The best I can do now satisfies the first 4 or 5, that it is alternating pairs of primes... ************************************************* 11-07-02, 07:47 PM silenteuphony The first six terms are the first six primes, in base 4 notation, but the pattern breaks down after that. (However, if you assume each digit represents a power of 4 instead of 10, then every term still represents a prime number, even though this isn't strict base 4 notation.)
11-07-02, 08:55 PM WiteoutKing yet "43" is missing...
The 10 terms yield 9 differences. For example, the first difference is 3 - 2 = 1, and the second difference is 11 - 3 = 8. All of the 9 differences are as follows:
1 8 2 10 8 10 18 8 34
Notice that the number "8" repeats. Hence we know that 2 terms beyond, there will be the number "8" again. Notice also that between the repeating number "8," are 2 terms whose differences are 8. For example, 10 - 2 = 8, and 18 - 10 = 8. (That is, the fourth term minus the third term, and the seventh term minus the sixth term, both differences yield 8.) Now, since the difference between the next missing term and 34 must be 8, then the next missing term must be 42. In short, we are expecting the following sequence:
1 8 2 10 8 10 18 8 34 42 8
Now, how do we get this sequence? Following is the sequence rewritten with the repeating number "8" in bold so that we can examine the sequence more readily.
1 8 2 10 8 10 18 8 34 42 8
Notice that 8 + 2 = 10, 8 + 10 = 18, and 8 + 34 = 42. Generally, any bold number "8" plus the first term to the right of the bold number "8" must equal the second term to the right of the bold number "8."
I leave you mathematicians to the task of writing any applicable formula.
[This message was edited by tsaeb on 11-07-02 at 10:42 PM.]
11-07-02, 10:59 PM tsaeb Here is a trick question for you fellows, a belated Halloween present from tsaeb.
What is the next number in sequence, given 4, 14, 23, 34, 42, 50, 59, 72, 81, . . ?
Why, it has to be the next stop on the C train in Manhattan, New York. Loved it, I know.
Answer: 86.
P.S. This is what you get for taking too long to solve a simple sequence problem.
11-08-02, 06:49 AM WiteoutKing So we have to find a formula to change 10 to 10 ( smile ) and 18 to 34.
What is the next number in sequence, given 4, 14, 23, 34, 42, 50, 59, 72, 81, . . ?
Why, it has to be the next stop on the C train in Manhattan, New York. Loved it, I know.
Answer: 86.
Actually, I think it's the A line train smile
Sequence: 4,14,23,34,42,50,59,72,81,86,96,103,110,116,125,135,145,155, 163,168,175,181,190,200,207 Name: Local stops on New York City A line subway. Comments: These are the numbered stops for the A train, as of Feb. 2000.
11-08-02, 08:21 AM FlyingHellfish Going back to the original sequence, it is quite obscure, but it does only involve simple algebra to figure. Would you like a hint yet?
11-08-02, 03:03 PM Pin~Jinx ....... it is the Prime#s, but, displaying two & skipping two.
For instance: 2, 3, 5, 7, 11, 13, 17, 19, 23....
11-08-02, 03:08 PM tsaeb Flying Hellfish: These days that A train makes only express stops in Manhattan. As for my "solution," I simply reasoned the way that I imagined a student who may have made up the problem would reason. I am glad that you claim that some of them know algebra. We await the unveiling.
Witeout King: Looking formidable, a bigger threat than tsaeb.
11-08-02, 04:10 PM WiteoutKing Just realized that 1 to 2 doesnt work though...
Let's try cubics...
2, 3, 11, 13, 23, 31, 41, 59, 67, 101
1 8 2 10 8 10 18 8 34 42 8
ax^2 + bx + c = y a(1^2) + b(1) + c = 2 a + b + c = 2
a(10^2) + b(10) + c = 10 100a + 10b + c = 10
This means a+b+8 = 100a+10b So 99a + 9b = 8
a(18^2) + b(18) + c = 34 324a + 18b + c = 34
So 323a + 17b = 32 323a + 17b = a + b + 32 323a + 17b = -17a - 17b + 68 306a = 68 a = 17/79
99(17/79) + 9b = 8 1683/79 + 9b = 632/79 1683/869 + b = 632/869 b = -1051/869
17/79 - 1051/869 + c = 2 187/869 -1051/869 + c = 1738/869 c = (1738 + 1051 - 187)/869 c = 2602/869
Let X = first term to right of each number "8." Let Y = second term to right of each number "8." Then, X = Y - 8, which X we can get from this formula by first finding each Y. So what is the formula for each successive Y?
Let Ysub2 be the next successive Y, and let Ysub1 be the present Y. Then, the formula is Ysub2 = 2Ysub1 - 2.
For example, given the X,Y pair of 2,10, find the next X,Y pair, which we already know to be 10,18 as follows.
Ysub2 = 2Ysub1 - 2 Ysub2 = (2)(10) - 2 Ysub2 = 18
X = Ysub2 - 8 X = 18 - 8 X = 10
Hence the next successive X,Y pair between the numbers "8" is 10,18, as we knew but sought to show.
Can we find the X,Y pair 34,42?
Ysub2 = 2Ysub1 - 2 Ysub2 = (2)(18) - 2 = 36 - 2 = 34 Hooray! Wait a minute! We have the next successive X term, not the next successive Y term! All right, how are we going to find the next successive Y term?
Since X = Y - 8, or Y = X + 8, Ysub2 = Xsub2 + 8 Ysub2 = 34 + 8 Ysub2 = 42.
Summarizing, we find the first next successive X,Y pair by using these equations to yield first Ysub2 then Xsub2 (the Ysub2 will be the Ysub2):
Ysub2 = 2Ysub1 - 2 and X = Ysub2 - 8.
However, we find the second next successive X,Y pair by using these equations to yield first Xsub2 then Ysub2 (the Ysub2 will be the Xsub2 in fact and in the formulas):
Xsub2 = 2Ysub1 - 2 and Y = Xsub2 + 8. Remember that in this second next successive X,Y pair, we will be getting the Xsub2 first instead of the Ysub2.
Following would be some more X,Y pairs, using this reasoning.
Okay, mathematicians, what is wrong (because this work does not jive with Witeout King's formulas), but it surely sounds good?! Did I go wrong by assuming the number "42," which may only be 8 more than the given number "34" by coincidence, or is there no coincidence such that the assumption of "42" is correct? I think that by switching the finding of Ysub2 to a finding of Xsub2, we substantiate the assumption of the number "42." Yet, what's with the number "1" in the beginning?
[This message was edited by tsaeb on 11-08-02 at 05:10 PM.]
This message has been edited. Last edited by: DorianGreyed,
Posts: 1363 | Location: Lowell, MA, USA | Registered: 06-03-02
Now that I know where to look, it all falls into place. The next three terms are 103, 113, and 131. I won't reveal the pattern yet, so other people have a chance to see the hint and figure it out.
If you need help translating FH's hint, check out my Math HTML page (scroll down to the Greek letters). Also, this list of primes may be helpful.
Posts: 265 | Location: Denver, Colorado, USA | Registered: 06-04-02
I would like to submit this sequence to The On-Line Encyclopedia of Integer Sequences, but I don't want to steal the credit. Do you know the original source, so I can give them credit? ************************************************** 11-08-02, 09:11 PM FlyingHellfish SilentEuphony has got it! And in far fewer days than it took me Lol. However, I will still keep the rule(s) a secret for those of you who want to keep thinking about it.
Tsaeb, however, has brought up an important issue. I cannot mathematically find fault in her method, although it is definitely much different from what I was looking for. The question, "Which is a more valid answer?" is not answerable. Both methods are equally valid, but the answer SilentEuphony came up with was the one I was looking for. This is the reason why sequences are no longer included on many standardized tests such as the SAT-there can be many valid answers for any particular sequence, so how does a supposedly objective grader determine which is the best?
SilentEuphony-
With regards to the originator of the sequence, all I know is that a student that I tutor came to me with it from his teacher. Where she got it from is unknown to me.
11-08-02, 09:29 PM tsaeb FlyingHellfish: I am not satisfied with my answers being different from those of silenteuphony, which you are implying are correct. My endeavor here is only to play the devil's advocate's role to stimulate a different mode of thinking. As I hinted, the number "1" does not fit into my way of seeing things, which is one reason why the scheme which I presented seems to be incorrect.
11-12-02, 12:47 PM FlyingHellfish Tsaeb,
I'm sorry for not being clearer in my previous post. The point that I was trying to make is that although SilentEuphony came up with the answer that I was thinking of, it doesn't mean that there aren't other answers that are just as defendable as the one's he gave. True, your method did not result in an answer yet, but that doesn't mean that SE's method is sounder than yours-it just happened to be the one I was thinking of. It is possible to mathematically justify any number as being the correct next number in the sequence-it's just a matter how the problem is approached.
11-12-02, 12:49 PM FlyingHellfish Does anyone else want to have a stab at determining the pattern of the sequence? Unless someone says otherwise, I think that SilentEuphony should post his answer tomorrow (or whenever he gets a chance to) since he is the one to get answer "right".
11-14-02, 05:19 PM silenteuphony This sequence is a subsequence of the prime numbers; it consists of those primes whose squares have square digit-sums. That is, if you square a term, then add up the digits of the resulting number, you get another square number.
Of course, this is just one solution, but I believe it is the simplest solution that generates the given terms and continues to generate definite terms for the sequence (i.e. there is a specific rule which determines a unique number for the nth term, given any n > 1).
All other things being equal, I suppose the simplest pattern for a sequence is the best solution, but then you have to figure out what simple means.
Here's another fun one: Find the next number in this sequence:
1, 2, 4, ...
1st answer: 8 (powers of 2)
2nd answer: 7 (difference increases by 1 each time)
3rd answer: 6 (prime numbers minus 1)
So which answer is right? I like the third one, because it's the least predictable, but in this case, I think it's a matter of personal preference.
This message has been edited. Last edited by: DorianGreyed,
Posts: 265 | Location: Denver, Colorado, USA | Registered: 06-04-02
