An Easy, Fun Number Sequence

[Chuck]

A somewhat longer example:

8562
6342
12780
14756
13856
26332
7991
23987
23781
37158
19322
48440
26031
52559
51185
46726
69094
87483
109886
138822
121211
253353
112883
39930
41421
54306
88497
101106
120401
113282
62713
124270
162669
8589
49157
30264
42548
11216
14343
12599
20093
23607
10610
7994
10659
15146
24189
26994
2057
9257
5880
11031
11471
19097
21687
34139
42812
25968
53052
56880
73136
112671
24080
62111
31951
73352
44831
109176
67656
40320
21479
26096
10359
21983
13797
17226
14066
560
3324
944
1415
1735
2669
3914
642
563
245
294
206
48
87
80
71
55
61
106
66
107
6
17
2
9
2
2
0

[Lepton]

Yay, more examples.
I'm all for the solution before I pull the rest of my hair out...

[Chuck]

It's too hard to believe that no one is as smart as Mathgrant so someone else will have to solve it.

[mathgrant]

Actually, it was a combination of math skill and some luck. I was lucky to get the right insight at the right time. Many people might have discarded that insight as "coincidence", but it was looking closely at the smaller numbers after getting the idea that led me into finding the answer.

And now, a word from the Hint Fairy.

[The Hint Fairy]

52
9999
107
737
459
722
773
97
452
163
235
280
264
479
431
333
639
266
230
171
126
230
80
175
76
75
75
0

I will now roll some dice to generate new numbers. Ready?

43625
51250
4594
4164
6287
5612
10923
6635
2936
5791
3247
2481
1563
842
437
718
166
163
162
162
0

Maybe I'm not a good fairy. I really can't think of a hint that will help without giving it away though. Let's just say that the new number you get is a number that will make something special happen to the others if applied correctly.

[Chuck]

Maybe the others could suggest numbers to start with.

[The Hint Fairy]

Good idea, Chuck. The Tooth Fairy doesn't pull out a random tooth and give the child a quarter, so why should I?

Leave some numbers under your pillow for me! I'll give you a quarter!

[ctrlaltdel]

ok hint fairy could you start with:

100
10

[The Hint Fairy]

100
10
39
29
42
62
3
17
2
9
2
2
0

Or:

10
100
39
56
68
87
140
80
175
76
75
75
0

By the way, if you gave me numbers of the form 8n and 18n, then the next number would always be zero.

[ctrlaltdel]

can i have a question? i guess if i can't it just won't be answered...

ok here goes: why do you stop at 0? is it zero from then on always? or is it not defined?

[dethwing]

It's zero from then on, I asked Chuck.

------------------
Jesus loves you
Chuck1863: Throw it on the ceiling and scream for ice cream.

[Celt]

I am really enjoying this puzzle. I just had to register and get involved. I'm looking forward to the second part of the puzzle, if only I could get past the first.

Here are the parts of the sequences that I find most interesting. Maybe others will find them useful.

• 21, 6 -> 5
  
• 6, 5 -> 5
  
• 11, 21 -> 6
  
• 2, 21 -> 6
  
• 107, 6 -> 17
  
• 62, 3 -> 17
  
• 39, 14 -> 17
  
• 189, 153 -> 17
  
• X, X -> 0
  
• X, 0 -> 0 dethwing:"It's zero from then on, I asked Chuck".
  
• 25, 1 -> 0
  
• 12, 3 -> 0
  
• 153, 17 -> 0
  
• 300, 675 -> 0
  

From looking at the data, I have come up with the following two hypotheses. There wasn't a lot of corroborating data but I also found no data to refute them. Are they just red herrings? Maybe The Hint Fairy could post a sequence that refutes them.
1. (X+1), X -> X
Corroborating data:

• 76, 75 -> 75
  
• 9, 8 -> 8
  
• 6, 5 -> 5
  
• 163, 162 -> 162
  

1. X, Y -> Z => Y, X -> Z
Corroborating data:

• 100, 10 -> 39
  
• 10, 100 -> 39
  
• 2, 9 -> 2
  
• 9, 2 -> 2
  
• 5, 6 -> 5
  
• 6, 5 -> 5
  

My last observation is that as the sequences converge to zero (if they do), the number 4 never appears.(Red Herring see next post)
All those who believe in psychokinesis, raise my hand.

------------------
For a good prime, call:
29819592777931214269172453467810429868925511217482600306406141434158089



[This message has been edited by Celt (edited 03-15-2002 02:40 PM).]

[The Hint Fairy]

5
8
4
7
3
5
6
5
5
0

[ctrlaltdel]

i have a bunch of those on paper too, celt but likewise no unifying theory so far... i guess mathgrant really deserves his books

[Chuck]

Ok, one more hint, the sequence starting with 1 googol and 2 googol.

[Lepton]

Good grief, Chuck!

[ctrlaltdel]

are you sure about that 72nd digit of step 68?

edit: so, who needs a calculator?

[This message has been edited by ctrlaltdel (edited 03-16-2002 06:17 PM).]

[Chuck]

I'll probably need a calculator if I do the googolplex version.

[The Hint Fairy]

Don't even think of it. I'll just post another sequence. I chose these starting numbers because Pokemon numbers 54 and 58 are Psyduck and Growlithe -- 16 distinct letters, none repeated! Are there two Pokemon that don't share letters with more letters total?

54
58
107
122
58
20
4
16
0

Now, a random starting set:

8858236
4664130
7067679
4029786
2444733
5532137
3228021
3929852
2003411
244336
904780
153319
217419
102732
287307
39123
71361
2159
10655
2529
14
62
27
74
62
99
54
17
18
17
17
0

One more:

5282760
4445667
8747336
9577103
5589567
13514285
10270106
361710
2522099
843161
1621243
1026847
223740
44019
34979
41480
4639
21079
12937
19854
18522
16812
10548
17120
1916
3391
4852
1996
48
327
71
113
102
77
110
6
35
14
6
3
2
2
0

One more (can you figure out the logic behind these numbers? it has to do with Pokemon):

3121525
1920518
3269101
199429
649233
418028
334760
214224
535391
296688
388304
558428
590512
1115111 (weird!)
1242991
2327776
216015
376976
428871
118527
130496
148703
99484
198868
73776
177692
70976
128188
164032
193980
216476
217919
131028
264203
39203
10225
10234
10209
10065
15089
14456
3223
11063
3208
4255
4404
7436
6860
2796
4919
4260
6011
3260
6384
9119
14055
2504
5096
1200
4416
4599
5148
7427
4707
7145
3114
8874
8100
2916
0

Whoa! I didn't know numbers based on a Pokemon could generate such a long sequence, and such a beautiful ending! What? You wanted a quarter? You'll need to see The Quarter Fairy for that!

[ctrlaltdel]

8100 ->modChuck-> 2916 = 0???

insane...

does the fact that 8100 = 81x4x5^2
and 2916 = 81x4x3^2 (or 3^6x4) mean anything?

[This message has been edited by ctrlaltdel (edited 03-18-2002 04:18 AM).]

[cubestudent]

my mind has slowed too far with age.

i think if i were 14, i could get it too.

remarkable puzzle, chuck

[The Hint Fairy]

It does in this case. Can you find the logic behind thestarters for that sequence?

[Jen Aside]

Considering:

Dunsparce, Cyndaquil, Dragonite, Poliwrath, and Growlithe all have nine letters each,

and

Kabutops, Clefairy, Articuno, Quagsire, Sunflora, Cloyster, Squirtle, and Slowking all have eight letters each,

and

any two pairs of Pokémon listed above share at least one letter,

and

there are no other Pokémon with more than seven letters in its name (non-repeating),

Growlithe and Psyduck have the most non-repeating letters between them of any pair (although other pairs might tie--I haven't bothered to check).

Is the Hint Fairy a Pokémon fan, or is it just a ready puzzle source?

...this seems unrelated, but only invisibly so

[ctrlaltdel]

im still clueless, im messing around with dividing each number with all of its primes... am i on the right track even...?

[Chuck]

Factoring the numbers isn't necessary. I don't know if it will help. Doing so might cause you to notice things, I suppose.

[mathgrant]

Well duh! I'm a Pokemon fan, and I'm The Hint Fairy, so The Hint Fairy is a Pokemon fan!

------------------
Sometimes life gives you a deck without any aces. Deal with it.

[The Hint Fairy]

I looked at Nintendo's Official Pokedex, and found these 7's with no repeats:

Ivysaur
Metapod
Pidgeot
Spearow
Pikachu
Dugtrio
Persian
Psyduck
Golduck
Poliwag
Machoke
Haunter
Kingler
Chansey
Seaking
Starmie
Scyther
Omanyte
Snorlax
Moltres
Umbreon
Shuckle
Kingdra
Tyrogue
Miltank

How many was that? 25? We need a 7 and a 9 to tie with Psyduck and Growlithe. That's 125 possibilities.

[Jen Aside]

I feel so silly but here are all the non-repeating Pokémon names I found, in order from highest number of letters to lowest, then alphabetized by letter content... problem letters seem to be A (mostly A), O, R, and sometimes I.

code:

---

Dunsparce	ACDENPRSU


Cyndaquil	ACDILNQUY


Dragonite	ADEGINORT


Poliwrath	AHILOPRTW


Growlithe	EGHILORTW





Kabutops	ABKOPSTU


Clefairy	ACEFILRY


Articuno	ACINORTU


Quagsire	AEGIQRSU


Sunflora	AFLNORSU


Cloyster	CELORSTY


Squirtle	EILQRSTU


Slowking	GIKLNOSW





Machoke		ACEHKMO


Chansey		ACEHNSY


Pikachu		ACHIKPU


Kingdra		ADGIKNR


Metapod		ADEMOPT


Flareon		AEFLNOR


Seaking		AEGIKNS


Haunter		AEHNRTU


Starmie		AEIMRST


Persian		AEINPRS


Omanyte		AEMNOTY


Spearow		AEOPRSW


Poliwag		AGILOPW


Miltank		AIKLMNT


Ivysaur		AIRSUVY


Snorlax		ALNORSX


Umbreon		BEMNORU


Golduck		CDGKLOU


Psyduck		CDKPSUY


Shuckle		CEHKLSU


Scyther		CEHRSTY


Pidgeot		DEGIOPT


Dugtrio		DGIORTU


Kingler		EGIKLNR


Tyrogue		EGORTUY


Moltres		ELMORST





Ledyba		ABDELY


Crobat		ABCORT


Golbat		ABGLOT


Kabuto		ABKOTU


Raichu		ACHIRU


Machop		ACHMOP


Ledian		ADEILN


Zapdos		ADOPSZ


Fearow		AEFORW


Horsea		AEHORS


Mankey		AEKMNY


Slugma		AGLMSU


Gastly		AGLSTY


Raikou		AIKORU


Ponyta		ANOPTY


Tauros		AORSTU


Staryu		ARSTUY


Cubone		BCENOU


Swinub		BINSUW


Pineco		CEINOP


Scizor		CIORSZ


Pidgey		DEGIPY


Rhydon		DHNORY


Togepi		EGIOPT


Meowth		EHMOTW


Vulpix		ILPUVX





Magby		ABGMY


Arbok		ABKOR


Zubat		ABTUZ


Ekans		AEKNS


Lugia		AGILU


Aipom		AIMOP


Pichu		CHIPU


Golem		EGLMO


Hypno		HNOPY





Natu		ANTU


Xatu		ATUX


Onix		INOX


Jynx		JNXY





Mew		EMW

---

I came across a puzzle idea (or three, actually) in doing this... but I haven't decided whether to bother formally posting anything... but no, Psyduck is the only one that doesn't conflict with any of the top five...

[ctrlaltdel]

what do you mean it does not conflict?

it does conflict with Dunsparce, Cyndaquil, and Dragonite

[This message has been edited by ctrlaltdel (edited 03-20-2002 07:05 AM).]

[ctrlaltdel]

can the sequence be worked the other way??

if you know the last digits are:

5
6
5
5
0

can you work your way back up the list? going up to infinity?

[Chuck]

You wouldn't get a unique series by working backward. When working forward, different starting numbers can fall into the same ending sequence.

[The Hint Fairy]

I can't believe I missed Flareon!

[The Hint Fairy]

Speaking of missing stuff, you missed Muk.

[ChienFou]

The fact that you are losing data as the sequence progresss (ie you can't backtrack) suggests to me that finally you will have no data. Hence the series will always end with zero. Is this a valid postulate? Alternatively can you construct a starting position in which no data is ever lost? Then if you can, there will be a series which does not converge on zero.

I am of course no nearer to solving the problem, and I'm very tempted to break up my partnership with Chuck in "Design-a-Universe Corp" because anyone who is this malevolent can't design a benign universe

[Chuck]

I wouldn't consider zero to be no data. It's just that applying the rule to zero and any other number always results in zero. The sequence continues. It's just not very interesting any more.

I was thinking of replacing you with Mathgrant in the next universe design team anyway.

[ctrlaltdel]

5 chuck 8 = 4
5 chuck 6 = 5 (and also 6 chuck 5 = 5)
5 chuck 5 = 0
5 chuck 1 = 1

it can't be that hard!!

[Mendoza]

~newbie tiptoes in, clears throat~

Well, some days ago I noticed that

if a chuck b = c (which we know to be equivalent to b chuck a = c) with a>b then
1/if a-b is even, then c=2b-(((a-b)/2)-1)^2 (if positive); and
2/if a-b is odd then c=b-((a-b-1)/2))^2 (if positive).

Unfortunately as soon as a-b is large enough compared to b, these expressions give negative quantities and then I don't know what's going on anymore!! Clearly a by-product of the actual rule which is simpler and more elegant...

Like most people here I'm sure, I reckon that as a 14-year old I would have cracked this in minutes.

[The Hint Fairy]

This puzzle is too hard, obviously. We need a vacation!

Let's go on down to Squaresville!

[The Hint Fairy]

Infernal Series Error

[Jen Aside]

any = one

It does not conflict with at least ONE, STOP BEING PICKY!!!!!!!111

Muk, in my boredom, I seem to have skipped in subconscious realization of its inconsequence, on several levels. However, I like Mew, so Mew is on the list.