Re: https://www.youtube.com/watch?v=094y1Z2wpJg
"The Simplest Math Problem No One Can Solve - Collatz Conjecture"
... but Pattern 616 suggests otherwise!
Re: https://en.wikipedia.org/wiki/Collatz_conjecture
"Jeffrey Lagarias stated in 2010 that the Collatz conjecture 'is an extraordinarily difficult problem,
completely out of reach of present day mathematics' "
Re: https://www.quora.com/Why-is-the-Collat ... not-proved
"The problem with the Collatz conjecture that makes it so intimately difficult
is the unpredictable nature of the cycles."
Now consider Pattern 616 resulting from 3X+1 or /2
applied to these descending numbers:
Code: Select all
16 = 8
15 = 46
14 = 7 1
13 = 40 6
12 = 6 1
11 = 34 6
10 = 5 1
9 = 28 6
8 = 4 1
7 = 22 6
6 = 3 1
5 = 16 6
4 = 2 1
3 = 10 6
2 = 1 1
1 = 4 6
Also note that the first mate to a number (to the right of '=') identifies a string ...
suggesting that every number begins a string and that strings contain strings
And Pattern 616 hints that, for many problems,
we should first come out of the rabbit hole.
Q: How does UB's 606 relate to 616?
A: 606 is one of the decimal planets:
"The number ten — the decimal system — is inherent
in the physical universe but not in the spiritual." (36:2.11)
Analysis: "3X+1" defines both infinity
and inevitable finality via linked "/2",
like the "Pi Corral" in another topic ...
making a solution a "God Equation"?
Pop Quiz:
If infinity -1 is odd, what is 3X+1
Collatz Conjecture Numbers As Strings
(3X+1 or /2 applied to beginning number)
Lower strings incorporate into higher strings;
higher strings incorporate into lower strings.
Every number in a string is the beginning
of a string that terminates with 4,2,1.
Evidence of a "God Equation" (infinity+finality)
lurking in the wild 'n crazy CC numbas
Code: Select all
308 > 154 = 77 232 116 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
306 > 153 = 460 230 115 346 173 520 260 130 65 196 98 49 148 74 37 112 56 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
296 > 148 = 74 37 112 56 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
294 > 147 = 442 221 664 332 166 83 250 125 376 188 94 47 142 71 214 107 322 161 484 242 121 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
284 > 142 = 71 214 107 322 161 484 242 121 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
282 > 141 = 424 212 106 53 160 80 40 20 10 5 16 8 4 2 1
272 > 136 = 68 34 17 52 26 13 40 20 10 5 16 8 4 2 1
270 > 135 = 406 203 610 305 916 458 229 688 344 172 86 43 130 65 196 98 49 148 74 37 112 56 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
248 > 124 = 62 31 94 47 142 71 214 107 322 161 484 242 121 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
246 > 123 = 370 185 556 278 139 418 209 628 314 157 472 236 118 59 178 89 268 134 67 202 101 304 152 76 38 19 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
244 > 122 = 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
242 > 121 = 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
236 > 118 = 59 178 89 268 134 67 202 101 304 152 76 38 19 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
234 > 117 = 352 176 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
224 > 112 = 56 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
222 > 111 = 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
220 > 110 = 55 166 83 250 125 376 188 94 47 142 71 214 107 322 161 484 242 121 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
218 > 109 = 328 164 82 41 124 62 31 94 47 142 71 214 107 322 161 484 242 121 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
216 > 108 = 54 27 82 41 124 62 31 94 47 142 71 214 107 322 161 484 242 121 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
214 > 107 = 322 161 484 242 121 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
212 > 106 = 53 160 80 40 20 10 5 16 8 4 2 1
210 > 105 = 316 158 79 238 119 358 179 538 269 808 404 202 101 304 152 76 38 19 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
208 > 104 = 52 26 13 40 20 10 5 16 8 4 2 1
206 > 103 = 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
204 > 102 = 51 154 77 232 116 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
202 > 101 = 304 152 76 38 19 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
200 > 100 = 50 25 76 38 19 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
198 > 99 = 298 149 448 224 112 56 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
196 > 98 = 49 148 74 37 112 56 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
194 > 97 = 292 146 73 220 110 55 166 83 250 125 376 188 94 47 142 71 214 107 322 161 484 242 121 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
192 > 96 = 48 24 12 6 3 10 5 16 8 4 2 1
190 > 95 = 286 143 430 215 646 323 970 485 1456 728 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
188 > 94 = 47 142 71 214 107 322 161 484 242 121 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
186 > 93 = 280 140 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
158 > 74 = 37 112 56 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
156 > 73 = 220 110 55 166 83 250 125 376 188 94 47 142 71 214 107 322 161 484 242 121 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
144 > 72 = 36 18 9 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
142 > 71 = 214 107 322 161 484 242 121 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
100 > 50 = 25 76 38 19 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
98 > 49 = 148 74 37 112 56 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
96 > 48 = 24 12 6 3 10 5 16 8 4 2 1
94 > 47 = 142 71 214 107 322 161 484 242 121 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
92 > 46 = 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
90 > 45 = 136 68 34 17 52 26 13 40 20 10 5 16 8 4 2 1
88 > 44 = 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
86 > 43 = 130 65 196 98 49 148 74 37 112 56 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
84 > 42 = 21 64 32 16 8 4 2 1
82 > 41 = 124 62 31 94 47 142 71 214 107 322 161 484 242 121 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
80 > 40 = 20 10 5 16 8 4 2 1
78 > 39 = 118 59 178 89 268 134 67 202 101 304 152 76 38 19 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
76 > 38 = 19 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
74 > 37 = 112 56 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
72 > 36 = 18 9 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
70 > 35 = 106 53 160 80 40 20 10 5 16 8 4 2 1
68 > 34 = 17 52 26 13 40 20 10 5 16 8 4 2 1
66 > 33 = 100 50 25 76 38 19 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
64 > 32 = 16 8 4 2 1
62 > 31 = 94 47 142 71 214 107 322 161 484 242 121 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
60 > 30 = 15 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
58 > 29 = 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
56 > 28 = 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
54 > 27 = 82 41 124 62 31 94 47 142 71 214 107 322 161 484 242 121 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
52 > 26 = 13 40 20 10 5 16 8 4 2 1
50 > 25 = 76 38 19 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
48 > 24 = 12 6 3 10 5 16 8 4 2 1
46 > 23 = 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
44 > 22 = 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
42 > 21 = 64 32 16 8 4 2 1
40 > 20 = 10 5 16 8 4 2 1
38 > 19 = 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
36 > 18 = 9 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
34 > 17 = 52 26 13 40 20 10 5 16 8 4 2 1
32 > 16 = 8 4 2 1
30 > 15 = 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
28 > 14 = 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
26 > 13 = 40 20 10 5 16 8 4 2 1
24 > 12 = 6 3 10 5 16 8 4 2 1
22 > 11 = 34 17 52 26 13 40 20 10 5 16 8 4 2 1
20 > 10 = 5 16 8 4 2 1
18 > 9 = 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
16 > 8 = 4 2 1
14 > 7 = 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
12 > 6 = 3 10 5 16 8 4 2 1
10 > 5 = 16 8 4 2 1
8 > 4 = 2 1
6 > 3 = 10 5 16 8 4 2 1
4 > 2 = 1 4 2 1
2 > 1 = 4 2 1
A curious pattern ...
Add 1 to every odd number; divide every even by 2.
(hints that all Collatz calculations resolve to 4,2,1
194 97 98 49 50 25 26 13 14 7 8 4 2 1
323 324 162 81 82 41 42 21 22 11 12 6 3 4 2 1
Collatz sequence for numbers 194, 323 ...
194 97 292 146 73 220 110 55 166 83 250 125 376 188 94 47 142 71 214 107 322 161 484 242 121 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
323 970 485 1456 728 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
Note the tight linkage of even and odd (194 + 98 = 292),
shouting every number in Collatz stairway resolves to 4,2,1:
200 > 100 = 50
198 > 99 = 298
196 > 98 = 49 148 74 37 112 56 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
194 > 97 = 292 146 73 220 110 55 166 83 250 125 376 188 94 47 142 71 214 107 322 161 484 242 121 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
192 > 96 = 48
190 > 95 = 286
Apparently, the seducing influence of Morbus Collatzmetricus:
More Pattern 616 linkage ('>' points to string's primary number;
note that every number in every string is in the stairway: 20, 22)
(46 x 2 = 22 + 70), (42 + 22 = 64), (70 - 64 = 6), (12 - 11 = 1)
48 > 24 = 12 6 3 10 5 16 8 4 2 1
46 > 23 = 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
44 > 22 = 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
42 > 21 = 64 32 16 8 4 2 1
40 > 20 = 10 5 16 8 4 2 1
38 > 19 = 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
(6 x 2 = 2 + 10), (2 x 2 = 4), (10 - 4 = 6), (2 - 1 = 1)
8 > 4 = 2 1
6 > 3 = 10 5 16 8 4 2 1
4 > 2 = 1 4 2 1
2 > 1 = 4 2 1
About this Collatzable Linkage ...
A fortuitous random selection of groups to contrast (24 - 21 vs 4 -1)
which highlights the +20 influence on '3X+1' and 'X/2' calculations
(further proving that all numbers resolve to 4,2,1).
Collatz Conjecture with application to X/Y axis:
These charts suggest "3X+1 or /2" of the Collatz Conjecture
effectively programs the consistent 4,2,1 ending loop
Code: Select all
144 72 36 18 9 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
72 36 18 9 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
36 18 9 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
18 9 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
9 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
34 17 52 26 13 40 20 10 5 16 8 4 2 1
17 52 26 13 40 20 10 5 16 8 4 2 1
52 26 13 40 20 10 5 16 8 4 2 1
26 13 40 20 10 5 16 8 4 2 1
13 40 20 10 5 16 8 4 2 1
40 20 10 5 16 8 4 2 1
20 10 5 16 8 4 2 1
10 5 16 8 4 2 1
5 16 8 4 2 1
16 8 4 2 1
8 4 2 1
4 2 1
2 1
1
4,2,1 consistency by powers of 2
Code: Select all
164 = 82 41 124 62 31 94 47 142 71 214 107 322 161 484 242 121 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
82 = 41 124 62 31 94 47 142 71 214 107 322 161 484 242 121 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
41 = 124 62 31 94 47 142 71 214 107 322 161 484 242 121 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
44 = 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
22 = 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
11 = 34 17 52 26 13 40 20 10 5 16 8 4 2 1
40 = 20 10 5 16 8 4 2 1
20 = 10 5 16 8 4 2 1
10 = 5 16 8 4 2 1
36 = 18 9 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
18 = 9 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
9 = 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
32 = 16 8 4 2 1
16 = 8 4 2 1
8 = 4 2 1
28 = 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
14 = 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
7 = 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
30 = 15 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
15 = 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
10 = 5 16 8 4 2 1
5 = 16 8 4 2 1
32 = 16 8 4 2 1
16 = 8 4 2 1
8 = 4 2 1
4 = 2 1
48 = 24 12 6 3 10 5 16 8 4 2 1
24 = 12 6 3 10 5 16 8 4 2 1
12 = 6 3 10 5 16 8 4 2 1
6 = 3 10 5 16 8 4 2 1
3 = 10 5 16 8 4 2 1
32 = 16 8 4 2 1
16 = 8 4 2 1
8 = 4 2 1
4 = 2 1
2 = 1 4 2 1
32 = 16 8 4 2 1
16 = 8 4 2 1
8 = 4 2 1
4 = 2 1
2 = 1 4 2 1
1 = 4 2 1
Apparently, the Collatz Matrix, a triangle!
Re: https://dictionary.cambridge.org/us/dic ... ish/matrix
"group of numbers or other symbols arranged in a rectangle used together
as a single unit to solve particular mathematical problems"
Code: Select all
12 6 3 10 5 16 8 4 2 1
6 3 10 5 16 8 4 2 1 1
3 10 5 16 8 4 2 1 3
10 5 16 8 4 2 1 7
5 16 8 4 2 1 15
16 8 4 2 1 31
8 4 2 1 36
4 2 1 46
2 1 49
1 55
67
This pattern suggests ALL numbers resolve with 4,2,1
regardless of the number of calculations required!
30 days hath September, April, June, and November; All the rest hath 31,
'cepting February alone (hath 28 or 29), thence all quiesce with 4,2,1.
Code: Select all
31 94 47 142 71 214 107 322 161 484 242 121 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1 - 31
30 15 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1 - 30
29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1 - 29
28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1 - 28
27 82 41 124 62 31 94 47 142 71 214 107 322 161 484 242 121 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1 - 27
26 13 40 20 10 5 16 8 4 2 1 - 26
25 76 38 19 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1 - 25
24 12 6 3 10 5 16 8 4 2 1 - 24
23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1 - 23
22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1 - 22
21 64 32 16 8 4 2 1 - 21
20 10 5 16 8 4 2 1 - 20
19 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1 - 19
18 9 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1 - 18
17 52 26 13 40 20 10 5 16 8 4 2 1 - 17
16 8 4 2 1 - 16
15 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1 - 15
14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1 - 14
13 40 20 10 5 16 8 4 2 1 - 13
12 6 3 10 5 16 8 4 2 1 - 12
11 34 17 52 26 13 40 20 10 5 16 8 4 2 1 - 11
10 5 16 8 4 2 1 - 10
9 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1 - 9
8 4 2 1 - 8
7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1 - 7
6 3 10 5 16 8 4 2 1 - 6
5 16 8 4 2 1 - 5
4 2 1 - 4
3 10 5 16 8 4 2 1 - 3
2 1 - 2
1 - 1
These patterns suggest 4,2,1 proof of any odd number
is proof for all odd numbers; 4,2,1 proof of any even number
is proof for all even numbers. HCIT
26 13 40 20 10 5 16 8 4 2 1
24 12 6 3 10 5 16 8 4 2 1
22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
20 10 5 16 8 4 2 1
18 9 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
16 8 4 2 1
14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
12 6 3 10 5 16 8 4 2 1
10 5 16 8 4 2 1
8 4 2 1
6 3 10 5 16 8 4 2 1
4 2 1
2 1 4 2 1
25 76 38 19 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
21 64 32 16 8 4 2 1
19 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
17 52 26 13 40 20 10 5 16 8 4 2 1
15 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
13 40 20 10 5 16 8 4 2 1
11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
9 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
5 16 8 4 2 1
3 10 5 16 8 4 2 1
1 4 2 1
48 8 12 6 3 10 5 16 8 4 2 1
24 12 6 3 10 5 16 8 4 2 1
12 6 3 10 5 16 8 4 2 1
6 3 10 5 16 8 4 2 1
88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
72 36 18 9 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
36 18 9 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
18 9 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
9 28 14 7 22 11 34 17 6 3 10 5 16 8 4 2 1
56 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
40 20 10 5 16 8 4 2 1
20 10 5 16 8 4 2 1
10 5 16 8 4 2 1
5 16 8 4 2 1
24 12 6 3 10 5 16 8 4 2 1
12 6 3 10 5 16 8 4 2 1
6 3 10 5 16 8 4 2 1
3 10 5 16 8 4 2 1
8 4 2 1 4 2 1
4 2 1 4 2 1
2 1 4 2 1
1 4 2 1
Every whole number is in Pattern 616
and always descends to 4,2,1 ... apparently.
Every number in every string is in Pattern 616
and every number in the Pattern 616 stairway
always descends to 4,2,1 ... apparently.
(2992 - 2986 = 6, 498 - 497 = 1, 2986 - 2980 = 6)
(1498 -1492 = 6, 249 - 248 = 1, 1492 - 1486 = 6)
(16 - 10 = 6, 2 - 1 = 1, 10 - 4 = 6)
997 2992 1496 .. 4 2 1
996 498 249 .. 4 2 1
995 2986 1493 .. 4 2 1
994 497 1492 .. 4 2 1
993 2980 1490 .. 4 2 1
499 1498 749 .. 4 2 1
498 249 748 .. 4 2 1
497 1492 746 .. 4 2 1
496 248 124 .. 4 2 1
495 1486 743 .. 4 2 1
5 16 8 4 2 1
4 2 1
3 10 5 16 8 4 2 1
2 1 4 2 1
1 4 2 1
The light at the end of the funnel
Re: https://www.quantamagazine.org/why-math ... -20200922/
"The Simple Math Problem We Still Can’t Solve: Collatz Conjecture"
"if you start with any positive integer, you’ll always end up in this 4,2,1 loop."
Consider strings created by '6' and '1' in Pattern 616:
(first mates of the sequential odd numbers differ by 6,
first mates of the sequential even numbers differ by 1)
Code: Select all
6 = 3 10 5 16 8 4 2 1
1 = 4 2 1
maintain the 616 pattern (differences) and are vulnerable to the 6,1 funnel
leading into the 4,2,1 trap.
How to know if number > 1 is Pattern 616
If number odd: ((3X+1)-4)/6 = no remainder
If number even: add 1, then ((3X+1)-4)/6 = no remainder
Example:
157 = 472 236 118 59 178 89 268 134 67 202 101 304 152 76 38 19 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
156 = 78 39 118 59 178 89 268 134 67 202 101 304 152 76 38 19 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
155 = 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
154 = 77 232 116 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
(((157 x 3) + 1) - 4) / 6 = 78
78 x 2 = 156 - 1 = 155
(((155 x 3) + 1) - 4) / 6 = 77
77 x 2 = 154 - 1 = 153
Apparently, every number in every string belongs to Pattern 616,
the defining pattern of the "unpredictable" Collatz Conjecture.
First number of every string downsteps predictably to next string
with strings 4,2,1 always the final strings.
How to verify that string is Pattern 616
Proceed backward from ending number 1.
If previous number even, multiply current number by 2.
If previous number odd, subtract 1 from current number, divide by 3.
Examples: (Go figure!)
6 = 3 10 5 16 8 4 2 1
How to prove these cycles are "unpredictable"?
"Completely out of reach of present day mathematics."
Collatz Conjecture, by definition (3X+1 or /2, depending on odd/even)
is "programmed predictability", confirmed by reverse calculations
(string cannot be confirmed if calculations were "unpredictable").
When this "programmed predictability" causes the ending loop
is unknown (unpredictable) without this program execution,
but hints all Collatz calculations create a string that funnels
into the ending loop (as suggested by Pattern 616).
Since the downstep string (from infinity -1 to 1) proves(?)
that every number ends in the 4,2,1 loop (does not escape),
then every numbered string must also end in a 4,2,1 loop
... because "3X+1 or /2" logic applies to all strings.
Actually, the Collatz Conjecture presents two independent problems:
1) Proving that all numbers from infinity -1 do not escape the 4,2,1 loop.
2) Predicting how many calculations are necessary for each number to reach its 4,2,1 loop
(precisely predictable for the downstep string, but like tossing a coin for other strings):
The complexity might be illustrated by starting with a bitcoin of fixed value
where one side = '3X+1' and the other side 'X/2'. The odd/even of starting value
determines both next value and then the next calculation.
These calculations seem to parallel bitcoin "investing" with a calculation
relative to time investing and ending value of 4,2,1 the possible final profit
... at the end of time.
Columnar Collatz (CC Stairway)
The two rightmost columns are the only ones needed to construct
this stairway to infinity (God?), with addition the only calculation.
Once constructed, the path of any number to its 4,2,1 resolution
becomes "child's play" to even the simplest of computers
This describes a computer pattern generator - not solution!
(refers to a 'linked list' with any number an index to the list;
the number's /2 or 3X+1 value becomes the next link.)
Leftmost columns define possible Collatz path of primary number
(all columns should be generated for this linked list).
How many entries to generate in the list?
Why not start with a googol (10^100)
Example: Just follow '7' in the two columns:
7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
Code: Select all
3X+1
/2 3X+1 ---> /2
200 33 = 100 = 50
198 = 99 = 298
196 = 98 = 49
194 = 97 = 292
192 = 96 = 48
190 = 95 = 286
188 31 = 94 = 47
186 = 93 = 280
184 = 92 = 46
182 = 91 = 274
180 = 90 = 45
178 = 89 = 268
176 29 = 88 = 44
174 = 87 = 262
104 17 = 52 = 26
102 = 51 = 154
100 = 50 = 25
98 = 49 = 148
96 = 48 = 24
94 = 47 = 142
92 15 = 46 = 23
90 = 45 = 136
88 = 44 = 22
86 = 43 = 130
84 = 42 = 21
82 = 41 = 124
80 13 = 40 = 20
78 = 39 = 118
76 = 38 = 19
74 = 37 = 112
72 = 36 = 18
70 = 35 = 106
68 11 = 34 = 17
66 = 33 = 100
64 = 32 = 16
62 = 31 = 94
60 = 30 = 15
58 = 29 = 88
56 9 = 28 = 14
54 = 27 = 82
52 = 26 = 13
50 = 25 = 76
48 = 24 = 12
46 = 23 = 70
44 7 = 22 = 11
42 = 21 = 64
40 = 20 = 10
38 = 19 = 58
36 = 18 = 9
34 = 17 = 52
32 5 = 16 = 8
30 = 15 = 46
28 = 14 = 7
26 = 13 = 40
24 = 12 = 6
22 = 11 = 34
20 3 = 10 = 5
18 = 9 = 28
16 = 8 = 4
14 = 7 = 22
12 = 6 = 3
10 = 5 = 16
8 1 = 4 = 2
6 = 3 = 10
4 = 2 = 1
2 = 1 = 4
Observation: '/2' downsteps more than '3X+1' upsteps,
suggesting that Conjecture is biased in 4,2,1 direction.
Also, '3X+1' immediately changes odd to even,
whereas '/2' eventually changes even to odd.
00. 155 466 233
01. 700 350 175
02. 526 263
03. 790 395
04. 1186 593
05. 1780 890 445
06. 1336 668 334 167
07. 502 251
08. 754 377
09. 1132 566 283
10. 850 425
11. 1276 638 319
12. 958 479
13. 1438 719
14. 2158 1079
15. 3238 1619
16. 4858 2429
17. 7288 3644 1822 911
18. 2734 1367
19. 4102 2051
20. 6154 3077
21. 9232 4616 2308 1154 577
22. 1732 866 433
23. 1300 650 325
24. 976 488 244 122 61
25. 184 92 46 23
26. 70 35
27. 106 53
28. 160 80 40 20 10 5
29. 16 8 4 2 1
Also explains Pattern 616, since '+1' changes odd to even,
but is otherwise insignificant (especially for larger numbers);
which leaves '3X' vs '/2' for math relationship: 30 / 5 = 6
Number = 3X+1 = /2
10000 = 30001 = 5000
1000 = 3001 = 500
100 = 301 = 50
10 = 31 = 5
While quantum string theory fades into the background,
this number string theory seems to be leading us to infinity
and suggesting a "God Equation" (we go to Number 1).
That 3X+1 cannot be known when X = infinity -1
suggests Collatz Conjecture can never be "proven".
Pattern 616 seems a necessary reference
since every number from 1 to infinity -1 exists
as a discrete step in the Pattern 616 stairway
and every number in the stairway downsteps
(calculates) to its 4,2,1 beginning.
If Pattern 616 is consistent and controlling,
'powers of 2' numbers predict 4,2,1 resolution
of all numbers in the Pattern 616 stairway:
512 = 256 128 64 32 16 8 4 2 1
Restated: '3X' is somewhat balanced by '/2', however
'+1' converts odd to even, giving /2 more influence.
Observation: Collatz Conjecture ('3X+1' or '/2')
defines constant math pressure to downstep
all calculations to their 4,2,1 resolution.
Every number in a string is the primary number of a string, therefore
4,2,1 proof of any number in the string extends proof to the primary number
All numbers in this '27' string are a primary number
... and every string ends with 4,2,1:
27 82 41 124 62 31 94 47 142 71 214 107 322 161 484 242 121 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
There's no string with primary number in Pattern 616 stairway,
a linked computer list, that goes all the way Uptown (all numbers
from infinity -1 to 1 are sequential in the linked list).
Descending the stairway from infinity is pre-ordained
(decisions based on odd or even), but ascending stairway
is traveler's choice (like ascension to Paradise).
Downtown path: 3X+1 if number odd, X/2 if even.
Uptown path: if (X-1) odd, choose (X-1)/3 or 2X; else choose 2X.
Interestingly, this Downtown with Uptown verification
hints that Collatz Conjecture is indeed a Downtown path!
https://www.youtube.com/watch?v=Zx06XNfDvk0
"Things will be great when you're downtown!"
Collatzable Linkage
These many patterns shout that no formula can define how every number resolves (or not) to 4,2,1,
but there's mathematical opportunity for development of rules associated with such "Collatzable Linkage"
(the infinity -1 to 1 "stairway"). That the pattern of a primary number's association with its neighbors
is similar to the stairway's 4,3,2,1 foundation supports the concept of Collatzable Linkage:
(6 x 2 = 2 + 10), (2 x 2 = 4), (10 - 4 = 6), (2 - 1 = 1)
8 > 4 = 2 1
6 > 3 = 10 5 16 8 4 2 1
4 > 2 = 1 4 2 1
2 > 1 = 4 2 1
Considering this Collatz Stairway (all numbers from infinity -1 to 1), proof of 4,2,1 resolution
may simply be proof that all numbers can be calculated as 3X+1 or X/2 (since these guarantee
that a number will eventually tag the primary number of a string which leads to the 4,2,1).
And consider the math ...
3X+1 converts odd to even, then X/2 divides by 2,
effectively contrasting (3X+1)/2 and X/2.
Since odd converts to even more than even to odd,
X/2 is the greater pressure to 4,2,1 resolution
and prevents 3X+1 domination (IMO).
Hollandaze Conjecture In Totality (HCIT)
"Since every whole number identified by the most super of computers
exists in the Pattern 616 stairway (and every number in the stairway
is the primary number of a Collatz string that resolves to 4,2,1),
the 3X+1 vs X/2 mathematical tension must be the Collatz Key
with X/2 dominating the true direction of resolution no matter
how many calculations are required because of 3X+1 drift."
Collatzable Linkage ...
might now become a legal term referring to a pattern of activity
so convincing that 'a,b,c' proves 'd,e,f' (beyond the shadow of doubt).
Rod