145 and the Melancoil - Numberphile

Matt Parker talks us through "Happification" and resulting structures it creates. Includes his fascinating Melancoil!?
More links & stuff in full description below ↓↓↓

Filmed at the Champagne Bar at St Pancras railway station... thanks to them for having us!

NUMBERPHILE

Videos by Brady Haran

[ericstoverink]

"I'm not going to do happy numbers". Immediately does happy numbers. A Parker promise.

[SuperStingray]

"Matt's Melancoil and Happification Tree" sounds like a children's book.

[AuroraDashPteriforever]

Tried this for cubes as well as 4th and 5th powers, here's what I found:

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.

[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.

[breathless]

it might be worth me just trying again

base 2: there are no coils all number are happy
base 3: (2, 11)[2, 4]  (12)[5]  (22)[8]
base 4: same as 2
base 5: (4, 31, 20)[4, 16, 10]  (23)[13]  (33)[18]
base 6: (5, 41, 25, 45, 105, 42, 32, 21)[5, 25, 17, 29, 41, 26, 20, 13]
base 7: (2, 4, 22, 11)[2, 4, 16, 8]  (34)[25]  (13)[10]  (16, 52, 41, 23)[13, 37, 29, 17]  (63)[45]  (44)[32] 
base 8: (4, 20)[4, 16]  (5, 31, 12)[5, 25, 10)  (32, 15)[26, 13]  (24)[20]  (64)-[52]
base 9: (55)[50]  (58, 108, 72)[53, 89, 65]  (45)[41]  (75, 82)[68, 74]

[scoldingMime]

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...

[egalomon]

I once tried to get my brother into this kind of stuff, because I personally find this really enjoyable. He was just shaking his head.

[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".

[numberphile]

See the full video description by clicking "show more" for a link to Matt's coil!!!

[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).

[breathless]

here's a list of coils in bases less than 10

note: I've put the numbers in a list in round brackets with commas starting with the smallest in the coil, but in the order of the coil not numerical, I've also put the base 10 translation next to it in square brackets seperated by a hyphen. also I've excluded the 1sq =1 coil which exists in all bases. some oscillate between two numbers and some, one numbers leads to itself, others have more numbers

base 2: there are no coils all number are happy
base 3: (2, 11)-[2, 4]  (12)-[5]  (22)-[8]
base 4: same as 2
base 5: (4, 31, 20)-[4, 16, 10]  (23)-[13]  (33)-[18]
base 6: (5, 41, 25, 45, 105, 42, 32, 21)-[5, 25, 17, 29, 41, 26, 20, 13]
base 7: (2, 4, 22, 11)-[2, 4, 16, 8]  (34)-[25]  (13)-[10]  (16, 52, 41, 23)-[13, 37, 29, 17]  (63)-[45]  (44)-[32] 
base 8: (4, 20)-[4, 16]  (5, 31, 12)-[5, 25, 10)  (32, 15)-[26, 13]  (24)-[20]  (64)-[52]
base 9: (55)-[50]  (58, 108, 72)-[53, 89, 65]  (45)-[41]  (75, 82)-[68, 74]
higher bases coming soon

[AlastairCarr]

The Collatz conjecture looks really interesting - especially because the conjecture seems kind of intuitive (maybe that's because I remember being told the sequence when I was about 10 and my teacher telling me the conjecture like it was brute fact, so I took it that way) but has been so hard to prove.

There's something particularly appealing in happification because it can go up or down even though it's not defined piecewise, but still the Collatz trees are awesome!

[mcol]

I don't understand why he emphasises 145, I don't think he explains it.

[shocklab]

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"]

[WylliamJudd]

The happy tree and the melancoil has a legendary ring to it.

[Booskop.]

Writes down 100...
"Those are all the 2-digit-numbers..."

[numberphile]

@TheMetalProxy there are scans of most of them on the numberphile flickr page!!!

[longevitee]

It requires a few three-digit numbers? Uh oh... he Parker squared it.

[vantrickpaughney]

Don't know what's the point of all of these procedures, but I find this whole story really charming and kinda thought-provoking

[jgilgorri]

You, sir, have inspired me. I'm going to try doing this in the Dozenal System.
Thank you for providing me with at least a few days worth of fun!