# Four ones and four zeros

Which is the smallest natural number that can not be expressed in base 2 with precisely four ones and four zeros using the four basic arithmetic operations, exponentiation, concatenation, brackets, fractions, factorials (including double factorials), fractional points (both with or without leading zeros, e.g. 0.0001111 or .00110011), and square roots?

(This question arose during a discussion with participants in the Soacha (Colombia) 2023 Math Circle.)

499 and the Gamma Function

• I am looking for the smallest positive integer not so expressible. I have a possible candidate, but perhaps others can do better. Commented Mar 14, 2023 at 23:02
• "Concatenation of the original digits" is a much better way of putting it, unless you really mean we can take any intermediate result and concatenate it with other stuff.
– Bass
Commented Mar 15, 2023 at 21:10
• No, not allowed. Commented Mar 17, 2023 at 14:21
• The current least candidate for the actual answer in the accepted answer is disqualified if, for example, binary fractions such as .1 can be represented by "." with (the result of) "0!" concatenated, where this generated "1" does not count against the four given ones. This possible interpretation and example emphasize the importance of the comment of @Bass (which I upvoted earlier). For clarity (for future readers), the question should explicitly say what exactly can be concatenated. It would take a lot of looking/time to see what everyone is actually (not) doing in each answer otherwise. Commented Mar 23, 2023 at 14:46
• MountainEucalyptus: Ashamed to say, a certain number in the 80's! Commented Mar 23, 2023 at 23:36

Edit: I upgraded the solver to include the pattern .1^-x. It can now solve more numbers.

I wrote a computer program to automatically search for solutions. My program has found solutions for most integers from 0 to 900, listed below. Here are the ten smallest integers it couldn't find solutions for (when I originally posted this):

$$462 = ((100!!)!!+0!)/(0!-1/11!)$$
$$462 = (11!)!-.1^{-100!!}-0!-0!$$
$$499 = ((11!)!!+0!+0!)^{(0!+.1)}/\sqrt{.1} - 0!$$ (loopy walt)
$$501 = ((11!)!!+0!+0!)^{(0!+.1)}/\sqrt{.1} + 0!$$ (loopy walt)
$$507 = .1^{-100!!-0!}-101$$
$$549 = ((0!+0!+0!)!+(11!)!*.1)*(0!+.1)$$ (Weather Vane)
$$607 = ?$$
$$652 = ?$$
$$653 = ?$$
$$787 = ?$$
$$795 = (100!!-.1)*((11!+0!)!!+0!)$$ (Weather Vane)

As a result, the smallest unsolved integer is currently

$$607$$

Here's it's full output for all integers from 0 to 900. I stopped at 900 due to the character limit. Solutions are ordered by fewest bits used, tiebroken by fewest ones used. s represents square root.

0: 0
1: 0!
2: 0!+0!
3: 11
4: 100
5: 101
6: 11!
7: 11!+0!
8: 100!!
9: 100!!+0!
10: 1010
11: s(101!+0!)
12: (11!)/.1
13: 1101
14: 101!!-(0!)
15: 101!!
16: .01^-(0!+0!)
17: .01^-(0!+0!)+0!
18: (11!)*11
19: s(((11!)!)*.1+0!)
20: 10100
21: 10101
22: 100!-(0!)-(0!)
23: 100!-(0!)
24: 100!
25: 100!+0!
26: 100!+0!+0!
27: 11^11
28: 11100
29: (101!!)/.1-(0!)
30: (101!!)/.1
31: .1^-101-(0!)
32: .1^-101
33: .1^-101+0!
34: 100010
35: (11!)^(0!+0!)-(0!)
36: (11!)^(0!+0!)
37: (11!)^(0!+0!)+0!
38: s(((11!)!+0!+0!)/.1)
39: 100!+101!!
40: (11!)!!-(100!!)
41: (11!)!!-(11!+0!)
42: (11!)!!-(11!)
43: (11!)!!-101
44: (11!)!!-100
45: (11!)!!-11
46: (11!)!!-(0!+0!)
47: (11!)!!-(0!)
48: (11!)!!
49: (11!)!!+0!
50: (11!)!!+0!+0!
51: (11!)!!+11
52: (11!)!!+100
53: ((11!+0!)!!+0!)*.1
54: (11!)!!+11!
55: (11!)!!+11!+0!
56: (11!)!!+100!!
57: (11!+0!)!!-((11!)!!)
58: (101!)*.1-(0!+0!)
59: (101!)*.1-(0!)
60: (101!)*.1
61: (101!)*.1+0!
62: .1^-(11!)-(0!+0!)
63: .1^-(11!)-(0!)
64: .1^-(11!)
65: .1^-(11!)+0!
66: .1^-(11!)+0!+0!
67: .1^-(11!)+0!+0!+0!
68: .1^-(11!)+100
69: s((11!+0!)!+0!)-(0!+0!)
70: s((11!+0!)!+0!)-(0!)
71: s((11!+0!)!+0!)
72: ((11!)!!)*1.1
73: ((11!)!!)*1.1+0!
74: ((11!)^(0!+0!)+0!)/.1
75: (100!+0!)*11
76: (s(((11!)!)*.1+0!))/.01
77: s((11!+0!)!+0!)+11!
78: ((11!+0!)!!-(0!))*.11
79: (101!)/1.1-(0!)
80: (101!)/1.1
81: 11^100
82: 11^100+0!
83: 11^100+0!+0!
84: ((11!)!!-((0!+0!+0!)!))/.1
85: ((11!)!!-((0!+0!+0!)!))/.1+0!
86: ((11!)!!+0!-((0!+0!+0!)!))/.1
87: .1^-(11!)+100!-(0!)
88: .1^-(11!)+100!
89: ((11!)!)/(100!!)-(0!)
90: ((11!)!)/(100!!)
91: ((11!)!)/(100!!)+0!
92: ((11!)!!-(0!+0!))/.1
93: ((11!)!!-(0!))/.1-(0!)
94: ((11!)!!-(0!))/.1
95: ((11!)!!)/.1-(0!)
96: ((11!)!!)/.1
97: ((11!)!!)/.1+0!
98: ((11!)!!+0!)/.1
99: ((11!)!!+0!)/.1+0!
100: ((11!)!!+0!+0!)/.1
101: (11!+0!)!!-100
102: (11!+0!)!!-11
103: (11!+0!)!!-(0!+0!)
104: (11!+0!)!!-(0!)
105: (11!+0!)!!
106: (11!+0!)!!+0!
107: (11!+0!)!!+0!+0!
108: (11!+0!)!!+11
109: (11!+0!)!!+100
110: (11!+0!)!!+101
111: (11!+0!)!!+11!
112: 101!-(100!!)
113: (11!+0!)!!+100!!
114: 101!-(11!)
115: 101!-101
116: 101!-100
117: 101!-11
118: 101!-(0!)-(0!)
119: 101!-(0!)
120: 101!
121: 101!+0!
122: 101!+0!+0!
123: 101!+11
124: 101!+100
125: 101^11
126: 101!+11!
127: .1^-(11!+0!)-(0!)
128: .1^-(11!+0!)
129: .1^-(11!+0!)+0!
130: (.1^-(11!)+0!)/.1
131: (.1^-(11!)+0!)/.1+0!
132: (.1^-(11!)+0!+0!)/.1
133: (.1^-(11!)+0!+0!)/.1+0!
134: .1^-(11!+0!)+(0!+0!+0!)!
135: 101!+101!!
136: 101!+.01^-(0!+0!)
137: (100!-(0!))*(11!)-(0!)
138: (100!-(0!))*(11!)
139: ((11!+0!)!!)/.11-(0!)
140: ((11!+0!)!!)/.11
141: ((11!)!!-(0!))*11
142: (s((11!+0!)!+0!))/.1
143: ((11!)!!)*11-(0!)
144: ((11!)!!)*11
145: ((11!)!!)*11+0!
146: ((11!)!!)*11+0!+0!
147: ((11!)!!+0!)*11
148: ((11!)!!+0!)*11+0!
149: (100!+0!)*(11!)-(0!)
150: (100!+0!)*(11!)
151: (100!+0!)*(11!)+0!
152: (11!+0!)!!+(11!)!!-(0!)
153: (11!+0!)!!+(11!)!!
154: (11!+0!)!!+(11!)!!+0!
155: (11!+0!)!!+(11!)!!+0!+0!
156: ((11!+0!)!!-(0!))*1.1
157: ((11!+0!)!!-(0!))*1.1+0!
158: (101!)/.11-(0!+0!)
159: (101!)/.11-(0!)
160: (101!)/.11
161: (101!)/.11+0!
162: (11^100)/.1
163: (11^100)/.1+0!
164: (11^100+0!)/.1
165: (101!!)*(s(101!+0!))
166: 101!-(0!)-(0!)+(11!)!!
167: 101!-(0!)+(11!)!!
168: 101!+(11!)!!
169: 1101^(0!+0!)
170: 1101^(0!+0!)+0!
171: 1101^10+0!+0!
172: ((11!)!)*.01-(100!!)
173: ((11!)!)*.01-(100!!-(0!))
174: ((11!)!)*.01-((0!+0!+0!)!)
175: (100!+0!)*(11!+0!)
176: .1^-(11!+0!)+((0!+0!+0!)!)!!
177: ((11!)!)*.01-(0!+0!+0!)
178: ((11!)!)*.01-(0!+0!)
179: ((11!)!)*.01-(0!)
180: ((11!)!)*.01
181: ((11!)!)*.01+0!
182: ((11!)!)*.01+0!+0!
183: ((11!)!)*.01+0!+0!+0!
184: ((11!)!!-(0!+0!))/.01
185: ((11!)!!-(0!+0!))/.01+0!
186: ((100!!)!!)*.1-(11!)
187: ((11!)!!-(0!))/.01-(0!)
188: ((11!)!!-(0!))/.01
189: ((11!)!!-(0!))/.01+0!
190: (((11!)!!)/.1-(0!))/.1
191: ((100!!)!!)*.1-(0!)
192: ((100!!)!!)*.1
193: ((100!!)!!)*.1+0!
194: (((11!)!!)/.1+0!)/.1
195: ((11!)!!+0!)/.01-(0!)
196: ((11!)!!+0!)/.01
197: ((11!)!!+0!)/.01+0!
198: ((100!!)!!)*.1+11!
199: s((100!!)!-((11!)!)+0!)
200: ((11!)!!+0!+0!)/.01
201: ((11!)!!+0!+0!)/.01+0!
202: ((11!+0!)!!-100)/.1
203: ((11!)!)*.01+100!-(0!)
204: ((100!!)!!+100!)*.1
205: ((11!+0!)!!-(0!+0!))/.1-(0!)
206: ((11!+0!)!!-(0!+0!))/.1
207: ((11!+0!)!!-(0!))/.1-(0!)
208: ((11!+0!)!!-(0!))/.1
209: ((11!+0!)!!)/.1-(0!)
210: ((11!+0!)!!)/.1
211: ((11!+0!)!!)/.1+0!
212: ((11!+0!)!!+0!)/.1
213: ((11!+0!)!!+0!)/.1+0!
214: ((11!+0!)!!+0!+0!)/.1
215: (11!)^11-(0!)
216: (11!)^11
217: (11!)^11+0!
218: (11!)^11+0!+0!
219: (11!)^11+0!+0!+0!
220: ((100!!)!!+0!)/1.11
221: (101!!)^10-100
222: ((11!+0!)!!+(0!+0!+0!)!)/.1
223: (101!!)^10-(0!+0!)
224: (101!!)^(0!+0!)-(0!)
225: (101!!)^(0!+0!)
226: (101!!)^(0!+0!)+0!
227: (101!!)^10+0!+0!
228: (101!-((0!+0!+0!)!))/.1
229: (101!!)^10+100
230: ((11!)!!-(0!+0!))*101
231: (101!!)^(0!+0!)+11!
232: .1^-(100!!)-(100!)
233: (101!!)^10+100!!
234: (101!-(0!)-(0!+0!))/.1
235: ((11!)!!-(0!))*101
236: (101!-(0!)-(0!))/.1
237: (101!-(0!))/.1-(0!)
238: (101!-(0!))/.1
239: (101!)/.1-(0!)
240: (101!)/.1
241: (101!)/.1+0!
242: (101!+0!)/.1
243: 11^101
244: (101!+0!+0!)/.1
245: ((11!)!!+0!)*101
246: (101!+0!+0!+0!)/.1
247: (101!)/.1+100!!-(0!)
248: .1^-(100!!)-(100!!)
249: .1^-(100!!)-(11!+0!)
250: .1^-(100!!)-(11!)
251: .1^-(100!!)-101
252: (.1^-(11!)-(0!))/.01
253: .1^-(100!!)-11
254: .1^-(100!!)-(0!+0!)
255: .1^-(100!!)-(0!)
256: .1^-(100!!)
257: .1^-(100!!)+0!
258: .1^-(100!!)+0!+0!
259: .1^-(100!!)+11
260: (.1^-(11!)+0!)/.01
261: .1^-(100!!)+101
262: .1^-(100!!)+11!
263: (100!!)!!-(101!+0!)
264: (100!!)!!-(101!)
265: (100!!)!!-(101!-(0!))
266: .1^-(100!!)+1010
267: .1^-(100!!)+s(101!+0!)
268: ((11!)!)*.011-(0!+0!)
269: ((11!)!)*.011-(0!)
270: ((11!)!)*.011
271: ((11!)!)*.011+0!
272: ((11!)!)*.011+0!+0!
273: (101!!)^(0!+0!)+(11!)!!
274: (101!!)^(0!+0!)+(11!)!!+0!
275: ((11!)!!-(0!+0!))*(11!)-(0!)
276: ((11!)!!-(0!+0!))*(11!)
277: ((11!)!!-(0!+0!))*(11!)+0!
278: (100!!)!!-(0!)-((11!+0!)!!)
279: (100!!)!!-((11!+0!)!!)
280: ((11!+0!)!!)/.011
281: ((11!)!!-(0!))*(11!)-(0!)
282: ((11!)!!-(0!))*(11!)
283: ((11!)!!-(0!))*(11!)+0!
284: (s((11!+0!)!+0!))/.01
285: (((11!)!!)/.1-(0!))*(0!+0!+0!)
286: ((11!)!!)*(11!)-(0!+0!)
287: ((11!)!!)*(11!)-(0!)
288: ((11!)!!)*(11!)
289: ((11!)!!)*(11!)+0!
290: ((11!)!!)*(11!)+0!+0!
291: (((11!)!!)/.1+0!)*(0!+0!+0!)
292: ((11!)!!+0!)*(11!)-(0!+0!)
293: ((11!)!!+0!)*(11!)-(0!)
294: ((11!)!!+0!)*(11!)
295: ((11!)!!+0!)*(11!)+0!
296: ((11!)!!+0!)*(11!)+0!+0!
297: ((100!!)!!)*.1+(11!+0!)!!
298: (101!)*10.1-(0!+0!)
299: (101!)*10.1-(0!)
300: (101!)*10.1
301: (101!)*10.1+0!
302: (101!)*10.1+0!+0!
303: .1^-(100!!)-(0!)+(11!)!!
304: .1^-(100!!)+(11!)!!
305: .1^-(100!!)+(11!)!!+0!
306: ((11!+0!)!!+((0!+0!+0!)!)!!)/.1
307: ((100!!)!!+1)/1.01-(0!)
308: ((100!!)!!+0!)/1.01
309: ((11!+0!)!!-(0!+0!))*11
310: ((11!)!)/(0!+0!)-((11!)!!+0!+0!)
311: ((11!)!)*.1-(0!)-(((0!+0!+0!)!)!!)
312: ((11!)!)*.1-(((0!+0!+0!)!)!!)
313: ((11!)!)*.1+0!-(((0!+0!+0!)!)!!)
314: ((11!+0!)!!)*11-(0!)
315: ((11!+0!)!!)*11
316: ((11!+0!)!!)*11+0!
317: ((11!+0!)!!)*11+0!+0!
318: ((11!+0!)!!+0!)*11
319: (101!)/.011-(0!)
320: (101!)/.011
321: (101!)/.011+0!
322: ((11!)!!-(0!+0!))*(11!+0!)
323: ((11!)*11)^(0!+0!)-(0!)
324: ((11!)*11)^(0!+0!)
325: ((11!)*11)^(0!+0!)+0!
326: ((11!)*11)^(0!+0!)+0!+0!
327: ((11!)!!-(0!))*(11!+0!)-(0!+0!)
328: ((11!)!!-(0!))*(11!+0!)-(0!)
329: ((11!)!!-(0!))*(11!+0!)
330: ((11!)!!-(0!))*(11!+0!)+0!
331: ((11!)!!-(0!))*(11!+0!)+0!+0!
332: (100!!)!!-((11!)!!)-100
333: ((11!)!!)*(11!+0!)-(0!+0!+0!)
334: (100!!)!!-((11!)!!)-(0!+0!)
335: (100!!)!!-((11!)!!)-(0!)
336: (100!!)!!-((11!)!!)
337: (100!!)!!-((11!)!!)+0!
338: (100!!)!!-((11!)!!)+0!+0!
339: ((11!)!!)*(11!+0!)+0!+0!+0!
340: (.1^-(100!!)-(0!))/.11
341: (11!+0!)^11-(0!+0!)
342: (11!+0!)^11-(0!)
343: (11!+0!)^11
344: (11!+0!)^11+0!
345: (100!-(0!))*(101!!)
346: (11!+0!)^11+0!+0!+0!
347: ((11!)!-(100!))*.1-(0!)
348: ((11!)!-(100!))*.1
349: ((11!)!-(100!))*.1+0!
350: ((11!)!!+0!+0!)*(11!+0!)
351: ((11!)!)*.1-(100!!+0!)
352: ((11!)!)*.1-(100!!)
353: ((11!)!)*.1-(0!)-((0!+0!+0!)!)
354: ((11!)!)*.1-((0!+0!+0!)!)
355: ((11!)!)*.1+0!-((0!+0!+0!)!)
356: ((11!)!)*.1-100
357: ((11!)!)*.1-(0!+0!+0!)
358: ((11!)!)*.1-(0!+0!)
359: ((11!)!)*.1-(0!)
360: ((11!)!)*.1
361: ((11!)!)*.1+0!
362: ((11!)!)*.1+0!+0!
363: ((11!)!)*.1+0!+0!+0!
364: ((11!)!)*.1+100
365: ((11!)!)*.1-(0!)+(0!+0!+0!)!
366: ((11!)!)*.1+(0!+0!+0!)!
367: ((11!)!)*.1+0!+(0!+0!+0!)!
368: ((11!)!)*.1+100!!
369: (100!!)!!-(101!!)
370: (100!!)!!-(101!!-(0!))
371: (100!!)!!-1101
372: (100!!)!!-((11!)/.1)
373: (100!!)!!-(s(101!+0!))
374: (100!!)!!-1010
375: (100!+0!)*(101!!)
376: (100!!)!!-(100!!)
377: (100!!)!!-(11!+0!)
378: (100!!)!!-(11!)
379: (100!!)!!-101
380: (100!!)!!-100
381: (100!!)!!-11
382: (100!!)!!-(0!+0!)
383: (100!!)!!-(0!)
384: (100!!)!!
385: (100!!)!!+0!
386: (100!!)!!+0!+0!
387: (100!!)!!+11
388: (100!!)!!+100
389: (100!!)!!+101
390: (100!!)!!+11!
391: (100!!)!!+11!+0!
392: (100!!)!!+100!!
393: (100!!)!!+1001
394: (100!!)!!+1010
395: (100!!)!!+s(101!+0!)
396: (100!!)!!+(11!)/.1
397: (100!!)!!+1101
398: (100!!)!!+101!!-(0!)
399: (100!!)!!+101!!
400: 10100^10
401: 10100^10+1
402: (100!!)!!+1001/.1
403: (100!!)!!+s(((11!)!)*.1+0!)
404: ((11!)!)*.1001-(0!)
405: ((11!)!)*.1001
406: ((11!)!)*.1001+0!
407: (100!!)!!+100!-1
408: (100!!)!!+100!
409: (100!!)!!+100!+1
410: (100!!)!!+11010
411: (100!!)!!+100!+11
412: ((11!+0!)!!-(0!+0!))/.01
413: (100!!)!!-(0!)+(101!!)/.1
414: (100!!)!!+(101!!)/.1
415: ((11!+0!)!!-(0!))/.01-(0!)
416: ((11!+0!)!!-(0!))/.01
417: ((11!+0!)!!-(0!))/.01+0!
418: (((11!+0!)!!)/.1-(0!))/.1
419: ((11!+0!)!!)/.01-(0!)
420: ((11!+0!)!!)/.01
421: ((11!+0!)!!)/.01+0!
422: (((11!+0!)!!)/.1+0!)/.1
423: ((11!)!!-(0!))*(100!!+0!)
424: ((11!+0!)!!+0!)/.01
425: ((11!+0!)!!+0!)/.01+0!
426: (s((11!+0!)!+0!))*(11!)
427: (s((11!+0!)!+0!))*(11!)+0!
428: ((11!+0!)!!+0!+0!)/.01
429: (100!!)!!-(10+0!)+(11!)!!
430: (100!!)!!-(0!+0!)+(11!)!!
431: (100!!)!!-(0!)+(11!)!!
432: (100!!)!!+(11!)!!
433: (100!!)!!+(11!)!!+0!
434: (100!!)!!+(11!)!!+0!+0!
435: (100!!)!!+(11!)!!+10+0!
436: ((11!)^11+0!+0!)*(0!+0!)
437: (100!-(0!))*(s(((11!)!)*.1+0!))
438: (100!!)!!+(11!)!!+(10+0!)!
439: ((100!!)!!+0!)/.111-(0!)
440: ((100!!)!!+0!)/.111
441: 10101^(0!+0!)
442: 10101^10+0!
443: (100!!)!!-(0!)+(101!)*.1
444: (100!!)!!+(101!)*.1
445: (100!!)!!+(101!)*.1+0!
446: ((101!!)^10-(0!+0!))/.1
447: (100!!)!!-(0!)+.1^-(11!)
448: (100!!)!!+.1^-(11!)
449: ((11!)!)*.101-(0!)
450: ((11!)!)*.101
451: ((11!)!)*.101+0!
452: ((101!!)^(0!+0!)+0!)/.1
453: ((11!)!)*.101+0!+0!+0!
454: ((101!!)^10+0!+0!)/.1
455: (100!!)!!+s((11!+0!)!+0!)
456: (100!)*(s(((11!)!)*.1+0!))
457: (((0!+0!+0!)!)!)*.1+((11!)!!)/.1+0!
458: (((0!+0!+0!)!)!)*.1+((11!)!!+0!)/.1
459: ((11!+0!)!!+(11!)!!)*(0!+0!+0!)
460: ((11!)!!-(0!+0!))*1010
461: (((11!)!!)^(0!+0!)+0!)/101
462: (11!)!-(.1^-(100!!))-(0!+0!)
463: (11!)!-(.1^-(100!!))-(0!)
464: (11!)!-(.1^-(100!!))
465: ((11!)!)*.1+((0!+0!+0!)!+0!)!!
466: (11!)!-(.1^-(100!!))+0!+0!
467: ((100!!+0!)!!+0!)*.1-(11!)
468: (101!-10-(0!))/.01
469: ((11!)!!-(0!))*1010-(0!)
470: ((11!)!!-(0!))*1010
471: (1001!!-(0!))*.1-(0!)
472: ((100!!+0!)!!-(0!))*.1
473: ((100!!+0!)!!+0!)*.1
474: (1001!!+0!)*.1+0!
475: (101!-(0!))/.01-(0!)
476: (101!-(0!))/.01
477: (101!-(0!))/.01+0!
478: ((101!)/.1-(0!))/.1
479: ((11!)!)/1.1-(0!)
480: ((11!)!)/1.1
481: ((11!)!)/1.1+0!
482: ((101!)/.1+0!)/.1
483: (101!+0!)/.01-(0!)
484: (101!+0!)/.01
485: (101!+0!)/.01+0!
486: ((0!+0!+0!)^101)/.1
487: (100!!)!!-10+(11!+0!)!!
488: (101!+0!+0!)/.01
489: (100!!)!!+(11!+0!)!!
490: (100!!)!!+0!+(11!+0!)!!
491: (100!!)!!+111!!+0!+0!
492: ((11!)!)*.11-(((0!+0!+0!)!)!!)
493: ((11!)!-(0!))/.1-((100!!+0!)!!)
494: (11!)!-(0!)-((101!!)^(0!+0!))
495: (((0!+0!+0!)!)!)*.1011
496: (101!+100)/.01
497: (s((11!+0!)!+0!))*(11!+0!)
498: ((11!)!)*.101+((0!+0!+0!)!)!!
500: (1010^11)/(0!+0!)
502: (100!!)!!+101!-10
503: (100!!)!!+101!-(0!)
504: (100!!)!!+101!
505: (100!!)!!+101!+0!
506: .1^-(100!!+0!)-(11!)
507: .1^-(100!!+0!)-101
508: (.1^-(100!!)-(0!+0!))/.1
509: (.1^-(100!!)-(0!))/.1-(0!)
510: (.1^-(100!!)-(0!))/.1
511: .1^-(100!!+0!)-(0!)
512: .1^-(100!!+0!)
513: .1^-(100!!+0!)+0!
514: (.1^-(100!!)+0!)/.1
515: (.1^-(100!!)+0!)/.1+0!
516: (.1^-(100!!)+0!+0!)/.1
517: ((11!)!!-(0!))*(s(101!+0!))
518: .1^-(100!!+0!)+11!
519: .1^-(100!!+0!)+11!+0!
520: (.1^-(11!)+0!)/.001
521: (.1^-(11!)+0!)/.001+0!
522: ((11!)!-(((0!+0!)^(0!+0!))!))*.11
523: (100!-1)^(0!+0!)-(11!)
524: ((11!+0!)!!)*101-(0!)
525: ((11!+0!)!!)*101
526: ((11!+0!)!!)*101+0!
527: (11!)!-(0!)-(((100!!)!!)*.1)
528: ((11!)!!)*(s(101!+0!))
529: (100!-1)^(0!+0!)
530: (100!-1)^10+0!
531: (100!-1)^(.1^-1)+0!+0!
532: (100!-1)^(0!+0!)+11
533: ((11!)!)*.11-(0!)-((0!+0!+0!)!)
534: ((11!)!)*.11-((0!+0!+0!)!)
535: (100!-1)^(0!+0!)+11!
536: .001^-11+100!
537: ((11!)!)*.11-(0!+0!+0!)
538: ((11!)!)*.11-(0!+0!)
539: ((11!)!)*.11-(0!)
540: ((11!)!)*.11
541: ((11!)!)*.11+0!
542: ((11!)!)*.11+0!+0!
543: ((11!)!)*.11+0!+0!+0!
544: ((100!!)!!+100!)/.11
545: ((11!)!)*.11-(0!)+(0!+0!+0!)!
546: ((11!)!)*.11+(0!+0!+0!)!
547: ((11!)!)*.11+0!+(0!+0!+0!)!
548: ((11!)!)*.11+((0!+0!)^(0!+0!))!!
550: ((11!)!!+0!+0!)*(s(101!+0!))
551: ((11!)!+(100!!)!!)*.1-(0!)
552: ((11!)!+(100!!)!!)*.1
553: ((11!)!+(100!!)!!)*.1+0!
554: ((11!)!+(100!!)!!)*.1+0!+0!
555: ((11!)^(0!+0!)+0!)*(101!!)
556: ((100!!)!!-(0!)-((11!+0!)!!))/.1
557: ((100!!)!!-((11!+0!)!!))/.1-(0!)
558: ((100!!)!!-((11!+0!)!!))/.1
559: .1^-(100!!+0!)+(11!)!!-(0!)
560: ((11!+0!)!)/(100!!+0!)
561: 1001!!-((100!!)!!)
562: 1001!!+1-((100!!)!!)
563: (100!!)!!-(0!)+((11!)!)*.01
564: (((11!)!!-(0!))/.1)*((0!+0!+0!)!)
565: (100!!)!!+0!+((11!)!)*.01
566: (((11!)!!-(0!))*(11!)+0!)*(0!+0!)
567: (.1^-(11!)-(0!))*(100!!+0!)
568: (s((11!+0!)!+0!))/.001
569: (100!)^10-(11!+0!)
570: (100!)^(0!+0!)-(11!)
571: (100!)^10-101
572: (100!)^(.1^-1)-100
573: ((100!!)!!-(0!+0!))*1.1
574: (100!)^(.1^-1)-(0!+0!)
575: (100!)^10-(0!)
576: (100!)^(0!+0!)
577: (100!)^10+0!
578: (100!)^(.1^-1)+0!+0!
579: ((100!!)!!+0!+0!)*1.1
580: (100!)^(.1^-1)+100
581: (100!)^10+101
582: (100!)^(0!+0!)+11!
583: (100!)^10+11!+0!
584: (100!)^(.1^-1)+100!!
585: (((0!+0!+0!)!)!)*.1101
586: (100!!+0!)!!-(((11!)!)*.1)+0!
587: ((11!)!!+0!)*1100-(0!)
588: (((11!)!!+0!)/.1)*((0!+0!+0!)!)
589: ((11!)!!+0!)*1100+0!
590: (1001!!-(0!))*.101
591: (100!)^10+101!!
592: ((0!+0!+0!)!)!-(.1^-(11!+0!))
593: (11!)!+0!-((0!+0!)^(11!+0!))
594: (100!!)!!+((11!+0!)!!)/.1
595: (101!-(0!))*101
596: (101!-(0!))*101+0!
597: (11!)!-(101!+0!)-(0!+0!)
598: (11!)!-(101!+0!)-(0!)
599: (11!)!-(101!+0!)
600: (11!)!-(101!)
601: (11!)!-(101!-(0!))
602: (11!)!-(101!)+0!+0!
603: (11!)!-(101!)+0!+0!+0!
604: (101!+0!)*101-(0!)
605: (101!+0!)*101
606: (101!+0!)*101+0!
608: (.1^-(100!!)+(11!)!!)*(0!+0!)
609: (100!!)!!+(101!!)^10
610: (101!+0!+0!)*101
611: ((11!)!!-(0!))*(1100+0!)
612: ((100!!)!!+100!)*1.1
613: (11!)!-(0!+0!)-((11!+0!)!!)
614: (11!)!-(0!)-((11!+0!)!!)
615: (11!)!-((11!+0!)!!)
616: (11!)!+0!-((11!+0!)!!)
617: (11!)!+0!+0!-((11!+0!)!!)
618: ((11!+0!)!!-(0!+0!))*(11!)
619: (100!+1)^(0!+0!)-(11!)
620: (11!)!-(1010^(0!+0!))
621: (11!)!-((11!+0!)!!)+(0!+0!+0!)!
622: ((0!+0!+0!)!)!-(((11!)!!+0!)/.1)
623: ((0!+0!+0!)!)!-(((11!)!!)/.1+0!)
624: ((0!+0!+0!)!)!-(((11!)!!)/.1)
625: (100!+1)^(0!+0!)
626: 101^100+0!
627: 101^100+10
628: ((11!+0!)!!)*(11!)-(0!+0!)
629: ((11!+0!)!!)*(11!)-(0!)
630: ((11!+0!)!!)*(11!)
631: ((11!+0!)!!)*(11!)+0!
632: ((11!+0!)!!)*(11!)+0!+0!
633: ((111!!)/.1+0!)*(0!+0!+0!)
634: ((11!+0!)!!+0!)*(11!)-(0!+0!)
635: ((11!+0!)!!+0!)*(11!)-(0!)
636: ((11!+0!)!!+0!)*(11!)
637: ((11!+0!)!!+0!)*(11!)+0!
638: (1010!!)/(11!)-(0!+0!)
639: (1010!!)/(11!)-(0!)
640: (1010!!)/(11!)
641: (1010!!)/(11!)+0!
642: ((11!+0!)!!+0!+0!)*(11!)
643: ((11!+0!)!!+0!+0!)*(11!)+0!
644: ((11!)!!-(0!+0!))*(101!!-(0!))
645: ((11!)^11-(0!))*(0!+0!+0!)
646: ((11!)^100)*.1-(0!+0!)
647: ((11!)^100)*.1-(0!)
648: ((11!)^100)*.1
649: (11!)!-(s((11!+0!)!+0!))
650: (11!)!+0!-(s((11!+0!)!+0!))
651: ((11!)^11+0!)*(0!+0!+0!)
654: (11!)!-(0!+0!)-((0!+0!)^(11!))
655: (11!)!-(0!)-(.1^-((0!+0!+0!)!))
656: (11!)!-(.1^-((0!+0!+0!)!))
657: ((0!+0!+0!)!)!-(.1^-(11!)-(0!))
658: ((11!)!!-(0!))*(101!!-(0!))
659: (11!)!-(0!)-((101!)/(0!+0!))
660: ((0!+0!+0!)!)!-((101!)*.1)
661: (11!)!+0!-((101!)/(0!+0!))
662: ((100!!)!!)/.1-((11!+0!)!!)-(0!)
663: ((100!!)!!)/.1-((11!+0!)!!)
664: ((100!!)!!)/.1-((11!+0!)!!)+0!
665: (((11!)!!)/.1-(0!))*(100!!-(0!))
666: (11!)!-((11!)!!)-((0!+0!+0!)!)
667: (11!)!-((11!)!!)+0!-((0!+0!+0!)!)
668: (11!)!-((11!)!!)-((0!+0!)^(0!+0!))
669: (11!)!-((11!)!!)-(0!+0!+0!)
670: (11!)!-((11!)!!)-(0!+0!)
671: (11!)!-((11!)!!)-(0!)
672: (11!)!-((11!)!!)
673: (11!)!-((11!)!!)+0!
674: (11!)!-((11!)!!)+0!+0!
675: ((101!!)^(0!+0!))*11
676: 11010^(0!+0!)
677: 11010^10+0!
678: (11!)!-((11!)!!)+(0!+0!+0!)!
679: (((11!)!!)/.1+0!)*(100!!-(0!))
680: (11!)!-101000
681: (0!+0!+0!)^(11!)-((11!)!!)
682: (11!)!-(0!+0!)-((11!)^(0!+0!))
683: (11!)!-(0!)-((11!)^(0!+0!))
684: (11!)!-((11!)^(0!+0!))
685: (11!)!+0!-((11!)^(0!+0!))
686: ((11!+0!)^(0!+0!+0!))/.1
687: (11!)!-100001
688: ((0!+0!+0!)!)!-(.1^-101)
689: 1001!!-(.1^-(100!!))
690: ((0!+0!+0!)!)!-((101!!)/.1)
691: (11!)!+0!-((101!!)*(0!+0!))
692: (11!)!-(100!+100)
693: (11!)!-(11^(0!+0!+0!))
694: (11!)!-(100!+0!)-(0!)
695: (11!)!-(100!+0!)
696: (11!)!-(100!)
697: (11!)!-(100!-(0!))
698: (11!)!-(100!)+0!+0!
699: (11!)!-(10100+0!)
700: (11!)!-10100
701: ((0!+0!+0!)!)!-(s(((11!)!)*.1+0!))
702: (11!)!-10010
703: (11!)!-(0!)-(.01^-(0!+0!))
704: (11!)!-(.01^-(0!+0!))
705: (11!)!-(101!!)
706: (11!)!-(101!!-(0!))
707: (11!)!-(0!)-(((0!+0!+0!)!)/.1)
708: (11!)!-(((0!+0!+0!)!)/.1)
709: (11!)!-(s(101!+0!))
710: (11!)!-1010
711: (11!)!-(100!!+0!)
712: (11!)!-(100!!)
713: (11!)!-(11!+0!)
714: (11!)!-(11!)
715: (11!)!-101
716: (11!)!-100
717: (11!)!-11
718: (11!)!-(0!+0!)
719: (11!)!-(0!)
720: (11!)!
721: (11!)!+0!
722: (11!)!+0!+0!
723: (11!)!+11
724: (11!)!+100
725: (11!)!+101
726: (11!)!+11!
727: (11!)!+11!+0!
728: (11!)!+100!!
729: 11^(11!)
730: 11^(11!)+0!
731: 11^(11!)+0!+0!
732: (11!)!+((0!+0!+0!)!)/.1
733: ((0!+0!+0!)!)!+1101
734: (11!)!+101!!-(0!)
735: (11!)!+101!!
736: (11!)!+.01^-(0!+0!)
737: (11!)!+0!+.01^-(0!+0!)
738: (11!)!+10010
739: ((0!+0!+0!)!)!+s(((11!)!)*.1+0!)
740: (11!)!+10100
741: (11!)!+10100+0!
742: (11!)!-(0!+0!)+100!
743: (11!)!+100!-(0!)
744: (11!)!+100!
745: (11!)!+100!+0!
746: (11!)!+100!+0!+0!
747: (11!)!+11^(0!+0!+0!)
748: (11!)!+100!+100
749: (11!)!-(0!)+(101!!)*(0!+0!)
750: ((0!+0!+0!)!)!+(101!!)/.1
751: (11!)!-(0!)+(100!!)/.01
752: ((11!)!!-(0!))/.0001
753: ((100!!)!!)/.1-(101!!)
754: ((100!!)!!-(11!+0!))/.1
755: (11!)!-(0!)+(11!)^(0!+0!)
756: ((100!!)!!-(11!))/.1
757: (11!)!+0!+(11!)^(0!+0!)
758: ((100!!)!!-101)/.1
759: ((100!!)!!)/.1-1001
760: ((100!!)!!-100)/.1
761: ((100!!)!!)/.1-(11!+0!)
762: ((100!!)!!-11)/.1
763: ((100!!)!!-10)/.1-(0!)
764: ((100!!)!!-(0!+0!))/.1
765: ((100!!)!!-(0!))/.1-(0!)
766: ((100!!)!!-(0!))/.1
767: ((100!!)!!)/.1-(0!)
768: ((100!!)!!)/.1
769: ((100!!)!!)/.1+0!
770: ((100!!)!!+0!)/.1
771: ((100!!)!!+0!)/.1+0!
772: ((100!!)!!+0!+0!)/.1
773: ((100!!)!!+10)/.1+0!
774: ((100!!)!!+11)/.1
775: ((100!!)!!)/.1+11!+0!
776: ((100!!)!!+100)/.1
777: (0!+0!+0!)^(11!)+(11!)!!
778: ((100!!)!!+101)/.1
779: ((100!!)!!+11!)/.1-(0!)
780: ((100!!)!!+11!)/.1
781: ((100!!)!!+11!)/.1+0!
782: ((100!!)!!+11!+0!)/.1
783: (11!)!-(0!)+.1^-((0!+0!+0!)!)
784: (11!)!+.1^-((0!+0!+0!)!)
785: ((0!+0!+0!)!)!+.1^-(11!)+0!
786: ((100!!)!!+1001)/.1
788: ((100!!)!!+1010)/.1
789: (11!)!-(0!+0!)+s((11!+0!)!+0!)
790: (11!)!-(0!)+s((11!+0!)!+0!)
791: (11!)!+s((11!+0!)!+0!)
792: ((100!!)!!)/.1+100!
793: ((100!!)!!)/.1+100!+1
794: ((100!!)!!+1)/.1+100!
796: ((100!!)!!+101!!-(0!))/.1
797: ((100!!)!!+101!!)/.1-(0!)
798: ((100!!)!!+101!!)/.1
799: ((11!)!!-(0!))*10001
800: (10100^10)/.1
801: (11!)!+(10+0!)^100
804: ((100!!)!!)/.1+(11!)^(0!+0!)
808: ((11!)!)*1.001-(0!+0!)
809: ((11!)!)*1.001-(0!)
810: ((11!)!)*1.001
811: ((11!)!)*1.001+0!
812: (11!)!+(100!-(0!))/.01
813: (11!)!-(0!)+((11!)!!-(0!))*(0!+0!)
814: ((0!+0!+0!)!)!+((11!)!!-(0!))/.1
815: ((0!+0!+0!)!)!+((11!)!!)/.1-(0!)
816: ((0!+0!+0!)!)!+((11!)!!)/.1
817: ((0!+0!+0!)!)!+((11!)!!)/.1+0!
818: ((0!+0!+0!)!)!+((11!)!!+0!)/.1
819: (11!)!+0!+((11!)!!+0!)*(0!+0!)
820: (11!)!+1010^(0!+0!)
822: (11!)!-(0!+0!)-(0!)+(11!+0!)!!
823: (11!)!-(0!+0!)+(11!+0!)!!
824: (11!)!-(0!)+(11!+0!)!!
825: (11!)!+(11!+0!)!!
826: (11!)!+0!+(11!+0!)!!
827: (11!)!+0!+0!+(11!+0!)!!
828: ((11!)^10)*(100!-(0!))
830: (((11!+0!)!!-(0!))/.01-(0!))/.1
831: (111!!-(0!))*(100!!)-(0!)
832: ((11!+0!)!!-(0!))/.001
833: (101!-(0!))*(11!+0!)
834: (101!-(0!))*(11!+0!)+0!
835: (((11!+0!)!!)/.1-(0!))/.01-(0!)
836: (((11!+0!)!!)/.1-(0!))/.01
837: (11!)!-(0!+0!)+101!-(0!)
838: ((100!!)!)/((11!)!!)-(0!+0!)
839: ((100!!)!)/((11!)!!)-(0!)
840: ((100!!)!)/((11!)!!)
841: ((100!!)!)/((11!)!!)+0!
842: ((100!!)!)/((11!)!!)+0!+0!
843: (11!)!+101!+0!+0!+0!
844: (((11!+0!)!!)/.1+0!)/.01
845: (11!)!+101^(0!+0!+0!)
846: ((11!)!!-(0!))*10010
847: (101!+0!)*(11!+0!)
848: ((11!+0!)!!+0!)/.001
849: (100!!+0!)!!-(((11!)!!)/.1)
850: (100!!+0!)!!-(((11!)!!)/.1)+0!
851: (100!!+0!)!!-(((11!)!!-(0!))/.1)
852: ((s(111!+0!))/.1)*((0!+0!+0!)!)
854: (101!+0!+0!)*(11!+0!)
855: (((11!)!!)/.1-(0!))*(100!!+0!)
856: (111!!+0!+0!)/.001
858: (((11!)!!)*11-(0!))*((0!+0!+0!)!)
860: ((11!)^(10+0!)-(0!))/.01
861: (((11!)!!)*(11!)-(0!))*(0!+0!+0!)
862: ((100!!)!!-(0!)+(11!)!!)/.1
863: ((100!!)!!+(11!)!!)/.1-(0!)
864: ((100!!)!!+(11!)!!)/.1
865: ((100!!)!!+(11!)!!)/.1+0!
866: ((100!!)!!+(11!)!!+0!)/.1
867: (((11!)!!)*(11!)+0!)*(0!+0!+0!)
868: ((11!)^(10+0!)+0!)/.01
870: (((11!)!!)*11+0!)*((0!+0!+0!)!)
871: ((100!!)!!-(0!))/.1+(11!+0!)!!
872: ((100!!)!!)/.1-(0!)+(11!+0!)!!
873: ((100!!)!!)/.1+(11!+0!)!!
874: (100!!+0!)!!-(s(111!+0!))
875: ((100!!)!!+0!)/.1+(11!+0!)!!
876: ((11!+0!)!)*.01-((100!!)!!)
880: ((100!!)!!+0!)/.0111
881: (100!!+0!)!!-(.1^-(11!))
882: ((11!)!!+0!)*10010
885: (100!!+0!)!!-((101!)*.1)
886: ((100!!)!!-(0!))/.1+101!
887: ((100!!)!!)/.1+101!-(0!)
888: ((100!!)!!)/.1+101!
889: ((100!!)!!)/.1+101!+0!
890: ((100!!)!!+0!)/.1+101!
893: ((11!)!!-(0!))*(s(((11!)!)/(0!+0!)+0!))
894: ((11!)!)*1.01-((0!+0!+0!)!)
895: 1001!!-(0!)-((11!)!!)-(0!)
896: (100!!+0!)!!-((11!)!!+0!)
897: (100!!+0!)!!-((11!)!!)
898: (100!!+0!)!!-((11!)!!-(0!))
899: ((11!)!)*1.01-(0!)
900: ((11!)!)*1.01


The solver can be found here: https://github.com/isaacg1/eight-bits/

Caveats:

• The solver does not consider intermediate steps with value above 65535.

• The solver does not consider intermediate steps with negative value, except in special cases such as .1^-x.

• The solver does not consider intermediate steps with fractional value. Literals with fractional value are used, but intermediate steps with fractional value are not.

I believe that my solver finds all possible solutions that do not go through one of the above three cases.

• Excellent work! Though I think the tiebreaker should also include something for "number of characters used." $0!$ instead of just $1$ seems kinda silly. :) Also... man, I think I have to learn Rust at some point. Commented Mar 20, 2023 at 0:00
• Awesome. :-). Because it cant find all cases do you want to put 500 - 1000 up as-well so we can all work together on the gaps ???
– Mike
Commented Mar 20, 2023 at 7:02
• @BernardoRecamánSantos so it's just a typo in the post. The 462 = ((100!!)!!+0!)*(0!-1/11!) should be ((100!!)!!+0!)/(0!-1/11!) - edited. Commented Mar 20, 2023 at 12:15
• @WeatherVane Thanks for the edit Commented Mar 20, 2023 at 17:02
• I extended my generator to use fractional intermediate values and have two new results $549 = ((0! + 0! + 0!)! + (11!)! \times .1) \times (.1 + 0!)$ and $795 = ((\frac{0! + 0!}{.1})!! - .1) \times ((11! + 0!)!! + 0!)$ Commented Mar 21, 2023 at 20:17

Continuing from @Dante . . .

$$26 = 100! + 10 + 0 + 1 - 1$$
$$27 = 11^{11} + 0 + 0 + 0 + 0$$
$$28 = 100! + 100 + 1 - 1$$
$$29 = 100! + 10 + 11 + 0$$
$$30 = 10^{101} - 10 + 0$$
$$31 = 10^{101} - 1 + 0 + 0$$
$$32 = 10^{(110 - 1)} + 0 + 0$$
$$33 = 10^{101} + 1 + 0 + 0$$
$$34 = 10^{101} + 10 + 0$$
$$35 = 100! + 1011 + 0$$
$$36 = 10^{101} + 100$$
$$37 = 100! + 1101 + 0$$
$$38 = 110^{10} + 10 + 0$$
$$39 = 100111 + 0 + 0$$
$$40 = 101000 + 1 - 1$$
$$41 = 101001^{1} + 0$$
$$42 = 101010^{1} + 0$$
$$43 = 101011 + 0 + 0$$
$$44 = 101100^{1} + 0$$
$$45 = 101101 + 0$$
$$46 = 101110 + 0$$
$$47 = 110000 - 1^{1}$$
$$48 = 110000 + 1 - 1$$
$$49 = 110000 + 1^{1}$$
$$50 = 110000 + 1 + 1$$
$$51 = 110000 + 11$$
$$52 = (100! + 10) \times 10^{1}$$
$$53 = ((100! + 10) \times 10) + 1$$
$$54 = 11^{11} \times (0^{0} + 0^{0})$$
$$55 = 111000 - 1 + 0$$
$$56 = 111000^{1} + 0$$
$$57 = 111000 + 1 + 0$$
$$58 = 111000 + 10$$
$$59 = \frac{101!}{10} - 1 + 0 + 0$$
$$60 = 111100 + 0 + 0$$
$$61 = \frac{101!}{10} + 1 + 0 + 0$$
$$62 = 100^{11} - 10 + 0$$
$$63 = 100^{11} - 1 + 0 + 0$$
$$64 = 100^{11} + 0^{1} + 0$$
$$65 = 100^{11} + 1 + 0$$
$$66 = 100^{11} + 10$$
$$67 = ?$$

Any more?

• You've overlooked the simple $67=1000011/1$ or $67=1000011^1$. Commented Mar 15, 2023 at 12:04
• Indeed I don't know how I missed that (and several other variants) e.g. $1000011 \times 1$. I'll let someone else pick up the baton. Commented Mar 15, 2023 at 12:07
• Only 3 $0$s were used in 66. Commented Mar 19, 2023 at 10:46
• @PlaceReporter99 somehow several others only have 3 zeros but as noted elsewhere you can trivially add 0 or multiply by 1 to use up the spare digits. Commented Mar 19, 2023 at 12:20

Continuing from Dante and Weather Vane:

(EDIT: Extended it further, but left some gaps. Either other users will fill them, or I will tomorrow, or one of them is the answer. Outstanding numbers are: 158, 159, 164, 173, 187, 188, 190, 199)

Note that 'precisely 4 1s and 4 0s' is equivalent to 'up to 4 1s and 4 0s', since we can trivially add 0 or multiply by 1 use up additional digits. These trivial digits have been omitted for readability.

$$67 = 1000011$$
$$68 = 1000011 + 1$$
$$69 = 1000101$$
$$70 = 1000110$$
$$71 = 1000111$$
$$72 = 1001001 - 1$$
$$73 = 1001001$$
$$74 = 1001010$$
$$75 = 1001011$$
$$76 = 1001100$$
$$77 = 1001101$$
$$78 = 1001110$$
$$79 = 11^{100} - 10$$
$$80 = 1010001 - 1$$
$$81 = 1010001$$
$$82 = 1010010$$
$$83 = 1010011$$
$$84 = 1010100$$
$$85 = 1010101$$
$$86 = 1010110$$
$$87 = 1010110 + 0!$$
$$88 = 1011000$$
$$89 = 1011001$$
$$90 = 1011010$$
$$91 = 1011010 + 0!$$
$$92 = 1011100$$
$$93 = 1011100 + 0!$$
$$94 = (100 + 0!)! - 11010$$
$$95 = (100 + 0!)! - 11001$$
$$96 = 101! - 11000$$
$$97 = 1100001$$
$$98 = 1100010$$
$$99 = 1100011$$
$$100 = 1100100$$
$$101 = 1100101$$
$$102 = 1100110$$
$$103 = 1100110 + 0!$$
$$104 = 1101000$$
$$105 = 1101001$$
$$106 = 1101010$$
$$107 = 1101010 + 0!$$
$$108 = 1101100$$
$$109 = 1101100 + 0!$$
$$110 = 101! - 1010$$
$$111 = 1110000 - 1$$
$$112 = 1110000$$
$$113 = 1110001$$
$$114 = 1110010$$
$$115 = 1110010 + 0!$$
$$116 = 1110100$$
$$117 = 1110100 + 0!$$
$$118 = 101! - 10$$
$$119 = 101! - 1$$
$$120 = 101!$$
$$121 = 101! + 1$$
$$122 = 101! + 10$$
$$123 = 101! + 11$$
$$124 = 101! + 100$$
$$125 = 101! + 101$$
$$126 = 101! + 110$$
$$127 = 101! + 110 + 0!$$
$$128 = 10^{111}$$
$$129 = 10^{111} + 0!$$
$$130 = 10^{111} + 0! + 0!$$
$$131 = 10^{111} + 0! + 0! + 0!$$
$$132 = 101! + 1100$$
$$133 = 101! + 1100 + 0!$$
$$134 = (100 + 0!)! + 1110$$
$$135 = 10000111\ \ \text{(Jaap Scherphuis)}$$
$$136 = 101! + 100^{10}$$
$$137 = 101! + 10001$$
$$138 = 101! + 10010$$
$$139 = 10001011$$
$$140 = 101! + 10100$$
$$141 = 10001101$$
$$142 = 10001110$$
$$143 = \frac{(11!)!}{101} - 0!$$
$$144 = \frac{(11!)!}{101}$$
$$145 = \frac{(11!)!}{101} + 0!$$
$$146 = \frac{(11!)!}{101} + 0! + 0!$$
$$147 = \frac{(11!)!}{101} + 0! + 0! + 0!$$
$$148 = (11!^{10} + 0!) \times 100\ \ \text{(loopy walt)}$$
$$149 = 10010101 (\text{ Bass})$$
$$150 = 10010110 (\text{ Bass})$$
$$151 = (100! + 0!) \times 110 + 1$$
$$152 = 101! + (0! + 0!)^{101}$$
$$153 = 10011001$$
$$154 = (100! - 0! - 0!) \times 111$$
$$155 = 101! + (11!)^{0! + 0!} - 0!$$
$$156 = \frac{(11!)!}{100} - 100!$$
$$157 = 101! + (11!)^{0! + 0!} + 0!$$
$$158 = \frac{(11!)!}{100.1}-(0!+0!)\ \ \text{(MountainEucalyptus)}$$
$$159 = \frac{(11!)!}{100.1}-(0!)\ \ \text{(MountainEucalyptus)}$$
$$160 = 101 \times (0! + 0!)^{101}$$
$$161 = (100! - 0!) \times 111$$
$$162 = (100! - 0!) \times 111 + 0!$$
$$163 = 10100011$$
$$164 = 10 \times (11^{100} + 0!)\ \ \text{(loopy walt)}$$
$$165 = 101! \times (0! + 0.011)$$
$$166 = 100! \times 111 - 0! - 0!$$
$$167 = 100! \times 111 - 0!$$
$$168 = 100! \times 111$$
$$169 = 100! \times 111 + 0!$$
$$170 = 100! \times 111 + 0! + 0!$$
$$171 = 1101^{10} + 0! + 0!$$
$$172 = 10101100$$
$$173 = (11!)! \times .01 - 1 - (0!+0!+0!)!\ \ \text{(Retudin)}$$
$$174 = (100! + 0!) \times 111 - 0!$$
$$175 = (100! + 0!) \times 111$$
$$176 = (100! + 0!) \times 111 + 0!$$
$$177 = \frac{(11!)!}{100} - 10 - 0!$$
$$178 = \frac{(11!)!}{100} - 10$$
$$179 = \frac{(11!)!}{100} - 0!$$
$$180 = \frac{(11!)!}{100}$$
$$181 = \frac{(11!)!}{100} + 0!$$
$$182 = \frac{(11!)!}{100} + 10$$
$$183 = \frac{(11!)!}{100} + 10 + 0!$$
$$184 = \frac{(11!)!}{100} + 100$$
$$185 = 101! + (0! + 0!)^{11!} + 0!$$
$$186 = \frac{(11!)!}{100} + (10 + 0!)!$$
$$187 = (((1 + 1 + 1)!)!! - 0!) \times 100 - 0!\ \ \text{(Weather Vane)}$$
$$188 = ((0! + 0!)^{11!} - 0!) \times 11 - 0!\ \ \text{(loopy walt)}$$
$$189 = (10^{11!} - 0!) \times (10 + 0!)$$
$$190 = (0! + 0!)^{11!} \times 11 - 0! - 0!\ \ \text{(loopy walt)}$$
$$191 = 10^{11!} \times (10 + 0!) - 0!$$
$$192 = 10^{11!} \times (10 + 0!)$$
$$193 = 10^{11!} \times (10 + 0!) + 0!$$
$$194 = 1110^{10} - 0! - 0!$$
$$195 = (10^{11!} + 0!) \times (10 + 0!)$$
$$196 = 1110^{10}$$
$$197 = 11000101$$
$$198 = 11000110$$
$$199 = (100! + 0!)/(.1^{11}) - 0!\ \ \text{(isaacg)}$$
$$200 = (100! + 0!) \times (111 + 0!)$$
$$201 = 11001001$$
$$202 = 11001010$$

• Like Weather Vane, you stopped just before an easy one: $135=10000111$. I got stuck at 107 cause I didn't think of the 0! trick. Very nice! Commented Mar 15, 2023 at 15:05
• @JaapScherphuis Great spot! Have updated. Commented Mar 15, 2023 at 16:12
• 148 = (11!^10+0!)x100 Commented Mar 15, 2023 at 19:41
• @loopywalt good one. Can you carry on? Commented Mar 15, 2023 at 19:42
• 149=10010101, 150=10010110 :-)
– Bass
Commented Mar 15, 2023 at 21:16

I tentatively claimed that 265 is the answer, since I thought I did a pretty good go at trying to solve it, and didn't manage it. Commenters have managed to fill all the gaps in this answer, from 263 to 307

Continuing from Dante, Weather Vane, BlueHairedMeerkat, and Mike's efforts. As with their answers, please comment or edit in any gaps you solve.

Gaps at: 265-267, 275-279, 281, 283, 293, 295, 297, and 303 onward

Note that 'precisely 4 1s and 4 0s' is equivalent to 'up to 4 1s and 4 0s', since we can trivially add 0 or multiply by 1 use up additional digits. These trivial digits have been omitted for readability.

$$263 = 10^{100!!}+11!+0!$$
$$264 = 10^{100!!}+(11+0!)!!$$
$$265 = 101!/.1+100!+0!$$
$$266 = (101!+0!)/.1+100!$$
$$267 = (11!)!*.011-0!-0!-0!$$
$$268 = (11!)!*.011-0!-0!$$
$$269 = 101! \times 10.01-0!$$
$$270 = 101! \times 10.01$$
$$271 = 101! \times 10.01+0!$$
$$272 = (101!!)^{0!+0!}+(11!)!!-0!$$
$$273 = (101!!)^{0!+0!}+(11!)!!$$
$$274 = (101!!)^{0!+0!}+(11!)!!+0!$$
$$275 = (100!-0!)*11!/.1-0!$$
$$276 = (100!-0!)*11!/.1$$
$$277 = (100!-0!)*11!/.1+0!$$
$$278 = (11!+0!)!!/.011-0!-0!?$$
$$279 = (11!+0!)!!/.011-0!$$
$$280 = (11!)!/(0!+0!+.1)-100!!$$
$$281 = (11!+0!)!!/.011+0!$$
$$282 = (11!)!/(0!+0!+.1)-(10+0!)!$$
$$283 = ((11!)!!-0!)*11!+0!$$
$$284 = (11!)!/(0!+0!+.1)-100$$
$$285 = (11!)!/(10.1)-0!-0!-0!$$
$$286 = (11!)!/(10.1)-0!-0!$$
$$287 = (11!)!/(10.1)-0!$$
$$288 = (11!)!/(10.1)$$
$$289 = (11!)!/(10.1)+0!$$
$$290 = (11!)!/(10.1)+0!+0!$$
$$291 = (11!)!/(10.1)+0!+0!+0!$$
$$292 = (11!)!/(0!+0!+.1)+100$$
$$293 = ((11!)!!+0!)*11!-0!$$
$$294 = (11!)!/(0!+0!+.1)+(10+0!)!$$
$$295 = ((11!)!!+0!)*11!+0!$$
$$296 = (11!)!/(0!+0!+.1)+100!!$$
$$297 = ((11!)!!+0!)*11!+0!+0!+0!$$
$$298 = 101! \times 10.1-0!-0!$$
$$299 = 101! \times 10.1-0!$$
$$300 = 101! \times 10.1$$
$$301 = 101! \times 10.1+0!$$
$$302 = 101! \times 10.1+0!+0!$$
$$303 = ((0!+0!+0!)!!+10.1)*11!$$
$$304 = (0!+1)^{100!!}+(11!)!!$$
$$305 = (0!+1)^{100!!}+(11!)!!+0!$$
$$306 = ((11!)!!+0!+0!+0!)*11!$$
$$307 = ((11!)!!+0!+0!+0!)*11!+0!$$

• I can't comment on Mike's answer, and I feel that it is inappropriate to solve the gaps from that answer in a separate place. Can someone please look through my rejected edit on Mike's answer and either comment the solutions I offered or edit them in? Commented Mar 17, 2023 at 6:35
• Mountain, maybe leave your proposed solutions for Mike's range in the comments on this answer? I can't see your rejected edit Commented Mar 17, 2023 at 7:03
• 265 = 101!/.1 + 100! + 0!. 266 = (101! +0!)/.1 +100! Commented Mar 17, 2023 at 7:07
• 267 = (11!)! x .011 - 0! - 0! - 0! Commented Mar 17, 2023 at 9:11
• 276 = (100!-0!) x 11! / .1; 275,277 = 276 +/- 0! Commented Mar 17, 2023 at 9:40

First 25 numbers can be written as

1:10-10+10-01

2:(11-01-00)^(01)

3:10+01+10-10

4:(10^10)*(01^01)

5:0111-0010

6:0111-0001

7:(10^11)-01-00

8:(11^10)-0001

9:0110+0011

10:0101+0101

11:0101+0110

12:(10*10)*11+00

13:1110-0001

14:1101+0001

15:1010+0101

16:10*10*10*10

17:((10^10)^10)+01

18:1001+1001

19:1000+1011

20:1010+1010

21:1100+1001

22:1010+1100

23:00010111

24:1000*011*1

25:((11+10)^10)+00

• How about 100!+10+0+1-1 Commented Mar 15, 2023 at 8:00
• Among others, I've just attempted to change the incorrect expression for 24 to 1000*011*1, and I'd also like to see the one for 16 changed to be 10*10*10*10, which already seemed to be there while I was creating my edit. However, my pending edit shows the first without the asterisks. I don't understand why. Maybe the second is in someone else's pending edit. I don't know; I'm new to this. Thanks. Commented Mar 15, 2023 at 23:53
• Apparently asterisks cause italics to be used. I'll learn later how to suppress that if possible -- unless someone wants to say so here. I believe one of the original expressions had partial italics already, too. Apologies for the comments here. Commented Mar 16, 2023 at 0:07
• @RickShepherd fixed. (use a backslash to escape asterisks, so they are not treated as italics markup)
– Rubio
Commented Mar 16, 2023 at 2:11
• @Rubio Thanks -- on both counts. Commented Mar 16, 2023 at 17:15

203 = 111!!/.1-(0!+0!+0!)!-0! (isaacg)
204 = 11001100
205 = 111!!/.1-(0!+0!+0!)!+0! (isaacg)
206 = 111!!/.1-0!-0!-0!-0! (isaacg)
207 = 111!!*10-0!-0!-0!
208 = 111!!*10-0!-0!
209 = 11010001
210 = 110^11-(0!+0!+0!)!
211 = 111!!*10+0! (P.-S. Park)
212 = 11010100
213 = 110^11-0!-0!-0!
214 = 110^11-0!-0!+0
215 = 110^11-0!+0+0!
216 = 110^11+0+0+0
217 = 110^11+0!+0+0
218 = 110^11+0!+0!+0
219 = 110^11+0!+0!+0!
220 = 11!^11+0!+0!+0!+0!
221 = 11!^11+(0!+0!+0!)!-0!
222 = 110^11+(0!+0!+0!)!
223 = 1111^(0!+0!)-0!-0!
224 = 1111^(0!+0!)-(0!+0)
225 = 1111^(0!+0!)+0+0
226 = 1111^(0!+0!)+(0!+0)
227 = 1111^(0!+0!)+0!-0!
228 = (101!-(10+0!)!*10
229 = (101!-(10+0!)!/.1+0!
230 = (101!-100-0!)/.1
231 = (101!-100)/.1-0!
232 = (101!-100)*10
233 = 101! / .1-(0!+0!+0!)!-1 (P.-S. Park)
234 = 101!*10-(10+0!)!
235 = 101!/.1-100-0! (P.-S. Park)
236 = 101!*10-100
237 = 101!*10-0!-10
238 = 101!*10-10+0
239 = 101!*10-1+0+0
240 = 101!*10*(0+0+1)
241 = 101!*10+1+0+0
242 = 101!*10+10+0
243 = 101!*10+10+0!?
244 = 101!*10+100
245 = 11^101+0!+0!+0
246 = 101!*10+(10+0!)!
247 = (101!+100)/.1-0! (P.-S. Park)
248 = (101^11-0!)*(0!+0!)
249 = (101^11)*(0!+0!)-0!
250 = 10^1000-11!
251 = (101^11)*(0!+0!)+0!
252 = (101^11+0!)*(0!+0!)
253 = 10^1000-11
254 = 10^1000-1-1
255 = 10^1000-1^1
256 = 10^1000+1-1
257 = 10^1000+1^1
258 = 10^1000+1+1
259 = 10^1000+11
260 = 10^(100!!)+11+0!
261 = 10^(100!!)+101
262 = 10^1000+11!

Someone can continue from here. If you contributed an answer feel free to put your username next to it.

• 211 = 111!! * 10 + 0! + 0 + 0 Commented Mar 16, 2023 at 21:22
• 203 = 111!!/.1 - (0! + 0! + 0!)! - 0!. The same idea works for 205 and 206. Commented Mar 17, 2023 at 7:05
• 233 = 101! / .1 - (0! + 0! + 0!)! - 1 Where is my name? Commented Mar 17, 2023 at 8:05
• 235 = 101!/.1-(0!+0!+0!)!+1 Commented Mar 17, 2023 at 8:12
• 247 = 101!/.1+(0!+0!+0!)!+1 Commented Mar 17, 2023 at 8:13

Some more: 311 to 409 with gaps.
Edit by @Joel Rondeau just leaves 326 unsolved in this range.
Update: Contribution by @isaacg completes the range.

$$311 = (11!)! \times .1 - 0! - ((1 + 0! + 0!)!)!!$$
$$312 = (11!)! \times .1 - ((1 + 0! + 0!)!)!!$$
$$313 = (11!)! \times .1 + 0! - ((1 + 0! + 0!)!)!!$$
$$314 = (\sqrt{((11!)!! + 0!)})!! \times 11 - 0!$$
$$315 = (\sqrt{((11!)!! + 0!)})!! \times 11$$
$$316 = (\sqrt{((11!)!! + 0!)})!! \times 11 + 0!$$
$$317 = (\sqrt{((11!)!! + 0!)})!! \times 11 + 0! + 0!$$
$$318 = ((\sqrt{((11!)!! + 0!)})!! + 0!) \times 11$$
$$319 = ((\sqrt{((11!)!! + 0!)})!! + 0!) \times 11 + 0!$$
$$320 = (0! + 0!) ^ {11!} \times 101$$
$$321 = ((\sqrt{((11!)!! + 0!)})!! + 0! + 0!) \times 11$$
$$322 = ((11!)!! - 0! - 0!) \times \sqrt{((11!)!! + 0!)}$$
$$323 = 10010 ^ {10} - 1$$
$$324 = 10010 ^ {10}$$
$$325 = 10010 ^ {10} + 1$$
$$326 = (11! \times 11)^{(0!+0!)}+0!+0!$$
$$327 = ((11!)!! - 0!) \times (11! + 0!) - 0! - 0!$$
$$328 = ((11!)!! - 0!) \times \sqrt{((11!)!! + 0!)} - 0!$$
$$329 = ((11!)!! - 0!) \times \sqrt{((11!)!! + 0!)}$$
$$330 = ((11!)!! - 0!) \times \sqrt{((11!)!! + 0!)} + 0!$$
$$331 = ((11!)!! - 0!) \times (11! + 0!) + 0! + 0!$$
$$332 = (100!!)!! - (11!)!! - 100$$
$$333 = (100!!)!! - (11!)!! - 10 - 0!$$
$$334 = (100!!)!! - 0! - 0! - (11!)!!$$
$$335 = (11!)! - 0! - ((1 + 0! + 0! + 0!)!!)!!$$
$$336 = (11!)! - ((1 + 0! + 0! + 0!)!!)!!$$
$$337 = (11!)! + 0! - ((1 + 0! + 0! + 0!)!!)!!$$
$$338 = (100!!)!! + 0! + 0! - (11!)!!$$
$$339 = (11!)! - (100!!)!! + 1 + 0! + 0!$$
$$340 = (11! + 0!) ^ {11} - 0! - 0! - 0!$$
$$341 = \sqrt{((11!)!! + 0!)} ^ {11} - 0! - 0!$$
$$342 = \sqrt{((11!)!! + 0!)} ^ {11} - 0!$$
$$343 = \sqrt{((11!)!! + 0!)} ^ {11}$$
$$344 = \sqrt{((11!)!! + 0!)} ^ {11} + 0!$$
$$345 = (100! - 0!) \times (11! - 0!)!!$$
$$346 = (11! + 0!) ^ {11} + 0! + 0! + 0!$$
$$347 = ((11!)! - (1 + 0! + 0! + 0!)!) \times .1 - 0!$$
$$348 = ((11!)! - (1 + 0! + 0! + 0!)!) \times .1$$
$$349 = ((11!)!! + 0! + 0!) \times \sqrt{((11!)!! + 0!)} - 0!$$
$$350 = ((11!)!! + 0! + 0!) \times \sqrt{((11!)!! + 0!)}$$
$$351 = ((11!)!! + 0! + 0!) \times \sqrt{((11!)!! + 0!)} + 0!$$
$$352 = (11!)! \times .1 - 1000$$
$$353 = (11!)! \times .1 - 0! - (1 + 0! + 0!)!$$
$$354 = (11!)! \times .1 - (1 + 0! + 0!)!$$
$$355 = ((1 + 0! + 0!)!)! \times .1 - 101$$
$$356 = (11!)! \times .1 - 100$$
$$357 = ((1 + 0! + 0!)!)! \times .1 - 11$$
$$358 = (11!)! \times .1 - 1 - 0!$$
$$359 = (11!)! \times .1 - 1$$
$$360 = (11!)! \times .1$$
$$361 = (11!)! \times .1 + 1$$
$$362 = (11!)! \times .1 + 1 + 0!$$
$$363 = ((1 + 0! + 0!)!)! \times .1 + 11$$
$$364 = (11!)! \times .1 + 100$$
$$365 = ((1 + 0! + 0!)!)! \times .1 + 101$$
$$366 = (11!)! \times .1 + (1 + 0! + 0!)!$$
$$367 = (11!)! \times .1 + 0! + (1 + 0! + 0!)!$$
$$368 = 100!! + (11!)! \times .1$$
$$369 = (100!!)!! - (11! - 0!)!!$$
$$370 = (100!!)!! + 0! - (11! - 0!)!!$$
$$371 = (100! + (11!)!) \times .1 - 0!$$
$$372 = (100! + ((1 + 0! + 0!)!)!) \times .1$$
$$373 = (100!!)!! - \sqrt{((11! - 0!)! + 0!)}$$
$$374 = (100!!)!! - 1010$$
$$375 = (100! + 0!) \times (11! - 0!)!!$$
$$376 = (100!!)!! - (11 + 0!)!!$$
$$377 = (100!!)!! - 0! - 11!$$
$$378 = (100!!)!! - (1 + 0! + 0!)!$$
$$379 = (100!!)!! - 101$$
$$380 = (100!!)!! - 100$$
$$381 = (100!!)!! - 11$$
$$382 = (100!!)!! - 0! - 0!$$
$$383 = (100!!)!! - 0!$$
$$384 = (100!!)!!$$
$$385 = (100!!)!! + 0!$$
$$386 = (100!!)!! + 0! + 0!$$
$$387 = (100!!)!! + 11$$
$$388 = (100!!)!! + 100$$
$$389 = (100!!)!! + 101$$
$$390 = 11! + ((1 + 0! + 0! + 0!)!!)!!$$
$$391 = (100!!)!! + 0! + 11!$$
$$392 = (100!!)!! + (11 + 0!)!!$$
$$393 = ((11 + 0!)!!)!! + 1001$$
$$394 = (100!!)!! + 1010$$
$$395 = (100!!)!! + \sqrt{((11! - 0!)! + 0!)}$$
$$396 = (100!!)!! + 1100$$
$$397 = (100!!)!! + 1101$$
$$398 = (100!!)!! - 0! + (11! - 0!)!!$$
$$399 = (100!!)!! + (11! - 0!)!!$$
$$400 = (100!!)!! + 0! + (11! - 0!)!!$$
$$401 = (100!!)!! + 101!! + 1 + 0!$$
$$402 = (100!!)!! + \sqrt{((11!)! \times .1 + 0!)} - 0!$$
$$403 = (100!!)!! + \sqrt{((11!)! \times .1 + 0!)}$$
$$404 = (11!)! \times .1001 - 0!$$
$$405 = (11!)! \times .1001$$
$$406 = (11!)! \times .1001 + 0!$$
$$407 = 100! - 0! + ((11 + 0!)!!)!!$$
$$408 = 100! + ((11 + 0!)!!)!!$$
$$409 = 100! + 0! + ((11 + 0!)!!)!!$$

• You know, the $\sqrt{((11!-0!)!+0!}$ to get $11$ (decimal) is cute, but unless you're short on $1$s you could just use $(100!!)!! + 1011$ for $395$. :) A ton of other nice tools in here though, especially for $11$ and $19$. Really would love a $13$ but that's harder... Commented Mar 18, 2023 at 2:43
• @EricSnyder my generator isn't optimised yet. Commented Mar 18, 2023 at 8:50
• 332 = $(100!!)!! - (11!)!! - 100$, 333 = $(100!!)!! - (11!)!! - 10 - 0!$, 346 = $\sqrt{((11!)!! + 0!)} ^ {11} + 0! + 0! + 0!$, 349 = $\sqrt{((11!)!! + 0!)} ^ {11} + (0! + 0! + 0!)!$, 401 = $(100!!)!!+0!+(11!−0!)!! + 1$ Commented Mar 18, 2023 at 8:58
• @WeatherVane Yeah, I've been optimizing some of the numbers to get an extra 0 or 1. Thought I had 326, but math error. Commented Mar 18, 2023 at 19:23
• 326 = (11!*11)^(0!+0!)+0!+0! Commented Mar 19, 2023 at 6:06

A bit of lateral thinking:

All numbers can technically be made using this equation:$$\frac{0}{0}+0+0*1*1*1*1$$ $$\frac{0}{0}$$ will get you any number you want.

• Not my DV, but the lack of definition of $\frac{0}{0}$ is about whether the quotient is $0$ ($0$ divided by anything), $1$ (any number divided by itself) or $\infty$ (any number divided by $0$) – not some arbitrary value. Commented Mar 19, 2023 at 17:00
• There are better ways to do "all numbers" without resorting to $0/0$, using logarithms and square roots. Adapting from the four fours problem gives this, which I think works: $$n=-\sqrt{10}\frac{\log\left[\left(\log\underbrace{\sqrt{\sqrt{\cdots\sqrt{10}}}}_{n}\right)/\log10\right]}{\log10}$$ Commented Mar 19, 2023 at 23:58
• Based on the given rules, I'm not sure if the usage of 'log' is allowed. Commented Mar 20, 2023 at 20:05
• @Mike It's definitely WAY outside the given rules. But it's at least mathematical, as opposed to $0/0$. Commented Mar 20, 2023 at 22:07
• @EricSnyder from a few trials (n = 2, 3, 5, 7 and 11) the first term $-\sqrt{10}$ should be just $-$ (minus). Commented Mar 20, 2023 at 23:28