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I once told my brother about the "Look and Say Sequence" (i.e. 1, 11, 21, 1211, 111221, ...). My brother then showed the sequence to his maths teacher at school and asked him to predict the next number in the sequence. The teacher, baffled, eventually asked my brother what the answer was, and—when told—told him that it wasn't a "proper" mathematics problem.

But is it in some sense true that it is not a "proper" maths problem, and—if not—what should my brother have said in riposte?

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closed as off-topic by Magdiragdag, TMM, AlexR, Claude Leibovici, Asaf Karagila Jan 28 at 8:33

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is not about mathematics, within the scope defined in the help center." – Magdiragdag, TMM, AlexR, Claude Leibovici, Asaf Karagila
If this question can be reworded to fit the rules in the help center, please edit the question.

    
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There is a difference between saying (1) "Here are the first few terms of a sequence. I will not tell you the rules for generating the sequence. Can you guess it?", and saying (2) "Here I give you a well-defined rule which will generate a sequence. I can't find out if the sequence possesses or does not possess the following mathematical property. (...)". The question (1) feels a bit like a riddle. It is not exactly clear what was asked, except "Can you guess what I am thinking?". The question (2) feels like "proper math". –  Jeppe Stig Nielsen Jan 27 at 14:57
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What you do to get the next term in the "Look and Say" sequence is processing a string of digits (the current term), keep count of successive occurences of the current digit, and save the results in a new string. That sounds like a CompSci problem. I refer you to math.stackexchange.com/questions/649408/… to decide whether that's part of mathematics or not. In any case, your brother's teacher could have taken the opportunity to start an interesting discussion with his student instead of being so dismissive. –  Jubobs Jan 27 at 15:36
    
I'd say it's not a "proper" maths problem because it deals with the representations of a number (digits individually), not the actual number itself. However, it's so close to run-length encoding that it'd be hard to argue it's not relevant to the purest end of CompSci. –  cloudfeet Jan 27 at 16:26
    
There was also a related discussion on meta: Guess the next number/guess the relation etc –  Martin Sleziak Jan 27 at 16:28

9 Answers 9

"Guess the next term" is very often not a proper mathematics problem, and I would agree with your teacher that it is not a proper mathematics problem in this case (but a diverting puzzle among friends).

However, the sequence has surprising properties about the limit proportions of 1s and 2s that would be "true" mathematics problems for any mathematician and I invite you to read up on this on the internet.

The instinctive and understandable disdain to any "guess the next term"-question is often clouding the judgment of people on the general interest of such a sequence.

Yet, calling the opinion of the teacher "belittling", your brother "bored" and asking for a "riposte" does not make it entirely believable that your brother asked this question out of enthusiasm. It is totally appropriate to call the guess the next term- question not a real maths problem.

I have some empathy for this situation, because my brother had a teacher who would not believe that the harmonic series diverges and we could not convince him otherwise, but your teacher did not say anything mathematically wrong at all, it is understandable that he did not like this puzzle and your entitled demands for "praise" are really not appropriate.

You and your brother are very privileged to be students in the times of the internet, why do you need your teacher to learn nice mathematics?

If your brother asked his teacher this puzzle to show him up then he should really, really let it go now. If he asked your teacher this puzzle to share his enthusiasm he might still try with the nicer properties of the sequence, but much better to find someone else to share his enthusiasm if an honest, but negative opinion seems belittling to your brother.

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Actually we weren't privileged with the internet, this was many years ago now. Honestly I don't know if my brother meant to show the teacher up or not. I certainly don't believe my brother "deserved" any praise for mathematical ability or effort, but I do believe that teachers should encourage, rather than discourage, any sign of interest or enthusiasm in a subject which many people are turned off by. And I think that a concern with protecting yourself and your reputation, rather than showing interest in a genuinely interesting puzzle, is (in my opinion) a sign of a poor teacher. –  carlu Jan 27 at 10:18
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So you weren't looking for an actual answer; you just wanted to find people to agree with your low opinion of the teacher. In that case, I think the correct response is, "Get over it already." –  Blazemonger Jan 27 at 21:54

There are some interesting patterns, so I would say that it is proper mathematics.

http://en.wikipedia.org/wiki/Look-and-say_sequence

http://www.ams.org/journals/era/1997-03-11/S1079-6762-97-00026-7/home.html

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According to Ian Stewart's Galois Theory, Conway proved in 1985 that if L(n) is the length of the n'th term in this sequence, L(n) satisfies a 72-term recurrence relation, which implies that L(n) is proportional to $\alpha^n$ (for large $n$) where $\alpha \cong 1.303577$ is the smallest solution of a polynomial with integer coefficients of degre 71. He gives this example as a 'natural' problem where one is interested in solutions of polynomials of large degree.

So there are certainly aspects about this sequence that every mathematician would enjoy and appreciate.

Also, I would like to add that even though the question "what is the next term in this sequence" is one that most people would not consider a well defined mathematical question, most research questions behave like this. (What's the next theorem in this sequence of theorems where each one is a generalisation of the previous one?)

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I'd say the teacher is both right and wrong. It is not a “proper” math problem because such problems rarely, if ever, have a unique solution. But as brain teasers, where the goal is to find the simplest possible formula or recipe for generating the sequence, such problems can lead to much fun and interesting mathematics, all of it quite “proper”. As a teaching tool in the right hands, I am sure such problems can be used to great effect.

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If the teacher's objection to the sequence was that the process of counting and quoting numbers is 'not real math', this is completely wrong. Counting is fundamental to math - there is nothing about the definition of this sequence that makes it less basic, mathematical or rigorous than say, the Fibonacci sequence.

If his objection was that the presentation of the sequence leads one to believe it refers to the integers eleven, twenty-one, etc., rather than "1,1", "2,1"..., this is valid. But this isn't the objection he made - he should have said that commas were missing between the digits, or that the sequence was actually a sequence of finite sequences, and should have been presented as such. However note that firstly the presentation of this sequence is standard, and used by no less an authority than the OEIS, and secondly that since two-digit numbers never occur, there is no ambiguity.

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Well, i imagine that the teacher's objection was that it was an ill-posed problem: given an answer, how do you know it is the correct one? There are many ways to extend the sequence, for example with zeros, or by finding some kind of a recurrence relation. It was not stated how the sequence had to be continued. –  Alexey Jan 27 at 19:12
    
More importantly: in mathematics (fortunately) there is no authority that counts, surely not OEIS. To prove that the question was well posed, one needs to reformulate it in an acceptable way, prove that the provided algorithm was the unique solution, and explain how it would be possible to guess the complete question from the original incomplete one. –  Alexey Jan 27 at 19:22
    
Note that I didn't cite the OEIS as an authority on a question of mathematical truth, although surely it is authoritative on many of those, nor on the question of what is important mathematics, but only on a question of what is acceptable mathematical notation. There are (many) authoritative sources for this, and OEIS is one of them. –  jwg Jan 27 at 20:36
    
Ah, ok, i didn't read carefully, sorry. –  Alexey Jan 27 at 20:43

The sequence is of the kind $x_{n+1}=f(x_n)$, where $f$ is most easily described by referring to the decimal digit representation of $x_n$. While such relying on the fact that we have ten fingers and not on the "proper" numbers themselves may be frowned upon as did your teacher, I don't think it is always justified. For one, the well-known divisibility rules using digit sums would then be improper as well, which sounds absurd. Also, the problem requires thinking out of the box, which is a major requirement in many math problems.

To amuse you with another digit related problem, fill the gap: $$ 10000, 121, 100, ???, 24, 22, 20, 17, 16, 15, 14, 13, 12, 11, 10$$

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You don't have to interpret each term as an integer $-$ you can interpret it as just a sequence of digits. So it doesn't rely on the decimal representation (and in fact, no digit greater than $3$ is ever needed, so we don't even need to invent special symbols for digits greater than $9$). –  TonyK Jan 27 at 10:05
    
I'm pretty curious about the sequence you provided, can you give a hint? –  Ivo Beckers Jan 27 at 12:02
    
Ok, I already found the solution. Google was my friend –  Ivo Beckers Jan 27 at 12:27

This sequence is proper enough mathematics to appear in the Online Encyclopedia of Integer Sequences, with many references to the literature.

http://oeis.org/search?q=1%2C+11%2C+21%2C+1211%2C+111221%2C&language=english&go=Search

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You could define the sequence completion problems in terms of “What continuation would be produced by the smallest (typed) λ-term¹ that produces $a_n$ on input $n$ for all the provided initial terms of the sequence?”.

Then it becomes a “proper math” problem.

Though, I'm not sure how reasonably this definition behave on basic sequence riddles. For example, with the provided example, the code for polynomial interpolation might be of smaller size than the code that produces the expected answer. But add some (many?) more initial terms and it should work out (…don't ask me to prove it!).


1. you could use any reasonably simple programming language

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As a mathematical problem it has two weaknesses:

(a) it relies for its difficulty on a notational trick; it uses a sequence of symbols (digits) that normally represents a decimal number, but in this instance represents a sequence of number pairs.

(b) for any finite sequence of numbers S, there's an infinite number of formulas that will generate the sequence S, and for each such formula the next number in the sequence is different. Solving it therefore relies on some non-mathematical notion of which of these formulas is interesting or elegant enough to be the one intended.

Both of these weaknesses are common in "mathematical" puzzles, and indeed in exam questions set for schoolchildren, but together they mean that the problem is not one of significant mathematical interest.

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Can you give a proof for (b)? –  sjy Jan 28 at 2:09
    
As stated, I think it contains a contradiction: if you take a finite sequence $S_n$ and extend it by one to get $S_{n+1}$, you still have a finite sequence, and by the first part of (b) there are infinitely many formulae that generate $S_{n+1}$. Each of those also generates $S_n$, so it is false that "for each formula which generates $S_n$, the next number in the sequence is different." –  sjy Jan 28 at 2:15

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