[math-fun] Add or subtract the result. Iterate.
Hello Math-Fun, Here is another stupid iteration - which leads nowhere, probably. Start with (my date of birth) for instance: 1951 This in "p"; now make the absolute differences between p's digits, from left to right, like this: 1-9 = 8 9-5 = 4 5-1 = 4 1-1 = 0 (this is the last digit of "p" minus the first one) You then get "q" = 8440. And now: if q > p add q to p and iterate If q < p subtract q from p and iterate. Here: 1951 + 8440 ------- 10391 step 1 + 13680 ------- 24071 step 2 + 24761 ------- 48832 step 3 - 40512 ------- 8320 step 4 - 5128 ------ 3192 step 5 - 2871 ------ 321 step 6 - 112 ----- 209 step 7 + 297 ----- 506 step 8 + 561 ----- 1067 step 9 + 1616 ------ 2683 step 10 + 4251 ------ 6934 step 11 - 3612 ------ 3322 step 12 - 101 ------ 3221 step 13 - 1012 ------ 2209 step 14 - 297 ------ 1912 step 15 + 8811 ------- 10723 step 16 ... Etc. Don't know where this goes: loops? Fixed points? Patterns? Errors above? Waste of time? (yes) Best, É. [Color enhanced: http://www.cetteadressecomportecinquantesignes.com/AddOrSubtract.htm]
Eric Angelini:
Start with (my date of birth) for instance: 1951 This in "p"; now make the absolute differences between p's digits, from left to right, like this: 1-9 = 8 9-5 = 4 5-1 = 4 1-1 = 0 (this is the last digit of "p" minus the first one) You then get "q" = 8440.
if q > p add q to p and iterate if q < p subtract q from p and iterate
1951 yields: {1951, 10391, 24071, 48832, 8320, 3192, 321, 209, 506, 1067, 2683, 6934, 3322, 3221, 2209, 1912, 10723, 28235, 94358, 43127, 30974, 70205, 142457, 464583, 243232, 32122, 21021, 9810, 8091, 17078, 84795, 41552, 7520, 5193, 331, 309, 705, 1457, 4583, 3232, 2121, 1010, 2121, ...}, so {1010, 2121}, a small loop. There are loops of assorted lengths but a difficulty in compiling them is that even some small numbers (like 199 which apparently loops to {30, 63}, if I've done this correctly) take their time in getting to them: http://chesswanks.com/num/AddSubtractIterate.txt
Update here: http://www.cetteadressecomportecinquantesignes.com/AddOrSubtract.htm Best, É.
Page updated. 13094, discovered by Hans Havermann, seems to be the smallest integer escaping gravity. Phil Carmody has computed nearly 75 million iterations of 13094 and found the last result to be over 100000 digit long (see bottom of page). Thanks to all contributors. Best, É. http://www.cetteadressecomportecinquantesignes.com/AddOrSubtract.htm.
13094... seems to be the smallest integer escaping gravity.
I had assumed that a plus and a minus were equiprobable but, empirically, I'm finding that a minus may actually be more probable than a plus, at least for small numbers. This helps keep things down. However, the long-term evolution of 13094 argues strongly that, on average n becomes a number greater than n, and the eventual downfalls in the evolution of 199, 10853, and 10886, are the exceptions, not the rule.
participants (2)
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Eric Angelini -
Hans Havermann