Let's start with a triplet of integers, say (1, 2, 5) and a set of mathematical operations (+, -, *, /, ^, sqrt, fact!, concatenation, brackets).

Our task will be to represent all (or almost all - as explained below) integers from 1 to n using some or all of the initial triplet and any quantity of operations defined above.

So:
1=1
6=1+5
9=5*2-1
13=15-2
27=51-4!
60=12*5 etc

Let's define n as the first occurrence of not being able to find a valid representation for n+1 and for n+2. I believe that in our case n=17 (15+2), since neither 18 nor 19 get valid solutions.

You are requested to find a triplet of integers (a,b,c) enabling a maximal n.

(In reply to Record by Ady TZIDON)

105 is not the limit. I still do not know what the limit is.

105: (9!! / 9)
106: (9!! / 9) + !(√4)
107: (4!!)!!! + 9!!!!!!
108: (((√9)!)!!!! × 9)
109: (((√9)!)!!!! × 9) + !(√4)
110: (((√9)!)!!!! × 9) + √4
111: (9!!!!! + !(√4)) × √9
112: (((√9)!)!!!! × 9) + 4
113:  (4!!)!!! + 9!!!!! - √9
114:  (4!!)!!! + 9!!!!! - !(√9)
115:  (4!!)!!! + 9!!!!! - !(!(√9))
116:  (4!!)!!! + 9!!!!!
117: (9!! - 9)/4!!
118: (((√9)!)! / (√9)!) - √4
119: (((√9)!)! / (√9)!) - !(√4)
120: (((√9)!)! / (√9)!)
121: (((√9)!)! / (√9)!) + !(√4)
122: (((√9)!)! / (√9)!) + √4
123: (4!!)!!! + 9!!!! - !(√9)
124: (4!!)!!! + 9!!!! - !(!(√9))
125: (4!!)!!! + 9!!!!
126: (4!!)!!! + 9!!!! + !(!(√9))
127: (4!!)!!! + 9!!!! + !(√9)
128: (4!!)!!! + ((√9)!)!!
129: (4!!)!!! + ((√9)!)!! + !(!(√9))
130: (4!!)!!! + ((√9)!)!! + !(√9)
131: (4!!)!!! + ((√9)!)!! + √9
132: (9!!!!! × 4) - ((√9)!)!!!!
133: 9!!! - (9!!!!!! + √4)
134: (4!!)!!! + ((√9)!)!! + (√9)!
135: (9!!!!! × 4) - 9
136: (((√9)!)!!!! × ((√9)!)!!!!) - 4!!
137: (4!!)!!! + ((√9)!)!! + 9
138: (9!!!!! × 4) - (√9)!
139: 9!!! - (9!!!!!! - 4)
140: (9!!!!! × 4) - 4
141: (9!!!!! × 4) - √9
142: (9!!!!! × 4) - !(√9)
143: (9!!!!! × 4) - !(!(√9))
144: (9!!!!! × 4)
145: (9!!!!! × 4) + !(!(√9))
146: (9!!!!! × 4) + !(√9)
147: (9!!!!! × 4) + √9
148: (((√9)!)!! × √9) + 4
149: 9!!! - (9 + 4)
150: 9!!! - ((√9)!)!!!!
151: 9!!! - (9 + √4)
152: 9!!! - (9 - 4)!!!
153: 9!!! - 9
154: 9!!! - 4!!
155: 9!!! - (√49)
156: 9!!! - (√9)!
157: 9!!! - (9 - 4)
158: 9!!! - 4
159: 9!!! - √9
160: 9!!! - √4
161: 9!!! - !(√4)
162: 9!!!
163: 9!!! + !(√4)
164: 9!!! + √4
165: 9!!! + √9
166: 9!!! + 4
167: 9!!! + 9 - 4
168: 9!!! + (√9)!
169: 9!!! + √(49)
170: 9!!! + 4!!
171: 9!!! + 9
172: 9!!! + (9 - 4)!!!
173: 9!!! + 9 + √4
174: 9!!! + ((√9)!)!!!!
175: 9!!! + (9 + 4)
176: 9!!! + ((√9)!)!!! - 4
177: 9!!! + (9 - 4)!!
178: 9!!! + ((√9)!)!!! - √4
179: 9!!! + 4!! + 9
180: 9!!! + ((√9)!)!!!
181: 9!!! + ((√9)!)!!! + !(√4)
182: 9!!! + ((√9)!)!!! + √4
183: (((√9)!)! / 4) + √9
184: 9!!! + ((√9)!)!!! + 4
185: 9!!! + 9!!!!!! - 4
186: 9!!! + 4!
187: 9!!! + 9!!!!!! - √4
188: 9!!! + 9!!!!!! - !(√4)
189: 9!!! + 9!!!!!!
190: 9!!! + 9!!!!!! + !(√4)
191: 9!!! + 9!!!!!! + √4
192: 9!!! + (√9)! + 4!
193: 9!!! + 9!!!!!! + 4
194: 9!!! + 9!!!!! - 4
195: 9!!! + 4! + 9
196: 9!!! + 9!!!!! - √4
197: 9!!! + 9!!!!! - !(√4)
198: 9!!! + 9!!!!!
199: 9!!! + 9!!!!! + !(√4)
200: 9!!! + 9!!!!! + √4
201: !((√9)!) - (4!!)^(!(√9))
202: 9!!! + 9!!!!! + 4
203: 9!!! + 9!!!! - 4
204: 9!!! + ((√9)!)!!! + 4!
205: 9!!! + 9!!!! - √4
206: 9!!! + 9!!!! - !(√4)
207: 9!!! + 9!!!!
208: 9!!! + 9!!!! + !(√4)
209: 9!!! + 9!!!! + √4
210: 9!!! + ((√9)!)!!
211: 9!!! + 49
212: 9!!! + ((√9)!)!! + √4
213: 9!!! + 9!!!!!! + 4!
214: 9!!! + ((√9)!)!! + 4
215: 9!!! + 9!!!! + 4!!
216: (9!!!! + 9) × 4
217: !((√9)!) - ((√9)!)!!
218: !((√9)!) - 9!!!! - √4
219: !((√9)!) - !(9 - 4)
220: !((√9)!) -  9!!!!
221: !((√9)!) - ((√9)!)!! - √4
222: !((√9)!) - (9!!!! + √4)
223: !9 - 4! - ((√9)!)!!!
224: (((√9)!)!!! - 4)!!!!!!
225: (((√9)!)!!! - 4)!!!!!! + !(!(√9))
226: (((√9)!)!!! - 4)!!!!!! + !(√9)
227: (((√9)!)!!! - 4)!!!!!! + √9
228: !((√9)!) - 9!!!! + 4!!
229: !((√9)!) - 9!!!!!
230: !((√9)!) - 9!!!!! + !(√4)
231: (9 + √4)!!!!
232: !((√9)!) - (9 + √4)!!!!!!!!
233: (((√9)!)!!! - 4)!!!!!! + 9
234: (9 + √4)!!!! + √9
235: (9 + !(√4))!!! - 9!!!!
236: (((√9)!)!!! - 4)!!!!!! + ((√9)!)!!!!
237: !((√9)!) - (9!!!!! - 4!!)
238: !((√9)!) - 9!!!!!!
239: !((√9)!) - (9!!!!!! - !(√4))
240: !((√9)!) - (9!!!!!! - √4)
241: !((√9)!) - 4!
242: !((√9)!) - (4! - !(!(√9)))
243: !((√9)!) - (4! - !(√9))
244: !((√9)!) - (4! - √9)
245: !((√9)!) - (((√9)!)!!! + √4)
246: !((√9)!) - (((√9)!)!!! + !(√4))
247: !((√9)!) - ((√9)!)!!!
248: !((√9)!) - (9 + 4!!)
249: !((√9)!) - (((√9)!)!!!! + 4)
250: !((√9)!) - (4! - 9)
251: !((√9)!) - ((√9)! + 4!!)
252: !((√9)!) - (9 + 4)
253: !((√9)!) - ((√9)!)!!!!
254: !((√9)!) - (9 + √4)
255: !((√9)!) - ((√9)! + 4)
256: !((√9)!) - 9
257: !((√9)!) - 4!!
258: !((√9)!) - (9 - √4)
259: !((√9)!) - (√9)!
260: !((√9)!) - (9 - 4)
261: !((√9)!) - 4
262: !((√9)!) - √9
263: !((√9)!) - √4
264: !((√9)!) - !(√4)
265: !((√9)!)
266: !((√9)!) + !(√4)
267: !((√9)!) + √4
268: !((√9)!) + √9
269: !((√9)!) + 4
270: !((√9)!) + (9 - 4)
271: !((√9)!) + (√9)!
272: !((√9)!) + (9 - √4)
273: !((√9)!) + 4!!
274: !((√9)!) + 9
275: !((√9)!) + ((√9)! + 4)
276: !((√9)!) + (9 + √4)
277: !((√9)!) + ((√9)!)!!!!
278: !((√9)!) + (9 + 4)
279: !((√9)!) + ((√9)! + 4!!)
280: !((√9)!) + (4! - 9)
281: !((√9)!) + (((√9)!)!!!! + 4)
282: !((√9)!) + (9 + 4!!)
283: !((√9)!) + ((√9)!)!!!
284: !((√9)!) + (((√9)!)!!! +√4
285: !((√9)!) + (((√9)!)!!! + √4)
286: !((√9)!) + (4! - √9)
287: !((√9)!) + (4! - !(√9))
288: !((√9)!) + (4! - !(!(√9)))
289: !((√9)!) + 4!
290: !((√9)!) + (9!!!!!! - √4)
291: !((√9)!) + (9!!!!!! - !(√4))
292: !((√9)!) + 9!!!!!!
293: !((√9)!) + (9!!!!! - 4!!)
294: !((√9)!) + (9!!!!!! + √4)
295: !((√9)!) + 4! + (√9)!
296: !((√9)!) + (9!!!!!! + 4)
297: !((√9)!) + (9!!!!! - 4)
298: !((√9)!) + 4! + 9
299: !((√9)!) + (9!!!!! - √4)
300: !((√9)!) + (9!!!!!! + 4!!)
301: !((√9)!) + 4! + ((√9)!)!!!!
302: !((√9)!) + (9!!!!! + !(√4))
303: !((√9)!) + (9!!!!! + √4)
304: (((√9)!)!!!!  + !(√9))!!!!!! + (4!!)!!!
305: !((√9)!) + (9!!!!! + 4) 
306: !((√9)!) + (9!!!! - 4)
307: !((√9)!) + ((√9)!)!!! + 4!
...

Posted by Dej Mar on 2020-09-12 11:51:16