Four Fours Puzzle: Get Each Integer with 4s

[Jimmy Snyder]

This one is just for fun, I do not have the answer myself. I was reminded of it by BicycleTree's procedure. The goal is to get each integer as the result of using any of the four operations, and exponentiation, operating on four fours. For instance:

1 = 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

I worked on this a few years back and got most numbers, but not all. I do not remember which ones I got and which ones I didn't. I think I was allowing two fours to be used as 44.

 

[jdavel]

10 = (44 - 4)/4

 

[nnnnnnnn]

2.0636505122486923685638272848301e+267 = 4^(444)

 

[BicycleTree]

Wow, nnnnnnnn--at the moment it says you have Posts: 44.

 

[NateTG]

2.0636505122486923685638272848301e+267 = 4^(444)

4^{4^{4^4}}
is a bit larger.

 

[Jimmy Snyder]

For definiteness, make that:

\HUGE {4^{(4^{(4^{4})})}}

 

[Kazza_765]

Heh. Just tried \HUGE {4^{(4^{(4^{4})})}} in the 512bit calculator on the computer and the whole thing crashed.

 

[Jimmy Snyder]

Heh. Just tried \HUGE {4^{(4^{(4^{4})})}} in the 512bit calculator on the computer and the whole thing crashed.

A 1024 bit calculator would fare no better. The number is very roughly googolplex \times \sqrt{googolplex}

 

[rachmaninoff]

4^{\left( 4^ \left (4^4 \right) \right)}=4^{\left( 4^{256} \right) }=4^{\left( 2^{512} \right) }
log_{10} 4^{\left( 4^{256} \right) } = 2^{512} log_{10} 4 \thickapprox 0.60206 \times 2^{512} \approxeq 8.0723 \times 10^{153}

It's a number with about 8e153 digits.

 

[rachmaninoff]

From 0 through 100:

0=((4-4)-4)+4
1=((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=(44-4)/4
12=(4-(4/4))*4
15=(44/4)+4
16=((4+4)+4)+4
17=(4*4)+(4/4)
20=((4/4)+4)*4
24=((4*4)+4)+4
28=44-(4*4)
32=(4*4)+(4*4)
36=((4+4)*4)+4
43=44-(4/4)
44=(44-4)+4
45=(4/4)+44
48=((4*4)-4)*4
52=(44+4)+4
60=(4*4)+44
63=((4^4)-4)/4
64=4^(4-(4/4))
65=((4^4)+4)/4
68=((4*4)*4)+4
80=((4*4)+4)*4
81=((4/4)-4)^4
88=44+44

 

[yt2005]

0 = 44 - 44
1 = 44 / 44
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 = (44 / 4) - 4
8 = 4 + 4 + 4 - 4
9 = 4 + 4 + (4 / 4)
10 = (44 - 4) / 4
11 = 44 / sqrt(4 * 4)
12 = (4 + 44) / 4
13 = sqrt(4) + (44 / 4)
14 = 4 + 4 + 4 + sqrt(4)
15 = 4 + (44 / 4)
16 = 4 + 4 + 4 + 4
17 = (4 * 4) + (4 / 4)
18 = (44 / sqrt(4)) - 4
19 = 4! - (4 + (4 / 4))
20 = 4 * (4 + (4 / 4))
21 = (44 - sqrt(4)) / sqrt(4)
22 = (sqrt(4) / 4) * 44
23 = (sqrt(4) + 44) / sqrt(4)
24 = 4 + 4 + (4 * 4)
25 = (4 + (4 * 4!)) / 4
26 = 4 + (44 / sqrt(4))
27 = 4 - (4 / 4) + 4!
28 = 44 - (4 * 4)
29 = 4 + (4 / 4) + 4!
30 = (4 * (4 + 4)) - sqrt(4)
31 = ((4 + 4!) / 4) + 4!
32 = (4 * 4) + (4 * 4)
33 = 44 / sqrt(4 * .4~)
34 = (4! + 44) / sqrt(4)
35 = 4! + (44 / 4)
36 = 44 - (4 + 4)
37 = ((sqrt(4) + 4!) / sqrt(4)) + 4!
38 = 44 - (4 + sqrt(4))
39 = 44 - (sqrt(4) / .4)
40 = 44 - sqrt(4 * 4)
41 = sqrt(((4 + 4)! + 4!) / 4!)
42 = sqrt(4) - 4 + 44
43 = 44 - (4 / 4)
44 = 4 - 4 + 44
45 = (4 / 4) + 44
46 = 4 - sqrt(4) + 44
47 = (sqrt(4) * 4!) - (4 / 4)
48 = 4 * (4 + 4 + 4)
49 = (4 / 4) + (sqrt(4) * 4!)
50 = 4 + sqrt(4) + 44
51 = (4! - 4 + .4) / .4
52 = 4 + 4 + 44
53 = (4 / .4~) + 44
54 = 4 + sqrt(4) + (sqrt(4) * 4!)
55 = 44 / (.4 + .4)
56 = (4! / sqrt(4)) + 44
57 = ((4! + .4) / .4) - 4
58 = ((4 + 4!) * sqrt(4)) + sqrt(4)
59 = (4! / .4) - (4 / 4)
60 = (4 * 4) + 44
61 = (4 / 4) + (4! / .4)
62 = (4 * 4 * 4) - sqrt(4)
63 = (44 - sqrt(4)) / sqrt(.4~)
64 = (4 + 4) * (4 + 4)
65 = 4 + ((4! + .4) / .4)
66 = 4! - sqrt(4) + 44
67 = 4 + ((4 + 4!) / .4~)
68 = 4 + (4 * 4 * 4)
69 = (sqrt(4) + 44) / sqrt(.4~)
70 = sqrt(4) + 4! + 44
71 = (4 + 4! + .4) / .4
72 = 4 + 4! + 44
73 = ((sqrt(4) * 4!) + sqrt(.4~)) / sqrt(.4~)
74 = (4 * 4!) + sqrt(4) - 4!
75 = (44 / .4~) - 4!
76 = (4 * (4! - 4)) - 4
77 = 4! + ((4! - .4~) / .4~)
78 = (4 * (4! - 4)) - sqrt(4)
79 = 4! - ((sqrt(4) - 4!) / .4)
80 = 4 * (4 + (4 * 4))
81 = (4 - .4) / (.4~ - .4)
82 = (4 * (4! - 4)) + sqrt(4)
83 = 4! + ((4! - .4) / .4)
84 = (sqrt(4) * 44) - 4
85 = ((4 / .4) + 4!) / .4
86 = (sqrt(4) * 44) - sqrt(4)
87 = (4 * 4!) - (4 / .4~)
88 = 44 + 44
89 = ((sqrt(4) + 4!) / .4) + 4!
90 = sqrt(4) + (sqrt(4) * 44)
91 = (4 * 4!) - (sqrt(4) / .4)
92 = 4 + (sqrt(4) * 44)
93 = (4 * 4!) - sqrt(4 / .4~)
94 = 4 * (4! - (sqrt(4) / 4))
95 = (4 * 4!) - (4 / 4)
96 = (4 + 44) * sqrt(4)
97 = (4 / 4) + (4 * 4!)
98 = 4 + (4 * 4!) - sqrt(4)
99 = 4.4 / (.4~ - .4)
100 = 4 * ((4 / 4) + 4!)


101 = sqrt(4) + (44 / .4~)
102 = 4 + (4 * 4!) + sqrt(4)
103 = 4 + (44 / .4~)
104 = 4 + 4 + (4 * 4!)
105 = (44 - sqrt(4)) / .4
106 = (4 * (sqrt(4) + 4!)) + sqrt(4)
107 = ((sqrt(4) * 4!) - .4~) / .4~
108 = (4 * (4 + 4!)) - 4
109 = (44 - .4) / .4
110 = (4 * (4 + 4!)) - sqrt(4)
111 = 444 / 4
112 = (sqrt(4) * 44) + 4!
113 = (sqrt(4) + (sqrt(4) / .4~%)) / 4
114 = (4 * (4 + 4!)) + sqrt(4)
115 = (sqrt(4) + 44) / .4
116 = 4 + (4 * (4 + 4!))
117 = (4 + (sqrt(4) * 4!)) / .4~
118 = (4 + (4 / 4))! - sqrt(4)
119 = (sqrt(4) / .4)! - (4 / 4)
120 = ((44 - 4!) / 4)!
121 = (4 / 4) + (sqrt(4) / .4)!
122 = (4 + (4 / 4))! + sqrt(4)
123 = 4! + (44 / .4~)
124 = 4 + (4 + (4 / 4))!
125 = (sqrt(4) + (sqrt(4) * 4!)) / .4
126 = ((4! / .4) - 4) / .4~
127 = (4 ^ 4 - sqrt(4)) / sqrt(4)
128 = 4 * 4 * (4 + 4)
129 = (4 / .4~) + (sqrt(4) / .4)!
130 = (4 + (sqrt(4) * 4!)) / .4
131 = (4! / (.4 * .4~)) - 4
132 = (4 + sqrt(4)) * (4! - sqrt(4))
133 = (4! / (.4 * .4~)) - sqrt(4)
134 = 4! + (44 / .4)
135 = (4 + sqrt(4)) / (.4~ - .4)
136 = sqrt(4) * (4! + 44)
137 = sqrt(4) + (4! / (.4 * .4~))
138 = ((4! * 4!) - 4!) / 4
139 = 4 + (4! / (.4 * .4~))
140 = (4 * 4!) + 44
141 = ((4 * 4!) - sqrt(4)) / sqrt(.4~)
142 = sqrt(4 + ((4 + 4)! / sqrt(4)))
143 = ((4! * 4!) - 4) / 4
144 = (4 + (4 / 4))! + 4!
145 = (4 + (4! * 4!)) / 4
146 = ((4 + sqrt(4)) * 4!) + sqrt(4)
147 = ((4 * 4!) + sqrt(4)) / sqrt(.4~)
148 = 4 + ((4 + sqrt(4)) * 4!)
149 = ((4! / .4) - .4) / .4
150 = (4! + (4! * 4!)) / 4
151 = ((4! / .4) + .4) / .4
152 = (4 * 44) - 4!
153 = (4! + 44) / .4~
154 = 4 + (4! * .4 ^ -sqrt(4))
155 = (sqrt(4) + (4! / .4)) / .4
156 = (4 + sqrt(4)) * (sqrt(4) + 4!)
157 = ((4% + sqrt(.4~)) / .4~%) - sqrt(4)
158 = (root(.4, 4) - .4) / sqrt(4%)
159 = 4! + (4! / (.4 * .4~))
160 = 4 * (44 - 4)
161 = ((4! + .4)% + .4) / .4%
162 = (4 + sqrt(4))! / 4.4~
163 = ((4 + 4!)% + .4~) / .4~%
164 = (sqrt(4) / .4)! + 44
165 = 44 / (.4 * sqrt(.4~))
166 = 4! * (.4 ^ -sqrt(4) + sqrt(.4~))
167 = sqrt(.4~ / 4) + (sqrt(.4~) / .4%)
168 = 4 * (44 - sqrt(4))
169 = sqrt((4 + (4 / .4~)) ^ 4)
170 = (4! + 44) / .4
171 = ((4 / 4%) - 4!) / .4~
172 = (4 * 44) - 4
173 = (4 - (4! / .4~)%) / sqrt(4)%
174 = (4 * 44) - sqrt(4)
175 = (4 + 4!) * .4 ^ -sqrt(4)
176 = sqrt(4 * 4) * 44
177 = ((sqrt(4) / .4)! - sqrt(4)) / sqrt(.4~)
178 = (4 * 44) + sqrt(4)
179 = ((4 + sqrt(4))! - 4) / 4
180 = 4 + (4 * 44)
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)
185 = 44.4 / 4!%
186 = ((4 + sqrt(4))! + 4!) / 4
187 = (4 - (sqrt(4) + 4!)%) / sqrt(4)%
188 = ((4 + 4) * 4!) - 4
189 = (4! + (4! / .4)) / .4~
190 = ((4 + 4) * 4!) - sqrt(4)
191 = (4 / sqrt(4)%) - (4 / .4~)
192 = 4 * (4 + 44)
193 = sqrt(.4~% ^ -sqrt(4)) - root(.4, 4)
194 = ((4 + 4) * 4!) + sqrt(4)
195 = (4! + (4! / .4~)) / .4
196 = 4 + ((4 + 4) * 4!)
197 = (4 - (4 + sqrt(4))%) / sqrt(4)%
198 = sqrt(4) * 44 / .4~
199 = (4 / sqrt(4)%) - (4 / 4)
200 = (4 * 44) + 4!


201 = (4 / 4) + (4 / sqrt(4)%)
202 = 4 ^ 4 - (4! / .4~)
203 = (4 + (4 + sqrt(4))%) / sqrt(4)%
204 = ((4 + sqrt(4))! / 4) + 4!
205 = (4 + (.4 / 4)) / sqrt(4)%
206 = 4 ^ 4 - (sqrt(4) / 4%)
207 = ((4 * 4!) - 4) / .4~
208 = 4 * (4 + (sqrt(4) * 4!))
209 = (4 / sqrt(4)%) + (4 / .4~)
210 = ((4 + 4!) / 4)! / 4!
211 = 4 ^ 4 - (sqrt(4%) / .4~%)
212 = (4 * 4! / .4~) - 4
213 = (4 + (sqrt(4) + 4!)%) / sqrt(4)%
214 = (4 * 4! / .4~) - sqrt(4)
215 = ((4 * 4!) - .4~) / .4~
216 = ((4 + 4) * 4!) + 4!
217 = ((4 * 4!) + .4~) / .4~
218 = (4 * 4! / .4~) + sqrt(4)
219 = (44 - sqrt(4%)) / sqrt(4%)
220 = sqrt(4) * 44 / .4
221 = (4 / (4 * .4~)%) - 4
222 = 444 / sqrt(4)
223 = (4 / (4 * .4~)%) - sqrt(4)
224 = (4 + 4) * (4 + 4!)
225 = (4 + (4 * 4!)) / .4~
226 = (4 + (4 / .4~%)) / 4
227 = (4 / (4 * .4~)%) + sqrt(4)
228 = 4 ^ 4 - (4 + 4!)
229 = 4 + (4 / (4 * .4~)%)
230 = ((4 * 4!) - 4) / .4
231 = ((4 / .4~%) + 4!) / 4
232 = 4 * (4 + (4! / .4~))
233 = 4 + 4 + sqrt(.4~% ^ -sqrt(4))
234 = 4 * (sqrt(4) + 4!) / .4~
235 = ((4 * 4!) - sqrt(4)) / .4
236 = (4 * 4! / .4) - 4
237 = ((4 * 4) - sqrt(4%)) / sqrt(.4~%)
238 = (4 * 4! / .4) - sqrt(4)
239 = ((4 * 4!) - .4) / .4
240 = (4 + (4 / 4))! * sqrt(4)
241 = ((4 * 4!) + .4) / .4
242 = (4 * 4! / .4) + sqrt(4)
243 = (sqrt(4) / .4~) * 4! / .4~
244 = 4 + (4 * 4! / .4)
245 = ((4 * 4!) + sqrt(4)) / .4
246 = ((sqrt(4) / .4)! / .4~) - 4!
247 = 4 ^ 4 - (4 / .4~)
248 = 4 * (sqrt(4) + (4! / .4))
249 = ((4 / .4%) - 4) / 4
250 = (4 + (4 * 4!)) / .4
251 = 4 ^ 4 - (sqrt(4) / .4)
252 = 4 * (4 + 4!) / .4~
253 = 4 ^ 4 - sqrt(4 / .4~)
254 = (4! * 4! * .4~) - sqrt(4)
255 = 4 ^ 4 - (4 / 4)
256 = 4 * 4 * 4 * 4
257 = (4 / 4) + 4 ^ 4
258 = sqrt(4) + (4! * 4! * .4~)
259 = 4 ^ 4 + sqrt(4 / .4~)
260 = 4 * (sqrt(4) + 4!) / .4
261 = ((sqrt(4) / .4)! - 4) / .4~
262 = 4 + 4 ^ 4 + sqrt(4)
263 = (sqrt(4) - (4! + sqrt(.4~))%) / sqrt(.4~)%
264 = (4 + sqrt(4)) * 44
265 = 4 ^ 4 + (4 / .4~)
266 = ((sqrt(4) / .4)! / .4~) - 4
267 = 4! + root(sqrt(4%), sqrt(4 / .4~))
268 = ((sqrt(4) / .4)! / .4~) - sqrt(4)
269 = ((sqrt(4) / .4)! - .4~) / .4~
270 = (4 + (4 / 4))! / .4~
271 = ((sqrt(4) / .4)! + .4~) / .4~
272 = 4 * (4! + 44)
273 = ((sqrt(4) - sqrt(4)%) / sqrt(.4~)%) - 4!
274 = 4 + ((sqrt(4) / .4)! / .4~)
275 = 44 * .4 ^ -sqrt(4)
276 = ((4! * 4!) - 4!) / sqrt(4)
277 = ((sqrt(4) + sqrt(.4~)%) / sqrt(.4~)%) - 4!
278 = 4 ^ 4 - sqrt(4) + 4!
279 = (4 + (sqrt(4) / .4)!) / .4~
280 = 4 * (4 + 4!) / .4
281 = 4 ^ 4 + sqrt(4% ^ -sqrt(4))
282 = 4 ^ 4 + sqrt(4) + 4!
283
284 = ((4! / sqrt(4)) * 4!) - 4
285 = (4! + (.4 / .4~%)) / .4
286 = ((4! * 4!) - 4) / sqrt(4)
287 = ((4! * 4!) - sqrt(4)) / sqrt(4)
288 = (4 + 4 + 4) * 4!
289 = (sqrt(4) + (4! * 4!)) / sqrt(4)
290 = (4 + (4! * 4!)) / sqrt(4)
291 = (sqrt(4 * .4~) - 4%) / .4~%
292 = 4 + ((4! / sqrt(4)) * 4!)
293 = ((sqrt(4) - sqrt(4)%) / sqrt(.4~)%) - 4
294 = ((sqrt(4) / .4)! / .4~) + 4!
295 = ((sqrt(4) / .4)! - sqrt(4)) / .4
296 = 444 * sqrt(.4~)
297 = (4! - 4!%) / (4 + 4)%
298 = ((sqrt(4) / .4)! / .4) - sqrt(4)
299 = ((sqrt(4) / .4)! - .4) / .4
300 = (4! + (4! * 4!)) / sqrt(4)


301 = ((sqrt(4) / .4)! + .4) / .4
302 = sqrt(4) + ((sqrt(4) / .4)! / .4)
303 = (4! + 4!%) / (4 + 4)%
304 = 4 + ((sqrt(4) / .4)! / .4)
305 = (sqrt(4) + (sqrt(4) / .4)!) / .4
306 = 4 ^ 4 + (sqrt(4) / 4%)
307 = 4 + ((sqrt(4) + sqrt(4)%) / sqrt(.4~)%)
308 = 4 + 4 + (sqrt(4) / sqrt(.4~)%)
309 = (sqrt(4 * .4~) + 4%) / .4~%
310 = (4 + (sqrt(4) / .4)!) / .4
311 = (sqrt(sqrt(4) - 4%) / .4~%) - 4
312 = ((sqrt(4) + 4!) / sqrt(4)) * 4!
313 = (sqrt(sqrt(4) - 4%) / .4~%) - sqrt(4)
314 = 4 + ((sqrt(4) + sqrt(.4~%)) / sqrt(.4~)%)
315 = ((4 / 4) + .4) / .4~%
316 = ((4 + sqrt(4))! * .4~) - 4
317 = sqrt(4) + (sqrt(sqrt(4) - 4%) / .4~%)
318 = ((4 + sqrt(4))! * .4~) - sqrt(4)
319 = 4 + (sqrt(sqrt(4) - 4%) / .4~%)
320 = 4 * 4 * (4! - 4)
321 = ((sqrt(.4~) / .4%) - 4!) / .4~
322 = ((4 + sqrt(4))! * .4~) + sqrt(4)
323 = ((sqrt(4) - sqrt(.4~)%) / sqrt(.4~)%) + 4!
324 = 4 + ((4 + sqrt(4))! * .4~)
325 = ((sqrt(4) + 4!)! / 4!) / sqrt(4)
326 = ((4! - sqrt(4)) / sqrt(.4~%)) - 4
327 = ((sqrt(4) + sqrt(4)%) / sqrt(.4~)%) + 4!
328 = (4 + (4 ^ 4)%) / sqrt(4)%
329 = (4! - (sqrt(4) + sqrt(.4~%))) / sqrt(.4~%)
330 = 44 / sqrt((4 * .4~)%)
331 = (4! - sqrt(4) + sqrt(.4~%)) / sqrt(.4~%)
332 = (sqrt(4) / sqrt(.4~)%) + root(.4, 4)
333 = (4 - .4%) / (sqrt(4) / .4)!%%
334 = 4 - ((sqrt(4) - 4!) / sqrt(.4~%))
335 = (4 + sqrt(4)%) / (sqrt(4) / .4)!%%
336 = ((4 * 4) - sqrt(4)) * 4!
337 = (sqrt(4) + (4! + sqrt(.4~))%) / sqrt(.4~)%
338 = sqrt((sqrt(4) + 4!) ^ 4 / 4)
339 = (sqrt(sqrt(4) - 4%) / .4~%) + 4!
340 = (4! + 44) / sqrt(4%)
341 = (sqrt(4) - (4% + .4~)) / .4~%
342 = (4 - sqrt(4%)) * .4 / .4~%
343 = ((sqrt(4) - sqrt(.4~%))% + sqrt(.4~)) / sqrt(4%)%
344 = ((4 + sqrt(4))! * .4~) + 4!
345 = (4! - (4 / 4)) / sqrt(.4~%)
346 = 4 ^ 4 + (.4 / .4~%)
347 = (4! - (sqrt(4%) + sqrt(.4~))) / sqrt(.4~%)
348 = ((4 + sqrt(4))! - 4!) / sqrt(4)
349 = (sqrt(sqrt(4) - 4%) - .4%) / .4%
350 = (4 + (4 / .4)) / 4%
351 = (sqrt(4) - 44%) / .4~%
352 = (4 + 4) * 44
353 = ((4! - sqrt(4%)) / sqrt(.4~%)) - 4
354 = (sqrt(4) / .4~%) - (4 * 4!)
355 = ((4 * .4~) - sqrt(4%)) / .4~%
356 = ((4 + sqrt(4))! / sqrt(4)) - 4
357 = (4! - (.4 / sqrt(4))) / sqrt(.4~%)
358 = ((4 + sqrt(4))! - 4) / sqrt(4)
359 = ((4 + sqrt(4))! - sqrt(4)) / sqrt(4)
360 = (4 * 4 * 4!) - 4!
361 = ((4 + sqrt(4))! + sqrt(4)) / sqrt(4)
362 = (4 + (4 + sqrt(4))!) / sqrt(4)
363 = (4! + (.4 / sqrt(4))) / sqrt(.4~%)
364 = 4 + ((4 + sqrt(4))! / sqrt(4))
365 = ((sqrt(.4~) / .4~%) - 4) / .4
366 = (sqrt(4) + 44%) / sqrt(.4~)%
367 = 4 + ((4! + sqrt(4%)) / sqrt(.4~%))
368 = 4 * ((4 * 4!) - 4)
369 = 4% ^ -sqrt(4) - 4 ^ 4
370 = 4 + ((4! + .4) / sqrt(.4~%))
371 = (4 / (.4 + sqrt(.4~))%) - 4
372 = ((4 + sqrt(4))! + 4!) / sqrt(4)
373 = (4 / (.4 + sqrt(.4~))%) - sqrt(4)
374 = ((sqrt(4) - .4~) / .4~%) + 4!
375 = (4! / .4) * .4 ^ -sqrt(4)
376 = 4 * ((4 * 4!) - sqrt(4))
377 = (4 / (.4 + sqrt(.4~))%) + sqrt(4)
378 = (4 ^ 4 - 4) / sqrt(.4~)
379 = 4 + (4 / (.4 + sqrt(.4~))%)
380 = (4 * 4 * 4!) - 4
381 = (4 ^ 4 - sqrt(4)) / sqrt(.4~)
382 = (4 * 4 * 4!) - sqrt(4)
383 = (4 ^ 4 - sqrt(.4~)) / sqrt(.4~)
384 = 4 * 4 * sqrt(4 * 4)!
385 = (4 ^ 4 + sqrt(.4~)) / sqrt(.4~)
386 = (4 * 4 * 4!) + sqrt(4)
387 = (4 ^ 4 + sqrt(4)) / sqrt(.4~)
388 = 4 + (4 * 4 * 4!)
389 = (sqrt(4) + 4! - sqrt(.4~%)) / sqrt(.4~%)
390 = (4 + 4 ^ 4) / sqrt(.4~)
391 = ((4 * .4~) - 4%) / .4~%
392 = 4 * ((4 * 4!) + sqrt(4))
393 = (sqrt(4) + 4! + sqrt(4%)) / sqrt(.4~%)
394 = ((4 * 4) - 4!%) / 4%
395 = ((4 * 4) - sqrt(4%)) / 4%
396 = 4 * 44 / .4~
397 = (sqrt(4) - (4! - .4~)%) / .4~%
398 = sqrt((4! - 4) ^ 4) - sqrt(4)
399 = ((4 * 4) - 4%) / 4%
400 = 4 * (4 + (4 * 4!))


401 = ((4 * 4) + 4%) / 4%
402 = sqrt(4) + sqrt((4! - 4) ^ 4)
403 = (4 + (4 + 4)! - 4!)%
404 = 4 + sqrt((4! - 4) ^ 4)
405 = (4 + sqrt(4))! / (4 * .4~)
406 = (sqrt(4) / .4~%) - 44
407 = 4 + (4 + 4)!% - sqrt(4%)
408 = (4 * 4 * 4!) + 4!
409 = ((4 * .4~) + 4%) / .4~%
410 = ((4 * 4) + .4) / 4%
411 = ((sqrt(4) - sqrt(.4~%)) / .4~%) - 4!
412 = (4 + 4 + 4!%) / sqrt(4)%
413
414 = (sqrt(4) - (4 * 4)%) / .4~%
415 = ((4 / 4%) - .4) / 4!%
416 = 4 * 4 * (sqrt(4) + 4!)
417 = ((sqrt(4) - 4%) / .4~%) - 4!
418 = (sqrt(4) / .4~%) - root(.4, 4)
419 = (4 + 4! - sqrt(.4~%)) / sqrt(.4~%)
420 = 444 - 4!
421 = (4 + 4! + sqrt(.4~%)) / sqrt(.4~%)
422 = (sqrt(4) / .4~%) - (4 + 4!)
423 = (4 - 4!%) / (sqrt(4) * .4~)%
424 = sqrt((4! - 4) ^ 4) + 4!
425 = ((4 / 4%) + sqrt(4)) / 4!%
426 = 4! * (4! - .4 ^ -sqrt(4))
427 = (4 + 4)!% + 4! - sqrt(4%)
428 = sqrt(4) + (sqrt(4) / .4~%) - 4!
429 = (4! + (sqrt(.4~) / .4%)) / .4~
430 = ((4 / .4~) - .4) / sqrt(4)%
431 = ((sqrt(4) - sqrt(.4~%)) / .4~%) - 4
432 = ((4 * 4) + sqrt(4)) * 4!
433 = ((sqrt(4) - sqrt(.4~%)) / .4~%) - sqrt(4)
434 = (sqrt(4) / .4~%) - (4 * 4)
435 = ((sqrt(4) + 4%) / .4~%) - 4!
436 = 4 * (4% + .4~) / .4~%
437 = ((sqrt(4) - 4%) / .4~%) - 4
438 = ((4 / .4~%) - 4!) / sqrt(4)
439 = ((sqrt(4) - 4%) / .4~%) - sqrt(4)
440 = 444 - 4
441 = sqrt((4! - sqrt(4 / .4~)) ^ 4)
442 = 444 - sqrt(4)
443 = sqrt(4) + ((sqrt(4) - 4%) / .4~%)
444 = sqrt(444 ^ sqrt(4))
445 = 4 + ((sqrt(4) - 4%) / .4~%)
446 = sqrt(4) + 444
447 = ((sqrt(4) + .4~%) / .4~%) - 4
448 = 4 + 444
449 = (sqrt(4) / .4~%) - (4 / 4)
450 = (4 + sqrt(4))! / (4 * .4)
451 = (4 / 4) + (sqrt(4) / .4~%)
452 = (4 + (4 / .4~%)) / sqrt(4)
453 = 4 + ((sqrt(4) - .4~%) / .4~%)
454 = 4 + (4 / (sqrt(4) * .4~)%)
455 = ((sqrt(4) + 4%) / .4~%) - 4
456 = ((4! - 4) * 4!) - 4!
457 = ((sqrt(4) + 4%) / .4~%) - sqrt(4)
458 = 4 + 4 + (sqrt(4) / .4~%)
459 = (4 + (4 / sqrt(4)%)) / .4~
460 = sqrt((4! - sqrt(4)) ^ 4) - 4!
461 = sqrt(4) + ((sqrt(4) + 4%) / .4~%)
462 = (4! - sqrt(4))! / (4! - 4)!
463 = 4 + ((sqrt(4) + 4%) / .4~%)
464 = 4 * ((sqrt(4) / .4)! - 4)
465 = (4 - (.4 / .4~)) / sqrt(.4~)%
466 = (4 * 4) + (sqrt(4) / .4~%)
467 = sqrt(4) + ((sqrt(4) + sqrt(.4~%)) / .4~%)
468 = 4! + 444
469 = 4 + ((sqrt(4) + sqrt(.4~%)) / .4~%)
470 = ((4 / .4~) + .4) / sqrt(4)%
471 = ((sqrt(4) - sqrt(4)%) / .4%) - 4!
472 = (4 - 4!) * (.4 - 4!)
473 = ((sqrt(4) - .4~%) / .4~%) + 4!
474 = (4 / (sqrt(4) * .4~)%) + 4!
475 = (sqrt(4) - (.4 / 4)) / .4%
476 = ((4! - 4) * 4!) - 4
477 = (4 + 4!%) / (sqrt(4) * .4~)%
478 = ((4! - 4) * 4!) - sqrt(4)
479 = ((4 * 4!) - sqrt(4%)) / sqrt(4%)
480 = 4! * (44 - 4!)
481 = ((4 * 4!) + sqrt(4%)) / sqrt(4%)
482 = sqrt(4) - ((4 - 4!) * 4!)
483 = ((sqrt(4) + 4%) / .4~%) + 4!
484 = 4 - ((4 - 4!) * 4!)
485 = (sqrt(4) - (4 + sqrt(4))%) / .4%
486 = (4 / .4~) * 4! / .4~
487
488 = (4! - 4) * (4! + .4)
489 = (sqrt(4) - 4.4%) / .4%
490 = ((4 / sqrt(4)%) - 4) / .4
491 = (sqrt(4) / .4%) - (4 / .4~)
492 = (sqrt(4) / .4%) - (4 + 4)
493 = ((sqrt(4) - sqrt(4)%) / .4%) - sqrt(4)
494 = (sqrt(4) / .4~%) + 44
495 = (sqrt(4) - 4!) / (.4 - .4~)
496 = 4 * (4 + (sqrt(4) / .4)!)
497 = ((sqrt(4) + .4%) / .4%) - 4
498 = ((4 / .4%) - 4) / sqrt(4)
499 = (sqrt(4) / .4%) - (4 / 4)
500 = (44 - 4!) / 4%

 

[davee123]

283
413
487

Once you start allowing these functions, the possibilities explode. Using only a single four, I was able to get: 4/10, 4/9, 1/2, 2/3, 2, 4, 5, 15, 20, 24, 25, 50, 120, 150, 200, 225, and 250. I stopped there, but I'm sure I could find more "reasonable" numbers with successive %s, sqrts, and !s. To fill in some holes using your notation:

283 = 4^4 + 4% + sqrt(4)
413 = sqrt(4%)%^sqrt(4) + sqrt(.4~%) - sqrt(4)
487 = 4*sqrt(4%)! + sqrt(4) + sqrt(4%)

DaveE

 

[Jimmy Snyder]

283 = 4^4 + 4% + sqrt(4)

Is n% being taken to mean 100 divided by n? It doesn't mean that. It means n divided by 100.

 

[davee123]

Is n% being taken to mean 100 divided by n? It doesn't mean that. It means n divided by 100.

Ooops! I think I was attempting to decrypt what was intended by the % sign, and got 25 rather than 1/25, so I went with 100/x rather than x/100. Hmm... that does eliminate some possibilities, although I think we can still get the missing values with other odd functions. I was able to get 23, 26, 66, 63, 14, 75, 21, and 23 from a single 4 using the floor function along with inverse trig functions (like [asin(.4)] = 23), and I know I can get 487 using those, although I haven't tried 287 and 413 yet. Hmmm...

DaveE

 

[davee123]

Ok, so a bit further playing around this morning showed me I could get quite a lot of numbers with a single 4. I was able to get 0-11 without much problem, and I suspect you may be able to get quite a lot more too. On my first pass, I got:

1/25, 1/5, 4/10, 4/9, 1/2, 2/3, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 23, 24, 26, 41, 43, 48, 63, 64, 66, 67, 75, 78, 90, 120, 200

Pretty easy to fill in the 3 gaps that way, although I haven't seen what the highest sequential integer is you could get to with these. I could easily see it going solidly to 1000. Perhaps the challenge ought to be to get the integers 1-100 using only a single 4!

For the record, the longest one to write out so far was 43:

[sqrt(sqrt([sqrt([sqrt([asin(.4~)])]!)]!))]

[EDIT]
Ok, with only a single 4, I've been able to get 0-90. So ALMOST all the integers 0-100. Figures that there'd be a natural gap once you hit 90 (since arcsin, etc have a natural cutoff there).

So I am I right in thinking that that means we could prove that using four 4's, that we could get every integer between 0-729000? Hmmm...
[/EDIT]

DaveE