 = 4 ^ 4 - 4 * 4
241 = (4! * 4 + .4) / .4

As a famous Genie once said: "Your wish is my command"

Where there are multiple solutions I list all the solutions except, if there are gamma/percent free solutions I list them only but if all solutions contain gamma/percent I list all these solutions even if one has more or less of these....see 113.

I substituted the gamma symbol (an upside down L) for the word gamma.

I proceeded all gamma/percent answers with an asterisk (*).

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 = (√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 = 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 = (√4 / .4)! - Γ(4) - Γ(√4) = (√4 + (√4 + 4!)%)) / (√4)% = Γ(Γ(4)) - √4 / .4 - √4 = Γ(Γ(4)) - 4 - (√4 / √.4~)
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 = ((4% + √(.4~)) / .4~%) - √4
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 = ((4 + 4!)% + .4~) / .4~% = (Γ(Γ(4))% + √(√(√(4^4!)))) / .4
164 = (√4 / .4)! + 44 = √(√(√(4^4!)))) / .4 + 4
165 = (√4+ √(√(√(4^4!))))) / .4 = 44 / (.4 * √.4~)
166 = 4! * (.4 ^ (-√4) + √.4~)
*167 = Γ(Γ(4)) + 4! * √4 - Γ(√4)
168 = ((√4 / .4~) + 4) * 4!
169 = ((4 / .4~) + 4) ^ √4
170 = (√(√(√(4^4!))) + 4) / .4 = (4! + 44) / .4
*171 = Γ(Γ(4)) / √.4~ - 4/.4~
172 = 44 * 4 - 4
*173 = Γ(Γ(4)) + (4! - .4~)/.4~
174 = 44 * 4 - √4
175 = ((4 + 4!) / .4) / .4
176 = 44 * (√4 + √4)
177 = ((√4 / .4)! - √4) / √.4~
178 = (4 * 44) + √4
179 = ((4 + √4)! - 4) / 4
180 = 4 + (4 * 44)
181 = ((√4 / .4)! + √.4~)/√.4~ = (4 + (4 + √(4))!) / 4
182 = ((4 + √(4))! / 4) + √(4)
183 = (√(4) + (√(4) / .4)!) / √(.4~)
184 = 4 * (√(4) + 44)
*185 = Γ(Γ(4)) + √(√(√(4^4!))) + Γ(√4))
186 = ((√4 / .4)! + 4) / √.4~
*187 = ((Γ(4))! + 4 + 4!) / 4
188 = (4 + 4) * 4! - 4
189 = (4! + 4! / .4) / .4~
190 = (4 + 4) * 4! - √4
*191 = ((Γ(4))! + 44) / 4
192 = (44 + 4) * 4
*193 = (4 + 4) * 4! + Γ(√4)
194 = (4 + 4) * 4! + √4
195 = (4! + (4! / .4~)) / .4
196 = (4! / √4 + √4) ^ √4
*197 = (4 - (4 + √4)%) / √4%
198 = (44 * √4) / .4~
*199 = (4 + 4) / 4% - Γ(√4)
200 = 44 * 4 + 4!
*201 = (4 + 4) / 4% + Γ(√4)
202 = 4^4 - ( 4! / .4~ )
*203 = Γ(Γ(4)) / √.4~ + 4! - Γ(√4)
204 = (√4 / .4)! / √.4~ + 4!
*205 = Γ(Γ(4)) / √.4~ + 4! + Γ(4)
*206 = Γ(Γ(4)) / √.4~ + 4! + √4
*207 = (4! * Γ(4) - Γ(4)) / √.4~
208 = 4 ^ 4 - 4! - 4!
*209 = ((Γ(4) + Γ(√(4)))! - 4!) / 4!
210 = (4! + 4! / .4) / .4
*211 = (Γ(4 + 4) + 4!) / 4!
212 = 4 ^ 4 - 44
*213 = (4! * Γ(4) - √4) / √.4~ = (Γ(Γ(4)) + 4! - √4) / √.4~
214 = 4 * 4! / .4~ - √4
215 = ((4 * 4!) - .4~) / .4~
216 = 4! * (4 + 4) + 4!
217 = (4 * 4! + .4~) / .4~
218 = 4 * 4! / .4~ + √4
*219 = (4! * Γ(4) + √4) / √.4~
220 = 4 * 4! / .4~ + 4 = 44 * √4 / .4
*221 = (Γ(4) / .4) ^ √4 - 4
222 = 444 / √4
*223 = (Γ(4) / .4) ^ √4 - √4
224 = (4 + 4) * (4 + 4!)
225 = ((4 + √4) / .4) ^ √4
*226 = (Γ(4) / .4)√4 + Γ(√4) = 44 - 4! - Γ(4)
*227 = (Γ(4) / .4)√4 + √4
228 = 44 - 4! - 4
*229 = (Γ(4) / .4)√4 + 4
230 = 4 ^ 4 - 4! - √4
*231 = 4 ^ 4 - 4! - Γ(√4)
232 = (4 * 4) ^ √4 - 4!
*233 = 4 ^ 4 - 4! + Γ(√4)

167 with less gammas:

167 = Γ(4) * (4! + 4) - Γ(√4)

173 and 185 with only one gamma:

173 = (Γ(4)! - 4! - 4) / 4
185 = (Γ(4)! + 4) / 4 + 4

[QUOTE=DJones;102023]207 = (4! * gamma(4) - gamma(4)) / √.4~
[/QUOTE]

Or, without gamma:

207 = (4 * 4! - 4) / .4~

P.S.: Is it allowed to use the n-th root?

219 = [sup].4[/sup]√(4 / .4~) - 4!

242 = 4 * 4! / .4 + √4
243 = (√4 / √.4~) ^ (√4 / .4)
244 = 4[sup]4[/sup] - 4! / √4
245 = (4 * 4! + √4) / .4

246 = 4 ^ 4 - (4 / .4)
247 = 4 ^ 4 - (4 / .4~)
248 = 4 ^ 4 - 4 - 4

249 = 4[sup]4[/sup] - Γ(4) - Γ(√4)
250 = 4[sup]4[/sup] - 4 - √4
251 = 4[sup]4[/sup] - √4 / .4
252 = 4[sup]4[/sup] - √4 - √4

[QUOTE=Andi47;102515]
P.S.: Is it allowed to use the n-th root?

219 = [sup].4[/sup]√(4 / .4~) - 4![/QUOTE]

I like it, I think it's very inventive ... anyone object?

253 = 4[sup]4[/sup] - √(4 / .4~)
254 = 4[sup]4[/sup] - (4 / √4)
255 = 4[sup]4[/sup] - (4 / 4)
256 = [sup]4[/sup]√((4[sup]4[/sup])[sup]4[/sup])
or
256 = 4 * 4 * 4 * 4

I think [sup]4[/sup]√ is just fine, since it doesn't imply a number other than 4.

257 = 4[sup]4[/sup] + 4 / 4
258 = 4[sup]4[/sup] + 4 - √4
259 = 4[sup]4[/sup] + √4 / √.4~
260 = 4[sup]4[/sup] + √4 + √4

261 = 4[sup]4[/sup] + √4 / .4
262 = 4[sup]4[/sup] + 4! / 4
263 = 44 * Γ(4) - Γ(√4)
264 = 44 * 4! / 4

265 = 4[sup]4[/sup] + 4 / .4~
266 = 4[sup]4[/sup] + 4 / .4
267 = [sup].4[/sup]√(4 / .4~) + 4!
268 = 4[sup]4[/sup] + 4! / √4

269 = ((√4/.4)! - .4~) / .4~
270 = (4 + 4/4)! / .4~
271 = ((√4/.4)! + .4~) / .4~
272 = 4[sup]4[/sup] + 4*4

273 = gamma(4)! * .4 - gamma(4) / .4
274 = (√4 / .4)! / .4~ + 4
275 = (44 / .4) / .4
276 = 4[sup]4[/sup] + 4! - 4

gamma gamma hey!

Yes, I was surprised when this thread dried up at a relatively easy bit.

But it's given me time to catch up. So:
277 = (gamma(gamma(4)) + 4)/.4~ - √4
278 = 4^4 + 4! - √4

errr I haven't come up with a 279 that is essentially different to those two.

But i did find the rather silly

276 = (4!)! / (4!-√4)! / √4

Richard

279 = 4[sup]4[/sup] + 4! - Γ(√4)

or gamma-less:
279 = ((√4 / .4)! + 4)/.4~
280 = 4[sup]4[/sup] + (√4+√4)!
281 = 4[sup]4[/sup] + 4! +Γ(√4)
282 = 4[sup]4[/sup] + 4! + √4

all variations on existing themes:

283 = (ΓΓ(4)-Γ(4))/.4 -√4
284 = 4!^√4/√4 -4
285 = ((√4/.4)!-Γ(4))/.4
286 = (ΓΓ(4)-4)/.4 -4

de-gamma'ing 286:

286 = (4![sup]√4[/sup] - 4) / √4

going on:
287 = (4! * 4! - √4) / √4
288 = (4! * 4!) / (4 - √4)
289 = (4! * 4! + √4) / √4
290 = (4! * 4! + 4) / √4

four days with no fours? 291 is up fo(u)r grabs...

the fun is somehow spoiled by knowing that there are web pages with solutions up to 40 000 and maybe more.

[QUOTE=m_f_h;104365]the fun is somehow spoiled by knowing that there are web pages with solutions up to 40 000 and maybe more.[/QUOTE]

Not really...

First, every website I have seen has a different set of allowed operations.

Second, I suggest that the fun for most of us is solving it for ourselves without checking the websites....who knows maybe we'll find a more interesting answer than they have.

[QUOTE=petrw1;104372]Not really...

First, every website I have seen has a different set of allowed operations.

Second, I suggest that the fun for most of us is solving it for ourselves without checking the websites....who knows maybe we'll find a more interesting answer than they have.[/QUOTE]

And we have already found at least one solution where the EXTREME four fours page did give a solution with our standard operations or gamma / %

291 still up fo(u)r grabs...

[quote=Andi47;104376]the EXTREME four fours page did give a solution with our standard operations or gamma / %[/quote]
??? did not ? AND gamma ?

PS: I said well " *somehow* spoiled ", which is my impression, even if I agree of course with what you replied (not looking at, and finding better)

It's sooooo easy to find a soulition with 200Up:

291=(-(4+√(4))% + √(4))/(√(.4~))%

nuggetprime

292=(4!*4!)/√(4)+4

The [i]odd[/i]s don't seem to be good up around this range.:unsure:

Aha. Things got easier once I got the hang of Γ and %.

293 = Γ(4)/((√4)%)-Γ(4)-Γ(√4)
294 = ((√4)/.4)!/.4~+4!
295 = Γ(4)/((√4)%)-4-Γ(√4)

Hint: I have solutions up to 308 at this point. But that's all four now. :cool:

296 = (√4 / .4)! / .4 - 4
297 = Γ(4) / ((√4)%) - 4 + Γ(√4) = Γ(4) / ((√4)%) - √4 / √.4~
298 = (√4 / .4)! / .4 - √4
299 = (√4 / .4)! / .4 - Γ(√4)

300=4<sup>4</sup>+44
301=(√(4/.4)!+.4)/.4
302=√(4/.4)!/.4+√4
303=(Γ(Γ(4))+.4+.4)/.4

304 = 4[sup]4[/sup] + 4! * √4
305 = ((√4 / .4)! + √4) / .4

[QUOTE=wpolly;105960]301=(√(4/.4)!+.4)/.4[/QUOTE]Then I can de-gammify 299 as:

299=(√(4/.4)!-.4)/.4

The next four I have all have percents, and some have gammas, but I guess I'll continue and let someone else de-gamma/percent-ify them:

306 = √4 / √(.4~)% + 4! / .4
307 = Γ(4) / √(4)% + Γ(4) + Γ(√4)
308 = √4 / √(.4~)% + 4 + 4
309 = √4 / √(.4~)% + 4 / .4~

Edit: I just checked with 200 Up, and found no solutions not involving % for those four numbers. :sad:

310 = ((√4 / .4)! + 4) / .4
311 = ΓΓ(4) / .4 + √(Γ(√4) + ΓΓ(4)) (EXTREME gamma...)
312 = 4! * (4! + √4) / √4
313 = √(√4 - 4%) / .4~% - √4

I think my 311 is a little better:
311 = (Γ(4))! * .4 - Γ(√4) + 4! (less EXTREME gamma...)

But I'll wait awhile before continuing, in case anyone else wants to.

OK, it's later...

314 = (Γ4)! * .4~ - 4 - √4
315 = (((√4)/.4)! + Γ4)/.4
316 = 4^4 + 4!/.4
317 = (Γ4)! * .4~ - 4 + Γ(√4) (Not very original, but over 300 I'll takes what I gets.)

318 = .4 * Γ(4)! + 4! + Γ(4) = (4!/4)! * .4~ - √4
319 = .4~ * Γ(4)! - 4/4
320 = .44~ * (4 + √4)! = .4~ * (√4 + √4 + √4)!

It's been a quiet week. Maybe things will pick up this weekend?

321 = .4~ * (Γ4)! + 4/4 = .4~ * (4!/4)! + Γ(√4) (Pick your gamma...)
322 = .4~ * (4!/4)! + √4
323 = .4~ * (Γ4)! + 4 - Γ(√4)
324 = (4! * 4!) / (4 * .4~)

One more 324, a bit more squared:
324 = (4 * 4 + √4) [sup]√4[/sup]

now it's getting loud:
325 = (√4 + 4!)! / 4!! / √4

pick your gamma:
326 = (4! - Γ(4)) [sup]√4[/sup] + √4 = Γ(4)[sup]4[/sup] / 4 + √4

Is no one else working on this?:unsure:

327 = (√4 + (√4)%) / (√.4~)% + 4! = (Γ4)!*.4~ + Γ4 + Γ(√4) (Either gamma or percent)

328 = (Γ4)!*.4~ + 4 + 4
329 = (Γ4)!*.4~ + 4/.4~

330 = (√(√4+√4)+√(4%))/(√.4~)% = (Γ4)!*.4~ + Γ4 + 4 (Again, either gamma or percent).

Latest complete list to 330

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 = (√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 = 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 = (√4 / .4)! - Γ(4) - Γ(√4) = (√4 + (√4 + 4!)%)) / (√4)% = Γ(Γ(4)) - √4 / .4 - √4 = Γ(Γ(4)) - 4 - (√4 / √.4~)
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 = ((4% + √(.4~)) / .4~%) - √4
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 = ((4 + 4!)% + .4~) / .4~% = (Γ(Γ(4))% + √(√(√(4^4!)))) / .4
164 = (√4 / .4)! + 44 = √(√(√(4^4!)))) / .4 + 4
165 = (√4+ √(√(√(4^4!))))) / .4 = 44 / (.4 * √.4~)
166 = 4! * (.4 ^ (-√4) + √.4~)
*167 = Γ(Γ(4)) + 4! * √4 - Γ(√4) = Γ(4) * (4! + 4) - Γ(√4)
168 = ((√4 / .4~) + 4) * 4!
169 = ((4 / .4~) + 4) ^ √4
170 = (√(√(√(4^4!))) + 4) / .4 = (4! + 44) / .4
*171 = Γ(Γ(4)) / √.4~ - 4/.4~
172 = 44 * 4 - 4
*173 = Γ(Γ(4)) + (4! - .4~)/.4~ = (Γ(4)! - 4! - 4) / 4
174 = 44 * 4 - √4
175 = ((4 + 4!) / .4) / .4
176 = 44 * (√4 + √4)
177 = ((√4 / .4)! - √4) / √.4~
178 = (4 * 44) + √4
179 = ((4 + √4)! - 4) / 4
180 = 4 + (4 * 44)
181 = ((√4 / .4)! + √.4~)/√.4~ = (4 + (4 + √(4))!) / 4
182 = ((4 + √(4))! / 4) + √(4)
183 = (√(4) + (√(4) / .4)!) / √(.4~)
184 = 4 * (√(4) + 44)
*185 = Γ(Γ(4)) + √(√(√(4^4!))) + Γ(√4)) = (Γ(4)! + 4) / 4 + 4
186 = ((√4 / .4)! + 4) / √.4~
*187 = ((Γ(4))! + 4 + 4!) / 4
188 = (4 + 4) * 4! - 4
189 = (4! + 4! / .4) / .4~
190 = (4 + 4) * 4! - √4
*191 = ((Γ(4))! + 44) / 4
192 = (44 + 4) * 4
*193 = (4 + 4) * 4! + Γ(√4)
194 = (4 + 4) * 4! + √4
195 = (4! + (4! / .4~)) / .4
196 = (4! / √4 + √4) ^ √4
*197 = (4 - (4 + √4)%) / √4%
198 = (44 * √4) / .4~
*199 = (4 + 4) / 4% - Γ(√4)
200 = 44 * 4 + 4!
*201 = (4 + 4) / 4% + Γ(√4)
202 = 4^4 - ( 4! / .4~ )
*203 = Γ(Γ(4)) / √.4~ + 4! - Γ(√4)
204 = (√4 / .4)! / √.4~ + 4!
*205 = Γ(Γ(4)) / √.4~ + 4! + Γ(4)
*206 = Γ(Γ(4)) / √.4~ + 4! + √4
207 = (4 * 4! - 4) / .4~
208 = 4 ^ 4 - 4! - 4!
*209 = ((Γ(4) + Γ(√(4)))! - 4!) / 4!
210 = (4! + 4! / .4) / .4
*211 = (Γ(4 + 4) + 4!) / 4!
212 = 4 ^ 4 - 44
*213 = (4! * Γ(4) - √4) / √.4~ = (Γ(Γ(4)) + 4! - √4) / √.4~
214 = 4 * 4! / .4~ - √4
215 = ((4 * 4!) - .4~) / .4~
216 = 4! * (4 + 4) + 4!
217 = (4 * 4! + .4~) / .4~
218 = 4 * 4! / .4~ + √4
219 = .4√(4 / .4~) - 4!
220 = 4 * 4! / .4~ + 4 = 44 * √4 / .4
*221 = (Γ(4) / .4) ^ √4 - 4
222 = 444 / √4
*223 = (Γ(4) / .4) ^ √4 - √4
224 = (4 + 4) * (4 + 4!)
225 = ((4 + √4) / .4) ^ √4
*226 = (Γ(4) / .4)√4 + Γ(√4) = 4^4 - 4! - Γ(4)
*227 = (Γ(4) / .4)√4 + √4
228 = 4^4 - 4! - 4
*229 = (Γ(4) / .4)√4 + 4
230 = 4 ^ 4 - 4! - √4
*231 = 4 ^ 4 - 4! - Γ(√4)
232 = (4 * 4) ^ √4 - 4!
*233 = 4 ^ 4 - 4! + Γ(√4)
234 = 4 ^ 4 - 4! + √4 = (4 * (√4 + 4!)) / .4~ = 44 - 4! + √4
235 = (4! * 4 - √4) / .4
236 = 4 ^ 4 - 4! + 4 = 44 - 4! + 4
*237 = Γ(Γ(4)) * √4 - √4 / √(.4~)
238 = 4! * 4 / .4 - √4
239 = (4! * 4 - .4) / .4
240 = 4 ^ 4 - 4 * 4
241 = (4! * 4 + .4) / .4
242 = 4 * 4! / .4 + √4
243 = (√4 / √.4~) ^ (√4 / .4)
244 = 4^4 - 4! / √4
245 = (4 * 4! + √4) / .4
246 = 4 ^ 4 - (4 / .4)
247 = 4 ^ 4 - (4 / .4~)
248 = 4 ^ 4 - 4 - 4
*249 = 4^4 - Γ(4) - Γ(√4)
250 = 4^4 - 4 - √4
251 = 4^4 - √4 / .4
252 = 4^4 - √4 - √4
253 = 4^4 - √(4 / .4~)
254 = 4^4 - (4 / √4)
255 = 4^4 - (4 / 4)
256 = 4 * 4 * 4 * 4 = 4√((4^4)4)
257 = 4^4 + 4 / 4
258 = 4^4 + 4 - √4
259 = 4^4 + √4 / √.4~
260 = 4^4 + √4 + √4
261 = 4^4 + √4 / .4
262 = 4^4 + 4! / 4
*263 = 4^4 * Γ(4) - Γ(√4)
264 = 4^4 * 4! / 4
265 = 4^4 + 4 / .4~
266 = 4^4 + 4 / .4
267 = .4√(4 / .4~) + 4!
268 = 4^4 + 4! / √4
269 = ((√4/.4)! - .4~) / .4~
270 = (4 + 4/4)! / .4~
271 = ((√4/.4)! + .4~) / .4~
272 = 4^4 + 4*4
*273 = Γ(4)! * .4 - Γ(4) / .4
274 = (√4 / .4)! / .4~ + 4
275 = (44 / .4) / .4
276 = (4!)! / (4!-√4)! / √4 = 4^4 + 4! - 4
*277 = (Γ(Γ(4)) + 4)/.4~ - √4
278 = 4^4 + 4! - √4
279 = ((√4 / .4)! + 4)/.4~
280 = 4^4 + (√4+√4)!
*281 = 4^4 + 4! +Γ(√4)
282 = 4^4 + 4! + √4
*283 = (ΓΓ(4)-Γ(4))/.4 -√4
284 = 4!^√4/√4 -4
*285 = ((√4/.4)!-Γ(4))/.4
286 = (4!√4 - 4) / √4
287 = (4! * 4! - √4) / √4
288 = (4! * 4!) / (4 - √4)
289 = (4! * 4! + √4) / √4
290 = (4! * 4! + 4) / √4
*291 = (-(4+√(4))% + √(4))/(√(.4~))%
292 = (4!*4!)/√(4)+4
*293 = Γ(4)/((√4)%)-Γ(4)-Γ(√4)
294 = ((√4)/.4)!/.4~+4!
*295 = Γ(4)/((√4)%)-4-Γ(√4)
296 = (√4 / .4)! / .4 - 4
*297 = Γ(4) / ((√4)%) - 4 + Γ(√4) = Γ(4) / ((√4)%) - √4 / √.4~
298 = (√4 / .4)! / .4 - √4
299 = (√(4/.4)!-.4)/.4
300 = 4^4+44
301 = (√(4/.4)!+.4)/.4
302 = √(4/.4)!/.4+√4
*303 = (Γ(Γ(4))+.4+.4)/.4
304 = 4^4 + 4! * √4
305 = ((√4 / .4)! + √4) / .4
*306 = √4 / √(.4~)% + 4! / .4
*307 = Γ(4) / √(4)% + Γ(4) + Γ(√4)
*308 = √4 / √(.4~)% + 4 + 4
*309 = √4 / √(.4~)% + 4 / .4~
310 = ((√4 / .4)! + 4) / .4
*311 = (Γ(4))! * .4 - Γ(√4) + 4! = ΓΓ(4) / .4 + √(Γ(√4) + ΓΓ(4))
312 = 4! * (4! + √4) / √4
*313 = √(√4 - 4%) / .4~% - √4
*314 = (Γ4)! * .4~ - 4 - √4
*315 = (((√4)/.4)! + Γ4)/.4
316 = 4^4 + 4!/.4
*317 = (Γ4)! * .4~ - 4 + Γ(√4)
*318 = .4 * Γ(4)! + 4! + Γ(4) = (4!/4)! * .4~ - √4
*319 = .4~ * Γ(4)! - 4/4
320 = .44~ * (4 + √4)! = .4~ * (√4 + √4 + √4)!
*321 = .4~ * (Γ4)! + 4/4 = .4~ * (4!/4)! + Γ(√4)
322 = .4~ * (4!/4)! + √4
*323 = .4~ * (Γ4)! + 4 - Γ(√4)
324 = (4 * 4 + √4) √4 = (4! * 4!) / (4 * .4~)
325 = (√4 + 4!)! / 4!! / √4
*326 = (4! - Γ(4)) √4 + √4 = Γ(4)4 / 4 + √4
*327 = (√4 + (√4)%) / (√.4~)% + 4! = (Γ4)!*.4~ + Γ4 + Γ(√4)
*328 = (Γ4)!*.4~ + 4 + 4
*329 = (Γ4)!*.4~ + 4/.4~
*330 = (√(√4+√4)+√(4%))/(√.4~)% = (Γ4)!*.4~ + Γ4 + 4

Numbers preceded by * contain either Gamma or Percent function (or both) and are therefore deemed as second class citizen solutions - when solutions are reported that do not contain either of these the new formula replaces the one(s) listed. I got a couple of these since the last list was posted. If all formulas reported for a single number contain Gamma or Percent they are all listed; likewise if all formulas do NOT contain Gamma or Percent they are all listed as well.

From 0 to 99: there are no formulas containing Gamma or Percent
100 - 199: 12 (12%)
200 - 299: 26 (26%)
300 - 330: 19 (61%)

well I've been out of the mersenne forum loop for a while, but this seems a nice place to hop back in

I haven't got anything for 331+ yet since I've spent more time looking through the posts than working out these numbers, but ..

a less gamma'ed 113:
113 = (4! + 4) * 4 + Γ(√4)

and a (badly) gamma'ed 157 without %s:
157 = Γ(Γ(4)) + √(Γ(4)[sup]4[/sup]) + Γ(√4)