An Easy, Fun Number Sequence

[Chuck]

The first two numbers in this list were chosen at random, sort of. Each of the others was derived by applying a simple rule to the previous two numbers above it.

1. What is the rule?

2. Will repeated application of the rule always lead to zero eventually, no matter what two numbers start the list?

1914
1939
1770
2126
3299
1770
2174
2459
522
2174
603
2186
254
219
401
203
178
34
123
86
174
80
231
255
341
519
579
197
494
597
69
389
272
183
47
137
39
14
17
13
25
1
0

[Chuck]

This is obviously too hard so I'll go back to offering prizes. The first person to post the rule in this thread can have any three of the following prizes. Of course, the winner will have to provide me with a postal address to which I can mail them.

• Chess Trivia by Peter Hotten and Herbert A Kenny
  
• Cosmic Adventure and Other Secrets Beyond The Night Sky by Bob Berman
  
• The Emperor's New Mind by Roger Penrose
  
• Extraterrestrial Intelligence by Jean Heidmann
  
• The Great Book of Teasers and Mind Puzzlers by George J Summers
  
• Great Essays in Science edited by Martin Gardner
  
• The Handy Science Answer Book compiled by the Science and Technology Department of the Carnegie Library of Pittsburgh
  
• Immortals, Eight Timeless Tales of Life Everlasting edited by Jack Dann and Gardner Dozois
  
• The Infinity Link by Jeffrey A Carver
  
• Mathematics and Chess by Miodrag Petkovic
  
• StarStrike by W Michael Gear
  
• Time Journeys by Paul Halpern
  
• Uncle Petros and Goldbach's Conjecture by Apostolos Doxiadis
  
• What Einstein Didn't Know, Scientific Answers to Everyday Questions by Robert L Wolke
  
• AMISTAD on VHS
  
• THE ARRIVAL on DVD
  
• FARSCAPE The first six episodes on VHS
  
• FINAL DESTINATION on VHS
  
• PAST PERFECT on VHS
  
• THE STUFF on VHS
  
• SUPERNOVA on VHS
  

[This message has been edited by Chuck (edited 03-16-2002 06:49 PM).]

[Chuck]

Ok, four prizes.

[pikachamp]

i';ll take 2 if i can have a hint

[This message has been edited by pikachamp (edited 03-05-2002 08:24 PM).]

[ChienFou]

Does it help to know the factors of 1770 are, 2, 3, 5, 59?

[Chuck]

I don't think that's any help.

[ChienFou]

I have tried myriads of mathematical functions with no success. I'm wondering whether there is a letter based algorithm involved "one nine one four" etc

[Chuck]

Just math. No letters.

[ctrlaltdel]

chienfou im with you on this one. now im thinking somwhere along the lines of simply disecting the numbers into individual numerals and playing around with them, or checking for parity, taking roots and scratching the decimals etc. etc. but still no luck...

[Lepton]

Calling a number sequence like this "easy" is why Chuck has gained a reputation for evilness. I have tried similar approaches to no avail.

[Chuck]

I think it's easier than previous evil.

Five prizes.

[ChienFou]

~ throws singularity at chuck ~

[mole]

The first part looks interesting, since there are some numbers that appear more than once...

1914, 1939, 1770, 2126, 3299, 1770, 2174, 2459, 522, 2174, 603, 2186

code:

---

1914 • 1939 -> 1770


2126 • 3299 -> 1770





1939 • 1770 -> 2126


3299 • 1770 -> 2174





1770 • 2126 -> 3299


1770 • 2174 -> 2459





3299 • 1770 -> 2174


2459 • 522  -> 2174





1770 • 2174 -> 2459


522  • 2174 -> 603





2174 • 2459 -> 522


2174 • 603  -> 2186

---

-looks for a pattern-

[ctrlaltdel]

yeah i know, if the first two (1914 • 1939 -> 1770 and 2126 • 3299 -> 1770) don't give it away for me, then i dont think theres any reason for me to look at any other couples...

edit: on second thought, maybe the number of each step is relevant, like 1 in first operation, 2 in second etc... in which case there might be no direct connection between the operation that gave the first 1770 and the operation that gave the second 1770... chuck IS evil...

[This message has been edited by ctrlaltdel (edited 03-08-2002 03:07 AM).]

[LoopyDog]

Do we need a calculator to solve this, or just a pencil and paper? If Chuck is as evil as I think he is it should just be pencil and paper.

[Chuck]

No one ever needs a calculator. They do speed things up, though.

[Lepton]

I've tried, with moderate success, reversing the order of the digits of the different of the two factors. I don't know if this is the right track or not; and if it is, how to incorporate this into the rest of it.

[Chuck]

I don't think messing with the individual digits will be productive.

[LoopyDog]

I would expect to use a calculator if I were having to take square roots, (co)sines or logs. I am happy to use pencil and paper for multiplication and division. I take it that Chuck implies I could use pencil and paper. btw, Chuck admitted in IM that he was *prepared* to upgrade this to moderate difficulty

[Lepton]

Is the fact that 1914 and 1939 are dates (start of the two world wars) significant?

[CzarJ]

Well, I ain't too good at sequences, but I doubt it does, seeing as how he's implied that you can start with pretty much any two numbers. If you start with two random numbers, than getting two related dates is probably just a coinkydink.

------------------
Unslumping yourself is not easily done.

[Chuck]

It's pure coincidence that they're world war dates. Any two starting numbers will do, such as:

1969
1492
3379
1443
633
1394
681
638
197
370
529
366
14
83
6
14
3
6
2
3
2
2
0

[mathgrant]

I don't know. I want to know so I can try it with random numbers. I probably wouldn't get the przes anyway. . .

IF you e-mail me, THEN I promise I won't tell.

PI DAY in 4 days!

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

[Chuck]

It would be too cruel to give you the answer and deprive you of the opportunity to solve it.

[mathgrant]

See? You didn't e-mail me. Since "If A then B" is always true when A is false, I already kept my promise!!! I told you I could be trusted!

[Chuck]

I think "If A then B" is irrelevent, not true, when A is false. Logic sure has a lot of stupid propositions.

[mathgrant]

Never mind, I know the rule!!!! I'm only 14, so I don't know if I can be allowed to post the rule here. What should I do?

What I will do right now, though, is try the rule for a couple numbers. Is this correct?

3141
2718
4997
2621
7213
169
181
313
9
8
8
0

?

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

[Chuck]

I'm convinced that you know the rule. I guess you shouldn't post it if you can't accept the prizes. Maybe someone else will solve it.

[mathgrant]

I didn't mean to imply that I couldn't accept them. I meant that I didn't know if I could accept them. But even if I were allowed to accept the prizes, I'd keep the rule a secret. I think it's fun!

I'd like to mention that a-Chuck's operation-b = b-Chuck's operation-a. I'ts commutitive. The Fibonacci sequence is commutative because addition, the operation to get from one pair of terms to the next term, is commutative.

I'll try to see if I have permissions. Either way I'll be glad knowing that I'm the first person to solve your diabolical puzzle and have the guts to post about it!

[edited because of error in list, which was removed]

[This message has been edited by mathgrant (edited 03-11-2002 01:21 AM).]

[Chuck]

Ok, you can have the prizes if you can email me an address I can send them to. I guess it would be more fun to keep the rule a secret.

What about part 2 of the puzzle? Does every sequence reach zero?

[Chuck]

You made an error in your sequence. 8172 and 632 produce 2720, not 2270.

[mathgrant]

Thanks. I removed the list. The next few numbers may have used that typoed number, so I removed the list altogether. I knew the rule, don't worry. It was an "honets tpyo". I'll try something a little shorter:

12
34
8
16
7
2
5
1
1
0

As for Part 2, I have no idea. It looks hard, if not impossible to prove. Perhaps he knows a counter example.

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

[This message has been edited by mathgrant (edited 03-11-2002 01:26 AM).]

[Chuck]

I've deleted Mathgrant's prize choices from the list. The next solver, if any, can have any two prizes. Mathgrant can continue to give hints to and/or inflict tortures upon the rest of you.

[Chuck]

Starting with 12 and 34 actually produces:

12
34
8
16
7
12
3
0

but your example starting with 3141 and 2718 is correct so I'm accepting that as proof.

[mathgrant]

I am working on programming my calculator to do this. That will avoid any errors. I don't know how that happened. I'll share my program for TI-83 when someone posts the rule.

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

[This message has been edited by mathgrant (edited 03-11-2002 11:59 AM).]

[The Hint Fairy]

I programmed it. Here's what it says for another set of starters:

1234
12345
321
2645
804
816
1583
1232
1440
2519
2144
4407
5279
3224
3871
5080
4324
7324
5847
12579
1637
7454
1149
3170
1866
596
1220
1216
2431
1135
264
431
215
249
174
62
179
73
71
142
82
195
114
29
57
53
105
89
129
32
32
0

Let's try:
11
1111
121
475
354
50
11
21
6
5
5
0

One more:
987654321
3
11474
197
1369
332
232
295
319
469
615
66
189
153
17
0

I'm probably giving away a lot by now. But that's why I'm the Hint Fairy!

[mole]

Thanks, but it's still not enough for me to work it out yet (i'm not very good at this sort of thing)

I have made another step, though. All the operations that resulted in 0 had the second number as a factor of the first...

Still working on it,
mole.

[This message has been edited by mole (edited 03-12-2002 05:57 AM).]

[Chuck]

Oh, all right. Here's another example:

183
238
290
376
140
199
304
471
300
675
0

[The Hint Fairy]

Leave an unsolved except by a "moron" named mathgrant under your pillow, and get a hint!

149
149149
1405
28720
3696
17916
6767
17472
10895
231
1349
255
599
645
714
846
315
234
269
446
258
147
290
194
91
230
194
99
162
162
0

[The Hint Fairy]

One more. . .

1987
526
678
219
257
114
57
98
110
171
41
122
102
123
2
21
6
5
5
0