145 and the Melancoil - Numberphile
video description
Using cubes, there are 5 trees and 4 coils. Out of the 2-digit numbers:
-- 1 and 10 are part of the 1 tree.
-- All multiples of 3 are part of the 153 tree.
-- 7, 19, 34, 37, 43, 58, 67, 70, 73, 76, 85, 88, and 91 are part of the 370 tree.
-- 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44, 50, 53, 56, 59, 62, 65, 68, 71, 80, 83, 86, 92, and 95 are part of the 371 tree.
-- 47, 74, 77, 89, and 98 are part of the 407 tree.
-- 49 and 94 are part of the 1459 --> 919 coil.
-- 16, 22, 61, 79, and 97 are part of the 217 --> 352 --> 160 coil.
-- 4, 13, 25, 28, 31, 40, 46, 52, 55, 64, and 82 are part of the 133 --> 55 --> 250 coil.
-- The 244 --> 136 coil has no 2-digit members. The smallest one is 136 itself.
Using fourth powers, there are 4 trees and at least 2 coils, although there may be more. Out of the 2-digit numbers:
-- 1 and 10 are part of the 1 tree.
-- 12, 17, 21, 46, 64, and 71 are part of the 8208 tree.
-- 66 is part of the 6514 --> 2178 coil.
-- All the others are part of the 13139 --> 6725 --> 4338 --> 4514 --> 1138 --> 4179 --> 9219 coil.
-- There are also the 1634 tree and 9474 tree, which have no 2-digit members.
Using fifth powers, there are 7 trees and at least 7 coils, although there may be more. Out of the 2-digit numbers:
-- 1 and 10 are part of the 1 tree.
-- 4, 37, 40, 55, and 73 are part of the 10933 --> 59536 --> 73318 --> 50062 coil.
-- 16 and 61 are part of the 44155 --> 8299 --> 150898 --> 127711 --> 33649 --> 68335 coil.
-- 17, 47, 71, 74, 77, 89, and 98 are part of a coil of size 10.
-- 5, 8, 26, 35, 44, 50, 53, 62, 68, 80, and 86 are part of a different coil of size 10.
-- 2, 11, 14, 20, 23, 29, 32, 38, 41, 56, 59, 65, 83, 92, and 95 are part of a coil of size 12.
-- All the multiples of 3 are part of a coil of size 22.
-- 7, 13, 19, 22, 25, 28, 31, 34, 43, 46, 49, 52, 58, 64, 67, 70, 76, 79, 82, 85, 88, 91, 94, and 97 are part of a coil of size 28.
-- The 4150 tree, 4151 tree, 54748 tree, 92727 tree, 93084 tree, and 194979 tree all have no 2-digit members.
Date: 2022-04-08
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Comments and reviews: 9
TruthNerds
Wikipedia features a nice proof why all numbers are either happy or -melancoiling- (Matt Parker):
All numbers >= 1000 must lose digits upon -happification- e. g. 9999 -> 324 has one digit less (in fact, very large numbers lose digits rapidly, roughly from k to log10(k.
All numbers from 244 to 999 happify to at most 243 (because 999 -> 243.
All numbers from 164 to 243 happify to at most 163 (because 199 -> 163.
All numbers from to 108 to 163 happify to at most 107 (because 159 -> 107.
All numbers from 100 to 107 happify to at most 50 (because 107 -> 50.
Therefore, all numbers must successively happify to less than 100, and it can be shown exhaustively (as Matt has done here) that all numbers less than 100 are either happy or -melancoiling-.
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Wikipedia features a nice proof why all numbers are either happy or -melancoiling- (Matt Parker):
All numbers >= 1000 must lose digits upon -happification- e. g. 9999 -> 324 has one digit less (in fact, very large numbers lose digits rapidly, roughly from k to log10(k.
All numbers from 244 to 999 happify to at most 243 (because 999 -> 243.
All numbers from 164 to 243 happify to at most 163 (because 199 -> 163.
All numbers from to 108 to 163 happify to at most 107 (because 159 -> 107.
All numbers from 100 to 107 happify to at most 50 (because 107 -> 50.
Therefore, all numbers must successively happify to less than 100, and it can be shown exhaustively (as Matt has done here) that all numbers less than 100 are either happy or -melancoiling-.
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David
Matt, you have a habit of showing up from time to time and destroying my leisure time, not to mention pocket money. Thanx to you I have a drawer full of clear plastic cubes, some of which contain three indistinguishable tiny red dice, some containing different-colored dice. Thanx to you I now have a rudimentary understanding of both Mandelbrot and Julia, imaginary numbers, iterative functions, and all the rest of it. (I blame you for that last bit because you used the word -fractal- when discussing the Menger Sponge, which led to another drawer-full of paper doo-dads, you sadistic. uh. sadist)
Cheers.
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Matt, you have a habit of showing up from time to time and destroying my leisure time, not to mention pocket money. Thanx to you I have a drawer full of clear plastic cubes, some of which contain three indistinguishable tiny red dice, some containing different-colored dice. Thanx to you I now have a rudimentary understanding of both Mandelbrot and Julia, imaginary numbers, iterative functions, and all the rest of it. (I blame you for that last bit because you used the word -fractal- when discussing the Menger Sponge, which led to another drawer-full of paper doo-dads, you sadistic. uh. sadist)
Cheers.
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smartbandwidth
The -top branches- of Matt's -Happification Tree- contain the numbers 1991, 44 and 707 which add up to 2742 which factors into 457 prime X 6.
457 is the 89th prime number
89 is the highest prime factor of Leo (356) = 89 prime X 4.
457 doubles to 914 which mirrors I (9) Am (14) = 23 prime.
914 corresponds to 9/14 which in non-leap years is always the 257th day of the year.
257 = KEY using reduced gematria - 1-9, 1-9, 1-8 = A-Z
257 is also prime
There are 257 (KEY) minutes that separate 4: 57 am and 9: 14 am.
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The -top branches- of Matt's -Happification Tree- contain the numbers 1991, 44 and 707 which add up to 2742 which factors into 457 prime X 6.
457 is the 89th prime number
89 is the highest prime factor of Leo (356) = 89 prime X 4.
457 doubles to 914 which mirrors I (9) Am (14) = 23 prime.
914 corresponds to 9/14 which in non-leap years is always the 257th day of the year.
257 = KEY using reduced gematria - 1-9, 1-9, 1-8 = A-Z
257 is also prime
There are 257 (KEY) minutes that separate 4: 57 am and 9: 14 am.
reply
thenamelessone83
I notices something at 5: 48, 42 has only one number going into it, all other numbers in the circle have numbers from outside the circle going into them. That must mean the ultimate question is: What is the sum of the squares of 1 4 and 5?
-universe end and is replace with something even more bizarre.
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I notices something at 5: 48, 42 has only one number going into it, all other numbers in the circle have numbers from outside the circle going into them. That must mean the ultimate question is: What is the sum of the squares of 1 4 and 5?
-universe end and is replace with something even more bizarre.
reply
Scolding
5: 54
Matt: All other three-digit or higher numbers filter into this diagram. This is the complete structure for all two digit numbers. So everything--
Brady: All right, so let's draw the three-digit numbers one then.
Matt: Okay, I'll get some-- No.
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5: 54
Matt: All other three-digit or higher numbers filter into this diagram. This is the complete structure for all two digit numbers. So everything--
Brady: All right, so let's draw the three-digit numbers one then.
Matt: Okay, I'll get some-- No.
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Holden
I also got bored, so I made the Melancoil Cubing Tree - only to find that certain numbers -happify by cubing- to themselves! 371 and 153 are examples of this, 27+343+1=371 and 1+125+27=153! Just wondering, are there any cases in the squaring case?
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I also got bored, so I made the Melancoil Cubing Tree - only to find that certain numbers -happify by cubing- to themselves! 371 and 153 are examples of this, 27+343+1=371 and 1+125+27=153! Just wondering, are there any cases in the squaring case?
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MrHatoi
This sounds like a classist dystopia. The rich inhabit the luxurious happy tree, a perfect hierarchical structure while the poor must suffer within the melancoil, doomed to forever toil away in the same cycle for all of eternity.
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This sounds like a classist dystopia. The rich inhabit the luxurious happy tree, a perfect hierarchical structure while the poor must suffer within the melancoil, doomed to forever toil away in the same cycle for all of eternity.
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EebstertheGreat
I like the idea of classifying representations of natural numbers in a particular base as either happy (with a tree rooted at 1, lonely (isolated points, or melancholic/melancoilic (with a cycle of order greater than 1.
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I like the idea of classifying representations of natural numbers in a particular base as either happy (with a tree rooted at 1, lonely (isolated points, or melancholic/melancoilic (with a cycle of order greater than 1.
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Jonathan
Two line code in Mathematica:
happy[n_]: = Total[IntegerDigits[n]-2]
Graph[Union[
Flatten[-# -> happy[#]- & /-
Join[Range[100], Select[happy[#] & /- Range[100], # > 100 &]]]],
VertexLabels -> -Name-]
reply
Two line code in Mathematica:
happy[n_]: = Total[IntegerDigits[n]-2]
Graph[Union[
Flatten[-# -> happy[#]- & /-
Join[Range[100], Select[happy[#] & /- Range[100], # > 100 &]]]],
VertexLabels -> -Name-]
reply
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