#### Challenge "A really long number sequence" ¶

By: admin on Sept. 5, 2011, 10:38 p.m.

This is already a really long number sequence. Can you find the 10001st element?

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By: admin on Sept. 5, 2011, 10:38 p.m.

This is already a really long number sequence. Can you find the 10001st element?

Read more...

Last edited by: admin on Oct. 31, 2021, 2:54 a.m., edited 1 time in total.

By: fretty on Sept. 6, 2011, 11:06 a.m.

I wouldn't even know what this sequence was representing if I wasn't a member of the same online maths forum as the author of the challenge (and so hadn't seen this sequence on the same forum a few months ago).

A sequence has to have context with it sometimes!

I am assuming that there is a simple formula for the nth term?

By: Veselovský on Sept. 6, 2011, 10:09 p.m.

This is a "real-life" sequence… This means that it was not constructed by author artificially but instead sequence appeared naturally when author was solving some other problem.

You do not have to know anything about its source or if it is representing something or not (same as in part 1 or 2 of number sequence)

Just find logical explanation why these numbers follow one after another.

As two people already solved it I think there is no need for more hints.

By: h3po on Sept. 9, 2011, 2:12 p.m.

Nice one! It took a whole night, 5 cups of coffee, 1 bottle of coke and 3 beer, but i solved it [HTML_REMOVED]

The neat thing about this sequence is that you can start without any knowledge on how to solve these things, make your first observations, dig deeper and deeper and try to make sense of the numbers you see. I learned a lot in the progress.

By: Veselovský on Sept. 9, 2011, 2:49 p.m.

Thanks :-)

hmm…. interesting sequence… 5 cups, 1 bottle, 3 beers … we should make another "sequence" challenge :-D

Since there were some requests for the source of the sequence, I want to assure everybody that knowing the source of the sequence is of no advantage compared to not knowing anything about it.

I am inclined to think that it is even a disadvantage, because the problem the author was solving was much more difficult than just the sequence, which was just one part of it.

So if you think you would be able to solve the sequence if you know the source of it then I highly doubt about it.

But if you want to know it anyway - the source of sequence was the series expansion of Dottie number

By: fretty on Sept. 15, 2011, 10:43 a.m.

I haven't solved it yet…but really haven't worked on it for a while. This is quite embarassing for me since I am a mathematician lol.

I have found a recurring pattern (which the author says is not the right pattern to spot) but apart from that I have nothing.

I don't see why this or any of the other number sequence puzzles are here…they have nothing to do with cryptanalysis (or even cryptography).

By: Veselovský on Sept. 16, 2011, 2:57 p.m.

There is always more than one way to solve any number sequence.

Take for example the well studied Fibonacci number sequence.

There are plenty of different formulas to generate the numbers in this sequence.

…recurrence formulas, close-form formulas (depending only on "n"), generating function formulas… and many others

So there is not only one "right" pattern to spot and I never said that some pattern is the "right" and some other "wrong".

One of successful solvers solved it by spotting a different pattern than the pattern I used to solve it.

So if you discovered for example a recurrence relation, then use it to generate the desired numbers instead of trying to find a close-form formula, because finding a close-form formula may be more difficult than using a recurrence relation (but also may be not).

The question why number sequences on this website is more appropriate for people who created this website so I will not answer it, although I have my own explanation why…

Anyway, there is plenty of different challenges from which everyone can choose those that satisfy his/her requirements.

If you have questions on how to provide solution or want to verify your solution before submitting it to the website or just need a small hint feel free to send me a PM.

But I am not giving hints like "do this, then do that and then you get the solution".

By: fretty on Sept. 16, 2011, 3:48 p.m.

Well I havent really tried much with it, but the best way is probably to use a recurrence relation…otherwise so many people wouldn't have solved it in such a short space of time.

What I am asking is, what do you get from solving this challenge or any of the other number sequence challenges? Finding the most likely pattern to a sequence of numbers is just a mathematical curiosity, nothing much to do with cryptanalysis.

I can easily generate some tough sequence of numbers by using some complicated method and say "here find the next few terms"…but there comes a point where it becomes worthless to the person trying to solve it, the gain is much less than the amount of work needed in solving (once I solve this challenge there is a big possibility that I will just think "ok so i'm done, the next 100 number are…so what?").

I am not saying these challenges are useless, I just fail to see where they fit in to the aim of the site (similarly with the Diophantine one, since this was also just a standard maths question…nothing really to do with cryptanalysis).

By: Veselovský on Sept. 16, 2011, 4:13 p.m.

You can use this sequence to shift letters of plaintext to produce a ciphertext if you wish. If you are not able to find out what is the next number you will not be able to decrypt the ciphertext.

Similarly as you can generate a tough sequence you can also generate a tough cipher (totally independent on any number sequence) which nobody would be able to decode.

The given sequence is not so tough since it is easily to see a pattern in it (which also confirms the number of people who already solve it)… and that is why it is in level II because it is not as though as you are describing it.

By: fretty on Sept. 16, 2011, 4:25 p.m.

I am not saying that it is tough, I am saying that sequences can be made tougher and tougher but in the end where is the enjoyment and gain in solving them?

Yes you "can" use a sequence of numbers to make a cipher, just like you "can" use Diophantine equations to make ciphers…but you "aren't" actually using them for this purpose, which makes the challenge a "brainteaser" or "maths puzzle" rather than a challenge in cryptanalysis.

There is a difference between decrypt and break.

If you were attempting to break a cipher using this method, you would have to know enough of the sequence to find its pattern anyway…and that would require lots of known plaintext! Basically this is of identical security as the OTP (knowing that there IS a pattern to the shifts does not tell you much without knowing some of the actual numbers).

By: Veselovský on Sept. 16, 2011, 4:36 p.m.

I do not see a much sense in continuing this discussion, everybody can choose a challenge according to his/her desires, if someone do not like or do not enjoy solving particular one, then there is no need to solve it.

By: fretty on Sept. 16, 2011, 4:46 p.m.

It is not a question of liking challenges, it is more a question of how they fit into the bigger picture.

I can make a challenge about flying pigs if I wish, one that has some very weak significance in crypto/cryptanalysis…but if I did make this, people would start asking "why is this even on the site, it isnt a crypto/cryptanalysis challenge?"

The mere fact that it "can" be used in cryptography doesn't make the challenge itself a crypto challenge.

I don't know if anyone else agrees but in my opinion, "solve this sequence" and "solve this diophantine equation" are not real problems in crypto/cryptanalysis, they are just straight forward mathematical problems (again, I know that they "can" be used to make up some kind of encryption algorithm, but they arent being used in this way).

By: fretty on Sept. 16, 2011, 4:49 p.m.

It is just like me making a challenge that simply asks:

"Complete this saying: A stitch in time saves ____."

and in my defence saying:

"Well this is a crypto challenge because if you wanted to you could use the answer as a number in a caesar shift."

But where is the actual cipher that we are using the answer on? Without it we have no crypto content!

By: be on Sept. 22, 2011, 1:05 a.m.

But where is the actual cipher that we are using the answer on? Without it we have no crypto content!

I interestingly followed this discussion as we had similar ones when the three universities decided to build such a contest.

The reason to add number sequences wasn't because they could be used in OTP schemes. The ideas were manifold:

Therefore some challenge like "calculating modulo" are accompanied with easy-to-understand explanations, and some just promote mathematical or structured thinking like number sequences (which one does this well can be discussed). As math is the basis for modern crypto algorithms understanding it and applying some mathematical "tricks" (like for the Bleichenbacher attack) is also possible as a challenge, but this is not the majority.

So we still learn, and your discussions help us to improve. Thanks for your active participation.

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