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On Friday I posted this puzzle.…..what number comes next in the sequence?

1    4    7    11    15    19

Clue: It is less than 19.

If you have not tried to solve it, have a go now.  For everyone else the answer is after the break….

Well, the word ONE has 3 letters, FOUR has 4 letters, SEVEN has 5 letters and so on…..so the answer is SEVENTEEN.

Did you solve it?  Any other answers?

1. Roland says:

Twenty one

1. M says:

that is not lower than 19

2. Ian says:

Given that the values in the question aren’t minimal (http://oeis.org/A001166) isn’t the “clue” really a constraint?

1. Mick says:

Yes, I think so too.

3. Considered that but then why isn’t the 6th term thirteen, the first number consisting of 8 letters?

1. tort says:

because it doesn’t have to be the lowest number, just a number with the correct number of letters. That’s why you need the “clue” which restricts the answer to one value (although I quite like the idea of using negative numbers, I thought that was clever).

4. I wrote this as comment on question as well, why did you take NINETEEN but not THIRTEEN in the sequence?

1. It doesn’t matter, that’s why.

5. M says:

Got it.

But it would be a better sequence if it was:
ONE
FOUR
THREE
ELEVEN
FIFTEEN
THIRTEEN
SEVENTEEN

1. M says:

Oh, and the FOUR could even be ZERO

2. In what way is that better?

3. M says:

In that case it is the lowest number with the right amount of letters.

6. Karthik Durvasula says:

The correct answer given you sequence and your constraint is… “there is no number that can come in that position”; 19 marks the end sequence

1. Chris says:

Why?

7. Matt says:

I think I’m being an idiot here, but what exactly is the reson for the sequence, your explanation doesn’t actually tell me why those numbers are in that sequence, just how many letters are in each number…

1. Bob O'H says:

I was being an idiot too. But the sequence is in the word lengths:
1 one 3
4 four 4
7 seven 5
11 eleven 6
15 fifteen 7
19 nineteen 8

2. Matt says:

Thanks Bob O’H, That kinda makes sense but now I think that the sequence is an idiot…
Basically if you order all the numbers by the length of the word then pick a random number from each length you get a “sequence”?
Why not 1, 5, 8, 11, 70, 13…
There’s no reason those numbers have been picked, so the only way to predict 17 as the next number would be if it was the only word with 9 letters? Is that the case?

3. M says:

I now what you mean… It could basicly be every number with 9 letters. But the clue is actually more like a rule. 17 is the only number below 19 with 9 letters. But the rest of the sequence is a bit inconsistent.

4. tort says:

It’s not meant to be a unique sequence, that’s why the clue is there. It doesn’t matter if other numbers could be placed in the sequences, you are given that sequence, a constraint and try to solve the problem.

I think maybe the reason so many people are struggling with this is unfamiliarity with maths.

5. michaelbeaton says:

Matt. It is not supposed to tell you the reason why for the sequence! It is a set of conditions that needs a response.

The point is to figure out the pattern based on the information embedded in the sequence that you have been presented. Not trying to figure out the underlying reason/rules/proposition for the sequence.

In fact there is a axiom in play here: a list of items itself cannot tell you how the list was formed.

apple, carrot, plum.

what is the underlying logic/rule that formed this list. You don’t know, and cannot. It could be any of 10 different underlying propositions…and until you have more information you cannot know.

As I note in a different post, what I enjoy and find interesting about all this is the profound systemic flaws in thinking that this simple puzzle surfaces. I think it is useful as a indicator of larger systemic issues.

8. Jimbo says:

It’s 41, because #6 is not 13.

What is the rule for determining which number comes next, out of the set of numbers which contain n letters when they are spelt out?

For example, the smallest numbers which contain 3 and 4 letters (ignoring zero) are 1 and 4. This would explain the first 2 parts of the sequence and why they are not 2 and 9 for example. But the smallest number with 5 letters is 3, not 7. Perhaps the sequence rule is “the next number is the smallest number with 1 letter more than the preceding number and is not smaller than the preceding number”. This would explain why 19 and not 13 is the 6th in sequence. But this obviously rules out anything lower than 19 following 19, which you have said is not the case. I can’t understand why the next number is not 41!

1. No, #6 is thirteen, the question says so. Duh.

2. Jimbo says:

The question does not mention 13. By #6 I mean the 6th number in the sequence, which is 19.

3. Anonymous says:

Dave you’re such a fucking idiot, stop commenting.

9. Jimbo says:

I can see why #6 would be 13 or 18, but not 19.

This might help:

Number Letters Word
1 03 one
2 03 two
6 03 six
10 03 ten
4 04 four
5 04 five
9 04 nine
3 05 three
7 05 seven
8 05 eight
40 05 forty
50 05 fifty
60 05 sixty
11 06 eleven
12 06 twelve
20 06 twenty
30 06 thirty
80 06 eighty
90 06 ninety
15 07 fifteen
16 07 sixteen
70 07 seventy
13 08 thirteen
14 08 fourteen
18 08 eighteen
19 08 nineteen
17 09 seventeen
41 09 forty-one
42 09 forty-two
46 09 forty-six
51 09 fifty-one
52 09 fifty-two
56 09 fifty-six
61 09 sixty-one
62 09 sixty-two
66 09 sixty-six
21 10 twenty-one
22 10 twenty-two
26 10 twenty-six
31 10 thirty-one
32 10 thirty-two
36 10 thirty-six
44 10 forty-four
45 10 forty-five
49 10 forty-nine
54 10 fifty-four
55 10 fifty-five
59 10 fifty-nine
64 10 sixty-four
65 10 sixty-five
69 10 sixty-nine
81 10 eighty-one
82 10 eighty-two
86 10 eighty-six
91 10 ninety-one
92 10 ninety-two
96 10 ninety-six

1. You have assumed the rule “and is not smaller than the preceding number” which, if you had to use a rule to create the numbers in the puzzle, is a perfectly sensible one.
However, given the puzzle also states that the next number is below nineteen, then this assumption is rendered incorrect. So you therefore need to discard it, based on the evidence provided in the puzzle itself.

2. Jimbo says:

The puzzle does not state that the next number is below 19. That is a clue to help solve the puzzle.

To solve the puzzle you have to find a set of rules which can be used to build the sequence, and there may be many sets of rules which work. For example, you might be able to replicate this sequence 3 different ways and complete it with numbers 23, 54 and 6, depending on which rules you use. The clue tells you that 6 is the answer Richard is looking for, but it does not mean that 23 and 54 are not also valid next numbers in the sequence.

If your rule involves the length of spelt words then you have to include the fact that the sequence increases (unless you want to allow randomness in the sequence (ignoring for now that 19 should be 18 under the “increasing sequence” condition)). Then you arrive at a number which the clue tells you is wrong, so you have to rule out the length of spelt words.

10. Chrissie says:

Will you shut up about 41! Even with your rules it would be TWENTY ONE! See my comment after puzzle and get over yourselves with the pedantic “what’s a sequence” stuff. Move on!

1. Jimbo says:

I don’t know what your rules are, but 21 has too many letters for what Richard’s rules appear to be. See the table I posted above.

11. I always love watching people talk about the answers to these ones. An idea that often comes up is “there is no rule that means you could build this sequence”, and while that is strictly true (in that the puzzle seems to have randomly chosen the numbers with a five or more letters) it is not what the puzzle has asked you to do.
The puzzle has asked “what is the next number in the sequence”? And, with the extra clue that it is below 19, then there is only one answer.
So the puzzle provides enough information to provide the right answer, but not enough to generate a rule which would replicate the options given. That the sequence itself is not repeatable makes the puzzle harder, but it doesn’t render it meaningless.

1. Jimbo says:

If there is no rule governing which of the set of numbers with a given number of letters should be used then we cannot be sure that the sequence involves the number of letters in each number’s spelling. It could be shirt numbers of FA Cup goalscorers, or something completely different.

2. Yes but if you see a pattern, then you know that the next number can follow that pattern. The rule might be “a random number with three letters, followed by a random number with four letters”. That rule works, but does not make for a number sequence that you can replicate.

I should point out that I didn’t get it.

3. Jimbo says:

For number sequence to be replicated the rules which govern cannot include randomness. But randomness is required when creating this sequence, if you believe that the length of the number when spelt out in words forms any part of the rules for the sequence. FA Cup scorer shirt numbers or something else might fit this sequence without introducing randomness, but the length of spelt numbers does not.

12. Lazy T says:

2+2 = 10

13. Anonymous says:

If the correct answer were 17, then the logic in the sequence is flawed.
For the correct answer to be 17, the sequence would need to be:
1 4 7 11 15 13 17
where the rule is the FIRST number, in numerical order, with the necessary ascending amount of letters in it…..

14. Kristian says:

Tricky puzzle, didn’t get it.
I was on about adding the character length of the previous number to the value for a period, but didn’t make sense to me.

15. Ian Menzies says:

Perhaps the rule is “the number that Richard like the most with the necessary number of letters in the word.”

16. gustav says:

tricky, especially if you are not a native english speaker like me. I read the puzzle in english but when a sequence like this comes along, I tend to read the numbers in swedish in my head: “ett, fyra sju elva…” not much of a clue then… =)

17. Anonymous says:

clue != constraint, that is misleading.

18. James says:

Sorry, didn’t like this one. The most “puzzling” thing about it is that it doesn’t really make sense. Once you remove that, it’s too easy.

19. michaelbeaton says:

I find it amazing the contortions and thrashing people are doing about this. Esp those who are intent on finding a rule that the sequence itself it at fault because it doesn’t use the first number meeting the letter count condition.

The sequence is what it is because that what it is. There is no rule that generated it, other than the convention of the puzzle author. And the answer is just the same… Why are you so intent on making rules that are not there?

I think there is a indicative here about some of our (very bad) problem solving ability in our society. We seem to be intent on solving problems the do not exist, or introduce complexities that are not there, for what reason? Who knows. (it is complicated, lots of reasons…self interest is a huge one)

We cannot solve problems on their own terms, introducing new and unrelated terms we confound the issue to the point of useless thrashing.

Our political system is rife with this dynamic. It is very interesting to see it played out here.

1. MagnumOpus says:

Well said sir. I find it perplexing that some people are saying that the sequence is flawed.

The sequence was created by the author of the puzzle, it is what the author of the puzzle want it to be. He gave us a puzzle and ther is only one solution. Why people started saying that the sequence is flawed because the author didn’t use the lowest number with the correct amount of letters? This is a puzzle, just solve it, don’t try to create another.

20. Chris Emerson says:

I got it, and I understood the reasons behind the answer, it’s just not very neat. As others have said – ‘below 19’ was a ‘clue’. This implies that it isn’t required to solve the puzzle, but unfortunately it is. Shame it doesn’t have neater rules to govern the sequence.

1. michaelbeaton says:

I guess I am going to attempt to hold this line in this thread! (Strange,because it really doesn’t matter…but there is a point below the point…)

If you will respond, let me ask you by what measure, by what rule, logic; or by what propositional assertion do you assert that “it is not very neat”?

Your statement illustrates my basic point… Your own context/believes have influenced your ablility to work the problem on its own terms.
This is not a bad thing, it is what we do. If we do not know we are doing it…it creates unnecessary complications. Especially when we bring good/bad judgements to a situation without understanding it. And…in this case, as I said, it is impossible for you to make this judgement in of “neat” except by some rule that you have external to the problem.

In this case..no big deal, and not worth the time for you to say it or me to respond to it. In more substantial issues the pattern is the same.

2. James says:

Michael, for goodness’s sake. I’m not sure why you’re determined that there should be some wider social aspect to people’s ambivalence to this puzzle. The actual reason is very simple: the “clue” is not a clue but a rule. If you take it to be a clue, you have to assume that it is not strictly necessary in order to solve the puzzle. Without this information, the sequence we’re looking at appears to follow more than one rule: the number of letters in the English word for each number in the sequence increases by one each time; the value of each number in the sequence is greater than the last; for any given number in the sequence, no smaller number fits both of the previous two rules. Following these rules, the next number would not be 17; depending on what you class as a character, it would either be 21 or 41.

I’m sorry if you can’t see why this makes the language of the puzzle misleading and “not very neat”, but I wish you wouldn’t imply that those who do see it are in some way deficient or socially maladjusted. You’ve simply missed the point.

3. michaelbeaton says:

@James…And you miss mine…Or perhaps I was inelegant myself. I did not mean to imply This group of people… may be, and may not be.
I did mean to extrapolate the exchange into a larger context..and in a more general way. And I think those comments still stand.

I also did mean to point out the nature of levels of things, the easiest being a set of things, and the rules that created the set. And the essential fact that you cannot deduce the rule from the set. (thus the puzzle… and the required hint)

And missing this essential understanding is cause of much mischief.

But…I take your specific point, and apologize for not being more precise in my own language… I do think the complaints over the form of the puzzle is puzzling, for reasons stated. But I would be exampling my own point were I to presume that because they were so complaining that they were also a member of this other set/mindset that I attempted to describe.

Again… they might be. And they might not be.

So there is a distinction to be made. Thanks for pointing it out.

To take the latter part of your letter… I believe you are indeed making the error of the sort I am addressing. To wit: you are asserting that there are two rules in operation when in fact there is only one plus a directive in the form of a hint. You make the point in your statement: “Without this information, the sequence we’re looking at appears to…”

With more information, in my case the solution to the puzzle as I did not get it, it is apparent there is one rule. The sequence exposes one clause of the rule, as you well state. The hint exposes the other clause “… and not > 20”.

It seems to me that this level of uncertainty is precisely what we deal with in everyday life all the time. It is the basis of all inductive reasoning, and essentially is the trial and error aspect of the scientific method.

Yes? No?

michael

4. James says:

With regard to making a mistake, the point is that what is flagged up as a “clue” is not a handy hint but an essential piece of information. I tend to agree with the view that, in this context, that isn’t what “clue” would normally be taken to mean. I also agree with whoever said that a sequence that includes an element of arbitrary selection is not really much of a sequence. Even with boundaries of 1 and 19 – a fairly contrived set of limits for this kind of puzzle – if we’d been asked to find the 3rd number in the sequence, it would have been impossible to narrow it down to a single one.

Since you’re keen on extrapolating, I’d just say that the fact that someone might take a certain approach to a puzzle doesn’t mean that they take that approach to every aspect of life. Puzzles often ask you to be rigorously logical and search for clear-cut patterns, which is a large part of what many people find appealing about them; life is rarely as straightforward and requires flexibility, approximation, guesswork and all the rest. But it’s worth noting that most of the people who found the puzzle unsatisfying still figured out what the answer was meant to be, so it doesn’t really follow that their inflexibility would hinder their ability to find solutions in a broader context, does it? They found the solution, and correctly judged that it was probably the right one in spite of the perceived inelegance of the problem.

I just don’t think there is a wider social aspect, frankly. There are certain conventions around puzzles and mathematical or pseudo-mathematical sequences which dictate people’s expectations, and those expectations provide some of the tacit rules which allow you to solve them. This puzzle bucked my expectations and seemingly those of a number of other people, and left me feeling that I might have solved it but had no way to be sure. That’s why it was unsatisfying. There’s really nothing more to it.

5. michaelbeaton says:

If you’ll allow me: you are again making a leap that I have not, and are imputing to me judgements I do not hold.
ie. “I’d just say that the fact that someone might take a certain approach to a puzzle doesn’t mean that they take that approach to every aspect of life.”

I agree, and thought I said so quite explicitly in my prior post you are responding to.

I also take your point as true, and as a good way to frame all of this when you say, in essence, ‘puzzles have a certain form, and a certain expectation of a form..this one was badly formed according to the general “rules” of puzzle making’.
And I would agree with no reservation.

My “extrapolation” is not about the nature of puzzles and the proper form of making a good one. I stipulate your points re that.
My point is the different relationship, does it make sense if I say the “meta” relationship, of the participants to the puzzle, and the irritation it surfaced. And more than that, the explicit attempts to add rules and change the structure of the problem so it fit the pattern of what a good puzzle looks like.

My point entirely is this: That beyond the experience of puzzles like this there is a level of “puzzle” that is the life and life conditions we encounter. And I think we encounter these life situations – puzzles then – with similar responses as we do in this smaller example.

There is a wisdom that says : As in the little things so the large. Which is just a fancy way to say the same thing. Which is to say, how we act, live, approach a “puzzle” gives indication of how we will approach all the puzzles we are faced with….

It is my contention, or extrapolation if you will, that much of what we are doing, say on the national stage, with economy, politics, social issues, and etc, are following a similar pattern… One where the factors and the complexities are unknown and seemingly unknowable… And sometimes what governs what we can see is what we are looking at. (Which by definition defines or governs what we cannot/do not see).
Thus, to take a now well understood recent example: The banks/financial institutions, ostensibly, had no idea that the loans they were making, in the way they were making them, would cause the problem they did… So the puzzle of our economy was seriously not “solved” correctly because the relevant factors were not taken into account. Many of which were inelegant and presented, even knowable only as “Hints” or “Clues” , as in the much simpler example at hand.
[ I do know it is far more complex than this simple analysis..but i think the example holds up.]

The lesson then, I extrapolate!, is that we wont be able to see or acknowledge those factors until we open up our thinking to allow for patterns and complexities and factors that we haven’t before. And to take full account of the “hints” “clues” that are available even if they are seemingly irrelevant, or even inconvenient.

This is the meaning, as I take it, of the commonplace phrase: “Think outside the box”.

Which…if I understand puzzles (and I may not) is what puzzles are in part for! To challenge our rules, and ways of seeing. And to teach us that the answer to a puzzle is often just a shift of perspective, or a shift in received and predetermined definitions and expectations so as to allow for new factors or possibilities.

I wonder if you get the distinction I am making.
What I specifically am not saying is that “these specific people” are a certain way because of how they responded. I am saying there are patterns that may or may not be applicable….

And again.
Thanks for the exchange.

21. duckfish says:

yeah,I solved it!
I can’t believe the real answer REALLY is that!!

22. b says:

thats no clue

23. Eddie says:

I’d just like to say that I tweeted the following puzzle to @RichardWiseman on 12 July that goes: “What’s next in the sequence 3, 3, 5, 4, 4, 3, 5…and it ain’t golf!”

I wish he’d used this one. It would have saved a lot of argument and soul searching.

For the record, for neatness I reckon that his sequence should have followed this simple rule: Nth term = lowest positive integer with (N+2) letters, such that, e.g., the sixth term is 13.

24. Gabriel says:

This is an interesting thought provoking discussion, but really I have to agree with those who were annoyed by the puzzle, since I was annoyed when I realized the answer (in the shower) and saw in the comments that many were similarly confused. But there are two distinct problems here: 1) whether the final clause was a “clue” or a “rule” and 2) whether it is fair to call this a sequence puzzle at all. With respect to the first question, I think that James and michaelbeaton pretty much beat that horse, but what I was annoyed about was the second issue. Because when you are asked to provide the next number in a sequence puzzle, it is assumed that the sequence was generated by a deterministic rule. michaelbeaton’s point that we cannot uniquely determine the rule is well taken, but such a rule should at least exist, and the person who proposes the puzzle should be able to provide it after we are done. The fact is this puzzle does not have a generating rule, IT IS A LOUSY PUZZLE.

1. michaelbeaton says:

I would be interested about this distinction between rule and hint and clue. In this context they all seem to be synonyms.

And I do not understand your pt 2. Why? What quality does it manifest/posses that disqualifies it?

Re your point “I think that James and michaelbeaton pretty much beat that horse, ” regarding pt 1.
I don’t think our discussion had anything to do with your point 1. ie, the clarification about the meanings of rule/clue.

What else might you have meant?

re your point: such a rule should at least exist
it does. I state it in my comments above and now below. What am I missing?

25. Gabriel says:

@michaelbeaton

The act of solving a sequence problem necessarily involves an attempt to deduce the rule that generated the sequence. You simply cannot get around that. When you propose an answer it is because you are making a guess at the rule! If then the author responds “wrong, there is no rule”, you’d be justified in giving him the finger. If then someone tries to argue that *you* are the irrational one, you’d be justified in going “phhh!! you can’t be serious”. This talk about “the sequence is what the author wants it to be, no more no less” is ridiculous. Then you should be equally entertained trying to find the next in the sequence: -32, 3872194, 83, pi, -453, 4.2453, … And when I tell you that, according to its author Salvador Dali, the next is “a rrrrhinocerous”, you should go “oh well, I’m glad I’m not one of those irrational thinkers who were expecting a number”.

1. michaelbeaton says:

Gabriel.
I agree w you, and all. This was a crappy puzzle according the rules and norms of puzzle.

What I wonder is whether my more general point, one that is not based upon puzzles of this sort, is making any beachhead? As it happens each response, including yours, to mine is on the quality of the puzzle…which I am not defending. Fingers and all I agree…the puzzle sucks. (though I don’t think this puzzle is of the irrational set.)

My point is more that puzzles of this sort (ie, ones made in accordance with rules and norms you allude to) are a reduction in form of the puzzles we have in real life.

And etc.

Is it the case that that line of discussion has no resonance with this puzzle solving crowd? Or that my argument is badly argued? It is a puzzle…

2. michaelbeaton says:

On your specific point…because you responded as you did, I’ll make or remake my point re the puzzle itself….And look for your reply:

This is not an sequence without a rule.
It is just a rule (or logic) that you, and others don’t seem to find elegant. I accept the judgement, but not that assessment that there is no rule.

I said this above:
With more information, in my case the solution to the puzzle as I did not get it, it is apparent there is one rule. The sequence exposes one clause of the rule,[length of the word of the number increases by 1] . The hint exposes the other clause “… and not > 20″.

it may not be pretty, but it is a rational rule.

And it seems to me that the author may have put in this hint just to trouble everyone…(or, more likely , to make it just a little harder to solve) since there is an answer in a number that is > 20 as was pointed out above.

Which, to me, challenges your statement here :
This talk about “the sequence is what the author wants it to be, no more no less” is ridiculous.

Why ridiculous? Arent all puzzles of this sort determined by the author? And according to some rule he has determined? Isnt that what the puzzle is about? Seems to be according to your own response…Ie, that it is the solvers job to deduce(or induce) the rule that author utilized to generate the list.

26. Gabriel says:

@michaelbeaton

“My point is more that puzzles of this sort (ie, ones made in accordance with rules and norms you allude to) are a reduction in form of the puzzles we have in real life.”

To this point I would say no they are not similar. A sequence puzzle carries with it certain guarantees that we should not expect from everyday life, unless we are of the believing sort, ie we think that even life puzzles have a guarantor of fairness.

Quoting what you say about the supposed rule: “The sequence exposes one clause of the rule,[length of the word of the number increases by 1] . The hint exposes the other clause “… and not > 20?.”

By definition, a rule is something that determines the answer. That is not a rule! There are, as many have pointed out, multiple sequences that obey those two clauses.

Of course one may speculate that there is a rule that generated the given sequence. One ridiculous possibility would be: first comes one, then comes four, then comes seven, … But that is not the type of rule I have in mind (it is of the Dali sort). In the world of sequence puzzles I would imagine that what is sought is a fairly simple general rule that can produce an infinite sequence, the beginning of which is given. The rule should not contain ad-hoc exceptions such as “choose the smallest unless it is 13, then choose 17” (or whatever it was). Get my drift? Perhaps there is some ingenious such rule BUT THAT IS NOT WHAT WAS GIVEN IN THE ANSWER BY MR WISEMAN!!! I would have been very interested to see the magic rule that nobody found. That would have been a very good puzzle, one that stumped everyone. But if the rule exists, it is certainly not known to the proponent of the puzzle. That’s my point.

27. Steve Ulven says:

I just got the e-mail from Amazon saying Paranormality is available because I’ve bought previous books from you (Quirkology and Guidelines For Testing Psychic Claimants). I already got the Kindle version and am about half through it (loving it), but I would like a hard copy so I may order it in the future.