Representation of numbers with four 4's

Remark

There is a convention I applied to identities below, especially for the numbers beyond 100. As the table grew, some awkward formulas became more common. In order to simplify the entries and shorten the file that contains this page, I introduced several shorthands as listed below. These are used to express numbers with a single 4:

  • [√√4] = 1
  • [√√√√4!!] = [24!1/16] = 30
  • [√30] = 5
  • 5! = 120
  • [√120] = 10
  • [√10] = 3
  • 3! = 6
  • [√(6!)] = 26

where brackets denote the whole part function. ([x] is the largest integer not exceeding x.)

Now, let's agree to use {1},{30},{5},{120},{3} as substitutes for the corresponding formulas. Thus for example, {3} means 3 expressed as [√[√[√√√√√4!!]!]]. Similarly, {5} and {120} stand for 5 and 120, respectively, in terms of the just described expressions.

1 4·4/4/4 = (4 + 4)/(4 + 4) = 44/44(1)
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) = 4!!/(4 + 4)·4
5 (4 + 4·4)/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 = 4!! + (4 - 4)/4 = (4 + 4)·4/4
9 4 + 4 + 4/4 = 4/.4 - 4/4 = 4/.(4) + 4 - 4 = (4!/4/√4)√4(1)
10 (44 - 4)/4 = 4/.4 - 4 + 4 = 44/4.4(1)
11 44/(√4 + √4) = 4/.4 + 4/4
12 (44 + 4)/4 = 4/.4 + 4/√4
13 44/4 + √4 = 4!! + 4 + 4/4
14 4!/4 + 4 + 4 = 4!! + 4 + 4/√4
15 4·4 - 4/4 = 44/4 + 4(1) = ((√√√4)4! - 4)/4(2)
16 4 + 4 + 4 + 4 = 4·4·4/4 = 44/4/4 = 4! - 4 - 4
17 4·4 + 4/4 = ((√√√4)4! + 4)/4(2)
18 4·4 + 4 - √4
19 4! - 4 - 4/4 = 4!!/.4 - 4/4
20 (4! - 4)·4/4 = (4! - 4) + 4 - 4 = (4 + 4/4)·4
21 4! - 4 + 4/4 = 4!!/.4 + 4/4
22 4·4 + 4 + √4 = 4/.(4)·√4 + 4
23 4! - √4 + 4/4 = 4!!/.4 + 4/4
24 4·4 + 4 + 4 = 4/.4·√4 + 4
25 4! + (√4 + √4)/4
26 4! + (4 + 4)/4
27 4! + 4 - 4/4 = (4·4 - 4)/.4(1)
28 4! + 4 + 4 - 4 = (4! + 4)·4/4
29 4! + 4 + 4/4
30 4·4·√4 - √4 = ((√√√4)4! - 4)/√4(2)
31 4!!·4 - 4/4 = ((√√√4)4! - √4)/√4(2)
32 4!!·4 + 4 - 4
33 4!!·4 + 4/4 = ((√√√4)4! + √4)/√4(2)
34 4/.(4)·4 - √4 = 4·4·√4 + √4 = ((√√√4)4! + 4)/√4(2)
35 4!!/.(√4) - 4/4 = 4! + 44/4
36 4·(4 + 4) + 4 = 4·4·√4 + 4
37 4!!/.(√4) + 4/4
38 4!!/.(√4) + 4/√4
39 4!!/.√4 - 4/4
40 4·4/.(4) + 4 = 4!!·(4 + 4/4) = 4!!/.√4 + 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 44 + √4 + √4
49 4!·√4 + 4/4
50 4!·√4 + 4/√4
51 4!·√4 + 4 - {1} = (4! - 4 + .4)/.4(1)
52 (4! + √4)·4/√4
53 4!·√4 + 4 + {1}
54 4!·√4 + 4 + √4
55 44/√4/.4(1)
56 4!·√4 + 4 + 4 = 4! + 4! + 4 + 4(1)
57 44 + [√(4!!·4!)]
58 4!/.4 - 4 + √4(1)
59 4!/.4 - 4/4(1)
60 4·4·4 - 4(1)
61 4!/.4 + 4/4(1)
62 4·4·4 - √4(1) = 4!!√4 - 4/√4
63 4!!√4 - 4/4
64 4! + 44 - 4(1)
65 4!!√4 + 4/4
66 4!/.4 + 4!/4(1) = 4·4·4 + √4(1)
67 4!!√4 + 4 - {1}
68 4·4·4 + 4(1)
69 4!!√4 + 4 + {1}
70 44 + 4! + √4(1)
71 (4! + 4 + .4 )/.4(1)
72 (4!·4!)/(4 + 4)(1)
73 4!!/.4 + [√(4!!·4!)]
74 4!·4 - 4! - √4(1)
75 (4! + 4 + √4)/.4(1)
76 4!·4 - 4! + 4(1) = (4! - 4)·4 - 4 = 4!!/.4·4 - 4
77 (4!!)√4 + [√(4!!·4!)]
78 (4! - 4)·4 - √4(1)
79 (4! - 4)·4 - {1}
80 (44 - 4!)·4(1) = 4!!/.4·4·{1}
81 (4 - 4/4)4(1)
82 (4! - 4)·4 + √4(1)
83 4!·4 - [√(4!!·4!)]
84 44·√4 - 4(1) = (4! - √4)·4 -4(1) = (4! - 4)·4 + 4 = 4!!/.4·4 + 4
85 (4! + 4/.4)/.4(1)
86 (4! - √4)·4 - √4(1)
87 (4! - √4)·4 - {1}
88 44·(4 - √4)(1) = 4·4·4 + 4!(1)
89 (4! - √4)·4 + {1}
90 (4! - √4)·4 + √4(1)
91 4!·4 - √4/.4
92 (4! - 4/4)·4(1)
93 4!·4 - [4/√√√4]
94 4!·4 - 4 + √4(1)
95 4!·4 - 4/4(1)
96 (4 + 4/4)! - 4!(1)
97 4!·4 + 4/4(1)
98 4!·4 + 4 - √4(1)
99 4!·4 + 4 - {1} = 44/.(44)
100 (4! + 4/4)·4(1) = 44/.44(1)
101 4!·4 + √4/.4
102 4!·4 + 4 + √4
103 4!·4 + 4!! - {1}
104 4!·4 + 4 + 4
105 4!·4 + 4!! + {1}
106 4!·4 + 4!! + √4 = 44/.4 - 4
107 44/.(4) + 4!!
108 4!·4 + 4!! + 4 = 44/.4 - √4
109 44/.4 - {1}
110 44/.4·{1}
111 44/.4 + {1}
112 44/.4 + √4
113 44/.4 + {3}
114 44/.4 + 4
115 44/.4 + {5}
116 4!/.4·√4 - 4
117 4!/.4·√4 - {3}
118 4!/.4·√4 - √4
119 4!/.4·√4 - {1}
120 4!/.4·√4·{1}
121 4!/.4·√4 + {1}
122 4!/.4·√4 + √4
123 4!/.4·√4 + {3}
124 4!/.4·√4 + 4 = (4!!√4 - √4)·√4 = 4!!√4·√4 - 4
125 4!!√4·√4 - {3}
126 4!!√4·√4 - √4
127 4!!√4·√4 - {1}
128 4!!√4·√4·{1}
129 4!!√4·√4 + {1}
130 4!!√4·√4 + √4
131 4!!√4·√4 + {3}
132 4!!√4·√4 + 4
133 4!!√4·√4 + {5}
134 {120} + 4/.4 + 4 = {120} + 4·4 - √4
135 {120} + 4·4 - [√√4]
136 {120} + (√4 + √4)·4 = 4!!√4·√4 + 4!!
137 {120} + 4/.(4) + 4!!
138 {120} + 4·4 + √4
139 {120} + [4·4!/√4!]
140 {120} + 4!! + 4!! + 4 = {120} + 4·4 + 4
141 {120} + 4! - 4 + [√√4]
142 {120} + 4! - 4/√4
143 {120} + 4! - 4/4
144 {120} + 4! + 4 - 4 = {120} + 4!! + 4!! + 4!!
145 {120} + 4! + 4/4
146 {120} + 4! + 4/√4
147 {120} + 4! + 4 - [√√4]
148 {120} + 4! + √(4·4)
149 {120} + 4! + 4 + [√√4]
150 {120} + 4! + 4 + √4
151 {120} + 4!!·4 - [√√4]
152 {120} + 4! + 4 + 4
153 {120} + 4!!·4 + [√√4] = {120} + 4! + 4/.(4)
154 {120} + 4! + 4/.4
155 {120} + {30} + 4 + {1}
156 {120} + {30} + 4 + √4
157 {120} + {30} + 4 + {3}
158 {120} + {30} + 4 + 4
159 {120} + {30} + {5} + 4
160 {120} + {30} + {10}·{1}
161 {120} + {30} + {10} + {1}
162 {120} + {30} + {10} + √4
163 {120} + {30} + {10} + {3}
164 {120} + {30} + {10} + 4
165 {120} + {30} + {10} + {5}
166 {120} + {30} + 4!! + 4!!
167 {120} + 4! + 4! - {1}
168 {120} + 4! + 4!·{1}
169 {120} + 4! + 4! + {1}
170 {120} + 4! + 4! + √4
171 {120} + 4! + 4! + {3}
172 {120} + 4! + 4! + 4
173 {120} + 4! + 4! + {5}
174 {120} + 4! + 4! + {3}!
175 {120} + {30} + 4! + {1}
176 {120} + 4! + 4! + 4!!
177 {120} + {30} + 4! + {3}
178 {120} + {30} + 4! + 4
179 {120} + {30} + 4! + {5}
180 {120} + {30} + {30}·{1} = {120} + {120} - {30} - {30} = {30}·{10} - {120}·{1}
181 {120} + {30} + {30} + {1}
182 {120} + {30} + {30} + √4
183 {120} + {30} + {30} + {3}
184 {120} + {30} + {30} + 4
185 {120} + {30} + {30} + {5}
186 {120} + {30} + {30} + {3}!
187 {120} + 4! · {3} - {5}
188 {120} + 4! · {3} - 4
189 {120} + 4! · {3} - {3}
190 {120} + 4! · {3} - √4
191 {120} + 4! · {3} - {1}
192 {120} + 4! + 4! + 4!
193 {120} + {26} · {3} - {5}
194 {120} + {26} · {3} - 4
195 {120} + {26} · {3} - {3}
196 {120} + {26} · {3} - √4
197 {120} + {26} · {3} - {1}
198 {120} + {26} · {3} · {1}
199 {120} + {26} · {3} + {1}
200 {120} + {26} · {3} + √4
201 {120} + {26} · {3} + {3}
202 {120} + {26} · {3} + 4
203 {120} + {26} · {3} + {5}
204 {120} + {26} · {3} + {3}!
205 {120} + {30} · {3} - {5}
206 {120} + {30} · {3} - 4
207 {120} + {30} · {3} - {3}
208 {120} + {30} · {3} - √4
209 {120} + {30} · {3} - {1}
210 {120} + {30} · {3} · {1}
211 {120} + {30} · {3} + {1}
212 {120} + {30} · {3} + √4
213 {120} + {30} · {3} + {3}
214 {120} + {30} · {3} + 4
215 {120} + {30} · {3} + {5}
216 {120} + {30} · {3} + {3}!
217 {120} + 4! · 4 + {1}
218 {120} + 4! · 4 + √4
219 {120} + 4! · 4 + {3}
220 {120} + {30} · {3} + {10} = {120} + 4! · 4 + 4
221 {120} + 4! · 4 + {5}
222 {120} + 4! · 4 + {3}!
223 {120} + {26} ·4 - {1}
224 {120} + {120} - 4*4
225 {120} + {120} - {5} · {3}
226 {120} + {120} - 4! + {10}
227 {120} + {26} ·4 + {3}
228 {120} + {26} ·4 + 4
229 {120} + {26} ·4 + {5}
230 {120} + {26} ·4 + {3}!
231 {120} + {120} - {3} - {3}!
232 {120} + {120} - {5} - {3}
233 {120} + {120} - 4 - {3}
234 {120} + {120} - {3} - {3}
235 {120} + {120} - 4 - {1}
236 {120} + {120} - 4 · {1}
237 {120} + {120} - 4 + {1}
238 {120} + {120} - {1} - {1}
239 {120} + {120} - {1} · {1}
240 {120} · {1} + {120} · {1}
241 {120} + {120} + {1} · {1}
242 {120} + {120} + {1} + {1}
243 {120} + {120} + 4 - {1}
244 {120} + {120} + 4 · {1}
245 {120} + {120} + 4 + {1}
246 {120} + {120} + {3} + {3}
247 {120} + {120} + 4 + {3}
248 {120} + {120} + {5} + {3}
249 {120} + {120} + {3} + {3}!
250 {120} + {120} + {5} + {5}
251 {120} + {120} + {5} + {3}!
252 {120} + {120} + {3} · 4
253 {26} · {5} + {120} + {3}
254 {120} + {120} + 4! - {10}
255 {120} + {120} + {3} · {5}
256 {120} + {120} + 4 · 4
257 {26} · {5} · {2} - {3}
258 {120} + {120} + 4! - {3}!
259 {120} + {120} + 4! - {5}
260 {120} + {120} + {10} + {10}
261 {26} · {10} + {1} · {1}
262 {26} · {10} + {1} + {1}
263 {26} · {10} + {3} · {1}
264 {26} · {10} + 4 · {1}
265 {26} · {10} + {5} · {1}
266 {26} · {10} + {3}! · {1}
267 {26} · {10} + {3} + 4
268 {26} · {10} + 4 + 4
269 {26} · {10} + {5} + 4
270 {26} · {10} + {10} · {1}
271 {26} · {10} + {10} + {1}
272 {26} · {10} + {3}! + {3}!
273 {26} · {10} + {5} + 4!!
274 {26} · {10} + {10} + 4
275 {26} · {10} + {10} + {5}
276 {26} · {10} + {10} + {3}!
278 {26} · {10} + 4! - {3}!
279 {26} · {10} + 4! - {5}
280 {26} · {10} + {10} + {10}
281 {26} · {10} + 4! - {3}
282 {26} · {10} + 4! - √4
283 {26} · {10} + 4! - {1}
284 {26} · {10} + 4! · {1}
285 {26} · {10} + 4! + {1}
286 {26} · {10} + 4! + √4
287 {26} · {10} + 4! + {3}
288 {26} · {10} + 4! + 4
289 {26} · {10} + 4! + {5}
290 {26} · {10} + 4! + {3}!
291 {26} · {10} + {30} + {1}
292 {26} · {10} + 4! + 4!!
293 {26} · {10} + {30} + {3}
294 {26} · {10} + {30} + 4
295 {26} · {10} + {30} + {5}
296 {26} · {10} + {30} + {3}!
297 {30} · {10} - {3} · {1}
298 {26} · {10} + {30} + 4!!
299 {30} · {10} - {1} · {1}
300 {30} · {10} · {1} · {1}
301 {30} · {10} + {1} · {1}
302 {30} · {10} + {1} + {1}
303 {30} · {10} + {3} · {1}
304 {30} · {10} + {3} + {1}
305 {30} · {10} + {5} · {1}
306 {30} · {10} + {5} + {1}
307 {30} · {10} + {10} - {3}
308 {30} · {10} + 4 + 4(1)
309 {30} · {10} + {10} - {1}
310 {30} · {10} + {10} · {1}
311 {30} · {10} + {10} + {1}(1)
312 {30} · {10} + {3}! + {3}!
313 {30} · {10} + {10} + {3}
314 {30} · {10} + {3}! + 4!!
315 {30} · {10} + {3}·{5}
316 {30} · {10} + 4·4
317 4! · {26} / √4 + {5}(1)
318 {30} · {10} + {3}·{3}! = ({3}·{6})√4 - {6}
319 {3}4 · 4 - {5}(1) = ({3}·{6})√4 - {5}
320 {30} · {10} + {10} + {10} = ({3}·{6})√4 - 4
321 {3}4 · 4 - {3}(1) = ({3}·{6})√4 - {3}
322 {3}4 · 4 - √4(1) = ({3}·{6})√4 - √4
323 {3}4 · 4 - {1}(1) = ({3}·{6})√4 - {1}
324 (4 - {1})4 · 4(1) = {3}·{6}·{3}·{6} = {3} · {120} - {30} - {6}
325 {3}4 · 4 + {1}(1) = ({3}·{6})√4 + {1}
326 {3}4 · 4 + √4(1) = ({3}·{6})√4 + √4
327 {3}4 · 4 + {3}(1) = ({3}·{6})√4 + {3}
328 {3}4 · 4 + 4(1) = ({3}·{6})√4 + 4
329 {3}4 · 4 + {5}(1) = ({3}·{6})√4 + {5}
330 {3}4 · 4 + {6}(1) = ({3}·{6})√4 + {6} = {3}·{120} - {30}·{1}
331{3}·{120} - {30} + {1}
332{3}·{120} - {30} + √4
333{3}·{120} - {30} + {3}
334 {3}·{120} - {30} + 4 = ({3}·{6})√4 + {10}
335{3}·{120} - {30} + {5}
336{3}·{120} - {30} + {6}
337{3}·{120} - {26} + {3}
338{3}·{120} - {26} + 4
339{3}·{120} - {26} + {5}
340{3}·{120} - {30} + {10} = {3}·{120} - {26} + {6}
341{3}·{120} - 4! + {5}
342{3}·{120} - 4! + {6}
343({6} + {1}){3}·{1}
344{3}·{120} - {26} + {10}
345({6} + {1}){3} + √4
346({6} + {1}){3} + {3}
347({6} + {1}){3} + 4
348({6} + {1}){3} + {5}
349({6} + {1}){3} + {6}
350{3}·{120} - {10}·{1}
351{3}·{120} - {10} + {1}
352{3}·{120} - {10} + √4
353{3}·{120} - {10} + {3}
354{3}·{120} - {10} + 4
355{3}·{120} - {10} + {5}
356{3}·{120} - {10} + {6}
357{3}·{120} - {3}·{1}
358{3}·{120} - {1} - {1}
359{3}·{120} - {1} · {1}
360{3}·{120} - {1} + {1}
361{3}·{120} + {1} · {1}
362{3}·{120} + {1} + {1}
363{3}·{120} + {3} · {1}
364{3}·{120} + 4 · {1}
365{3}·{120} + {5} · {1}
366{3}·{120} + {6} · {1}
366{3}·{120} + {6} · {1}
367{3}·{120} + {6} + {1} = ({6} + {1}){3} + 4!
368{3}·{120} + {6} + √4
369{3}·{120} + {6} + {3} = ({6} + {1}){3} + {26}
370{3}·{120} + {10}·{1}
371{3}·{120} + {10} + {1}
372{3}·{120} + {10} + √4
373{3}·{120} + {10} + {3}
374{3}·{120} + {10} + 4
375{3}·{120} + {10} + {5}
376{3}·{120} + {10} + {6}
377{3}({120} + 4) + {5}(3)
378{3}·{120} + 4! - {6} = {3}({120} + 4) + {6}
379{3}·{120} + 4! - {5}
380{3}·{120} + {10} + {10}
381{3}{120} + {26}- {5}
382{3}({120} + 4) + {10}
3834!·4·4 - [√√4](4)
384{3}({120} + {30}) - {6}
385{3}({120} + {5}) + {10}  =  4!·4·4 + [√√4](4)
3864!·4·4 + √4(4)

(1) By Andre Gustavo dos Santos, Brasil

(2) By Richard Tschumpel, Vienna, Austria

(3) By Don Gosiewski

(4) By Byron Igoe

On Internet

  1. Four 4s puzzle

Related material
Read more...

  • Funny Arithmetic
  • A single formula to express all numbers
  • Any Integer with Three 2s
  • Representation of numbers with a single 4. The rules and the possibilities
  • One 4, a story
  • Three 3's
  • Representation of numbers with three 4's
  • Representation of numbers with three 5's
  • Representation of numbers with four 3's
  • Representation of numbers with four 5's
  • A problem of representing 4 by four identical digits
  • A problem of representing 6 by three identical digits
  • A 9's Fan's Clock
  • Make an Identity
  • Fun with numbers: place plus/minus signs between the digits
  • 24 with One Digit
  • Representation of 3 with Three Equal Digits

    • |Contact| |Front page| |Contents| |Up| |Algebra|

     

    Copyright © 1996-2018 Alexander Bogomolny

    72083757