Jul 03 2015

A Quick Logic Lesson

Try your hand at this quick puzzle, then come back and read the rest of this post.

How did you do? This is a great little test with a very important lesson.

The discussion that follows the puzzle is a fairly good explanation of confirmation bias, which is a partial explanation for why people might fail to solve the puzzle. It is a partial explanation only, however, and therefore missed an opportunity to  teach a critical lesson in scientific reasoning.

Confirmation bias is the tendency to seek out, perceive, accept, and remember information that confirms beliefs we already hold, coupled with the tendency to miss, ignore, forget, or explain away information that contradicts our beliefs.

How many times have you either said yourself or heard someone else say, “well, that’s an  exception?” Is it, or is it just data? By calling an example an “exception” you are assuming that there is a rule it violates. This is a way of dismissing information that contradicts your beliefs.

The puzzle article explains that people seek our information that confirms their hypothesis, rather than seeking out information that contradicts their hypothesis (confirmation bias). Therefore, they come up with a hypothesis about the rule governing the number sequence, they enter in a sequence that should yield a positive answer if their hypothesis is correct, and if it is correct they believe their hypothesis to be confirmed.

In testing a hypothesis there are actually three things a good scientist should do, and the article only discusses one of them – testing your hypothesis against information that should yield a negative result.

Another critical step that the article ignores, however, is the need to test alternate hypotheses – try to come up with a hypothesis that is also consistent with the existing data and then test that. Specifically you should have entered in a number sequence that would fulfill the alternate hypothesis but not your original hypothesis.

Failure to consider or test alternate hypotheses is called the congruence bias, and it is a type of heuristic. This is less well known than confirmation bias, but in many situations is just as important to understand.

The third step, which is not really relevant to this particular test, is to consider the effects of a negative result from any of your tests. In this case, since you are trying to figure out a mathematical rule, results are definitive – if a result breaks the rule, the rule is wrong, period. When testing scientific hypotheses, however, results are not always definitive and simply increase or decrease the chance that the hypothesis is correct.

To give a real-world example of this type of reasoning, let’s consider medical diagnosis. One of the reasons this puzzle was trivial for me is because I am familiar with confirmation bias and congruence bias, and the need to look for negative outcomes and to test alternate hypotheses. Hypothesis testing like this is a daily part of the practice of medicine.

When confronted with a patient with a set of signs and symptoms, physicians should create a differential diagnosis – a list of possible diagnoses from most likely to least likely. It would be a supreme mistake to only consider your first guess or only the most likely diagnosis.

Physicians then need to order tests; each test (physical exam findings, blood test, X-ray, biopsy, whatever) is a test of their diagnostic hypothesis. The pitfall physicians need to learn to avoid is to test only their pet diagnostic hypothesis, and interpret a positive outcome as absolute confirmation of their diagnosis.

They should also order workup to test other possible diagnoses, and they also need to consider the real predictive value of a positive or negative outcome of each test on each diagnosis they are considering.

This logic does not only apply to professional fields like medicine (although it is critical to any investigational profession). We could use this logic in everyday life. Consider political opinions, for example. We tend to seek out examples which confirm our political beliefs, and fail to consider the impact if those examples were negative, the effect those examples have on alternative views, and examples that contradict our views.

The combination of confirmation bias and the congruence bias can create a powerful sense that the world confirms our ideology, when in fact that ideology can be partly, mostly, or even completely wrong.

Conclusion

I love it that a somewhat viral article in the New York Times is teaching a core lesson of critical thinking – confirmation bias. A more nuanced discussion, however, would have included the congruence bias as well, which is even more pertinent, in my opinion, to why people might fail that puzzle.

The real challenge, however, is to get people to internalize these logic lessons and consistently apply them to themselves in everyday life.

110 responses so far

110 thoughts on “A Quick Logic Lesson”

  1. wfr says:

    I decided to test the assertion that “you won’t get a second chance” to take the test. It turns out that this claim is false. I reloaded the webpage and took the test a second time.

    This makes me suspicious that the authors of the test are either unqualified to design such a test, or are working some kind of con.

    In either case, I have decided not to believe anything they say on that website. This confirms my strongly held belief that everything I read on the internet is wrong.

  2. dohashi says:

    Just for fun, here are the 6 sequences I tested to get to the right answer

    1,1,1
    1,2,3
    2,1,3
    2,3,4
    3,4,2
    4,2,3

    After 2,1,3 I was fairly certain I knew the right answer, but I did think the answer seemed a little too simple. I think this one was pretty easy for me because I am a computer programmer, and this is a lot like testing a piece of code. With a certain amount of experience you learn to test for things that should/could go wrong just as much as things that should go right. Of course, I’ve also seen enough of these “figure out this sequence” puzzles to not assume all the obvious properties of the provided example were relevant.

  3. John Pieret says:

    Took me three yesses and two noes.

  4. FacelessMan says:

    Unfortunately I already knew the answer from the fanfiction “Harry Potter and the Methods of Rationality” (which I would really really recommend to any skeptic btw), so I couldn`t test myself. But I remember that it really made me think when I read it the first time, so I probably wouldn`t fall in the same trap again. But I`m sure I would`ve fallen for it before the lesson about the importance of falsifying your hypothesis.

  5. wfr says:

    The people who found the puzzle from this blog are probably wreaking havoc on the statistics (if they’re keeping them). It’s fair to say that fans of critical thinking aren’t typical of the NYT readership demographic.

  6. ccbowers says:

    I made 8 guesses with 3 that were “No” before committing to the answer. Interestingly, the article stated that people attempting to solve the puzzle were avoiding a “no” response. I wonder what the evidence for this is (versus a lack of motivation to explore the puzzle, or perhaps incompetence). When I was figuring out the puzzle, I was trying to get “No” responses to see how the rule could work. But I was motivated to do so. I imagine for those who were not would have given up early.

    Confirmation bias occurs most strongly when people are invested in a particular outcome (not just accepting the apparently obvious, like this puzzle). The results of this puzzle may be just a lack of a questioning attitude or lack of motivation, versus actually motivated to confirm their preconceived notion. I know these are related concepts, but I think there are important differences between those scenarios.

  7. mumadadd says:

    I got it, though I was primed, by the very fact that the puzzle was the subject of a post on this blog, that I shouldn’t treat it as I would normally approach this kind of puzzle. I agree with CCBowers in that what this puzzle demonstrates isn’t necessarily confirmation/congruency bias – I think it may be more a case of how this kind of puzzle is usually presented (e.g. in an IQ or general numeracy test) affecting how people respond in this instance. In these cases the correct answer would be the obvious one that crossed all our minds first (doubling).

    How about this for meta: has Steve, in his presentation of this as an effect of congruency bias, actually failed to test other hypotheses that might also explain the effect, and therefore fallen victim to congruency bias?

    🙂

  8. leo100 says:

    I got it on my first try. 3,5,7.

  9. leo100 says:

    By the way 1,3 and 7 is another correct number sequence.

  10. mumadadd says:

    Jesus…

  11. Willy says:

    H Christ

  12. tmac57 says:

    leo- What if the rule was ‘list any three numbers, but limited to single digit positive integers”?

  13. Willy says:

    Leo: It is impossible to be certain you “got it” with only one entry.

  14. mumadadd says:

    I don’t think he gets that you’re supposed to be finding a rule and not just a set of numbers that spits out a “Yes!” result. Seriously.

  15. mumadadd says:

    So when he says he “got it” he means that 3,5,7 is the correct answer.

  16. tmac57 says:

    “One Rule to rule them all!!!”

  17. leo100 says:

    HI Mumadadd I get it. Yes the correct answer was 3,5,7. The other entry was 1,3,7 Willy. Sure there are other possible sequences as well.

  18. leo100 says:

    Other sequences were

    3,5,9
    1,5,7

  19. mumadadd says:

    And what do these sequences have in common, leo?

  20. mumadadd says:

    leo, sometimes when you’re not here, I miss you. But then when you are here I just end up being mean to you. We’re trapped in an abusive relationship, although right now I’m unsure as to which of us is the abused.

  21. Bronze Dog says:

    I spent a few tests confirming the “obvious” answer, mostly to see if I could find such a sequence that didn’t get a yes. Then I proceeded to test other sequences, looking for nos, and it turned out my second, simpler hypothesis was correct.

  22. delphi_ote says:

    It would be very interesting to try this with randomly generated rules and see how people fare. I’d be fascinated to explore both how we pick examples and how we generalize.

    Picking some way subjects could represent their rules would be difficult. Anything too expressive would be far too difficult for a layperson to understand.

  23. Willy says:

    Leo doesn’t get it.

    Hey, here’s another correct answer– negative 3,234,982 and 5,073659, and 8,101,524.

    I wonder just how many correct answers there are?

  24. Gib says:

    How many correct answers are there?

    My guess is somewhere around 2 to the power of (3*(n-1))
    where ‘n’ is the number of bits in the size of the unit used for the number.

  25. tmac57 says:

    leo, “correct sequences” was not the goal. Deducing what the ‘rules’ were was the goal, and seeing how people went about that task was the thing being explored.

  26. tmac57 says:

    Apparently you can use decimal fractions, negative and positive numbers, 0, and very large place numbers (not sure if it is limited), so my guess is that there are near infinite possibilities.

  27. Willy says:

    Just to be clear, I was being sarcastic. We do need a sarcasm font.

  28. leo100 says:

    Tmac57,

    You are correct. I know that people can cheat. I wouldn’t be surprised if some took this test and hacked into it.

  29. mumadadd says:

    Quoting myself (sorry):

    “I think it may be more a case of how this kind of puzzle is usually presented (e.g. in an IQ or general numeracy test) affecting how people respond in this instance. In these cases the correct answer would be the obvious one that crossed all our minds first (doubling).”

    Mmmm, this is related to something I have rattling around in my head. While I attempted this puzzle I was fighting an urge to try and get to the right answer in as few moves as possible. I can’t recall ever seeing a question in any quiz or test where it was possible to do this kind of hypothesis testing without being penalised for it.

    It seems as though, for whatever reason, we place value in getting to the right answer quickly, but without regard for process. If you get there fast and you’re right, you’re talented; if you get there fast and you’re wrong, you lost but at least you’re decisive.

  30. mumadadd says:

    wfr,

    “It’s fair to say that fans of critical thinking aren’t typical of the NYT readership demographic.”

    I thought the NYT was a very well respected paper, in terms of journalistic quality and integrity? As I live in the UK it isn’t one of my regular news sources–in fact I’m unsure as to whether I’ve ever read one of their articles–so who knows from where I’ve absorbed that impression from (probably the liberal American media). I just spent about 10 minutes on the Internet trying to find any statistical information about the comparative accuracy of media outlets (generally, but also specifically the NYT) and drew a blank, so I’m curious as to the basis of this statement.

  31. mumadadd says:

    wrf,

    Please disregard my last comment. Clearly, readers of any national, mainstream media outlet are unlikely on the average to be fans of critical thinking. I can see now that you weren’t denigrating this particular outlet but making a general observation.

    I got this totally wrong so please accept my apology. 🙂

  32. BillyJoe7 says:

    Did anyone test fractions?

    No?

    Did anyone say CONFIRMATION BIAS!
    (Fractions are not even acceptable as an answer)

  33. mumadadd says:

    Decimals are though. Same difference.

  34. BillyJoe7 says:

    Aw, leo is so cute isn’t he?

  35. BillyJoe7 says:

    mumadadd, come on, you didn’t test fractions did you. 🙂

  36. mumadadd says:

    No, it never occurred to me test fractions, but I did try decimals. Same difference….? (I know, not quite). 🙂

  37. Damlowet says:

    @ Dohashi,

    There is no ‘correct’ answer.

    With those examples of questions you have asked, you have only determined a very small possibility of what the ‘rules’ are. You didn’t use negative numbers, you didn’t use decimals, you didn’t push number size limit, you didn’t use zeros. You presume you are ‘right’ based on the questions you asked. That is what the test was determining.

    Using your logic, Australian electricians are better at understanding more comprehensiveley a simple number sequence than ‘computer programers’ because my conclusion based on the 20 odd questions I asked the test showed a more plauseable rule than the one you came up with in your 6.

    There is no possible way to ‘know’ all the rules for this test, and that is part of the point. When you find enough (different for each individual) ‘yes’ or ‘no’ answers, you confirm your bias, and you feel happy to call it answer found.

    CCbowers and mumandadd, what if another stipulation of the rule is that ‘once 100 questions have been asked, the rule reverses’, or, the rule is valid for all numbers which don’t contain the sequence 3.141593, ect ect. There is no way to know all possibilities. 😉

    Damien

  38. tmac57 says:

    mumadadd- Attempting to solve a puzzle in as few moves as possible does not necessarily imply doing it as quickly as possible. What I like to do in such instances, is make a guess based on what I know, and then when I get feedback, I start to think more deeply about what other possibilities are out there, and then how to effectively test them in the least amount of guesses (think Minesweeper or Mastermind), but mostly as a challenge for fun, not because of time constraints (unless that is part of the stated challenge).

    I use to watch my grandson play adventure video games (like Zelda) and his approach would be to just randomly do things over and over without any reflection on what cause and effect was going on, hoping something good would happen. It use to make me so antsy, and I tried to occasionally steer him toward a more effective strategy, but mostly I just let him flounder, and hope that he could work it out, or get something fun out of the activity. Sadly he lost interest quickly on many of the games, but did later get good at the first person shooter types and such that had less to do with puzzles, and more to do with reflexes and awareness.

  39. leo100 says:

    Of course their is no answer. As there is an infinite number of answers that would be yes and no.

  40. Willy says:

    Leo–You’ll know you really did finally get it when you start to feel embarrassed over your posts wherein you claim you got it. Don’t feel embarrassed that you didn’t get it, but do feel embarrassed that you ignored ample evidence that suggested you didn’t get it. This could be a good learning opportunity for you if you’ll ponder on it a while.

    Jeez, Leo, I hope you are a twenty-something or younger.

  41. Damlowet says:

    @leo100

    The idea of the test was to find the ‘rule’, not a correct answer.

    But, you seem to contradict what you wrote in your earlier posts with that last reply. ?

    “(I got it on my first try. 3,5,7.

    By the way 1,3 and 7 is another correct number sequence.

    HI Mumadadd I get it. Yes the correct answer was 3,5,7. The other entry was 1,3,7 Willy. Sure there are other possible sequences as well.

    Other sequences were

    3,5,9
    1,5,7

    Tmac57,

    You are correct. I know that people can cheat. I wouldn’t be surprised if some took this test and hacked into it.)”

    No hacking is need to understand the test.

    Damien

  42. leo100 says:

    Willy,

    My point was to demonstrate that I am smart enough to know how to do number sequences. Because I am sure many of you here think I am probably stupid. That I cannot use critical thinking, which couldn’t be farther from the truth. Plus the ample evidence your talking about is hearsay (anecdotal evidence) which you materialist’s claim must be all bogus when it comes to the so called paranormal.

  43. Willy says:

    Leo: You are clueless, absolutely clueless. I suggest no one on this thread should ever respond to a post of yours again.

    Hearsay? Anecdotal evidence?

  44. petrossa says:

    The funny part is that the principle psychology itself is an extreme form confirmation bias. We see patterns in behavior and than obviously see them again when repeating test to show the pattern. My simple analogy is: the brain is uniform colored lamp and psychology is holding a punchcard before it which hey presto gives you pattern. Next ‘behavior pattern’ another punchcard. We make the punchcards based upon out own preconceptions. Hence psychology will forever be a soft science with no intrinsic value.

  45. BillyJoe7 says:

    leo,

    “I am smart enough to know how to do number sequences”

    What is the next number in this sequence:

    6, 3, 10, 7, 9, 2, 15, 14, 21, 8, 13, 0, 1, ?

  46. BillyJoe7 says:

    “psychology will forever be a soft science with no intrinsic value.”

    To high functioning autistics, who knows? But, to the general riff raff, it seems to have a few clues. |:

  47. a stray cat says:

    This is quite an interesting metaphor for science in general.

    I also first thought it was simply a matter of doubling, then any factor greater than one, then expanded my test cases until I had settled on the rule being any three increasing numbers, after tests runs like
    (-9, 99998.9, 99999).
    But still, I hesitated. Might I be missing something? Is there any pattern that includes a counterexample I might have overlooked? After all, I only get one guess….

    Of course, as Damien said, there is no way to be completely sure of any rule just from the information given. We can only be absolutely sure of what we tested, and increase our confidence in a rule as it passes more tests.
    I didn’t hesitate so long after that. After all, we know this test was made by people and intended for the general public, so it must have a sensible, fairly straightforward answer. Indeed, it did.

    However, if this was a real puzzle of nature, there would be no reason to expect such satisfying simplicity, aside from the general observation that things tend to be logical. We have to test extensively, and our tests are limited by what’s possible for us at the time. Even after arriving at a rule for something, we have to be cautious when extrapolating beyond the extent or precision of our tests. Maybe it changes once we get start looking very small (QM), or very fast (relativity). Or maybe there are effects we simply can’t see yet. We never get a final answer.

  48. ccbowers says:

    “CCbowers and mumandadd, what if another stipulation of the rule is that ‘once 100 questions have been asked, the rule reverses’, or, the rule is valid for all numbers which don’t contain the sequence 3.141593, ect ect. There is no way to know all possibilities. ;)”

    I assume your winky eye was meant to be tongue-in-cheek, but yes there are a theoretically infinite number of rules to rule out, but of course the game implied a simple rule to discover. And yes, in my 8 sequences I tried negative numbers, and decimals, odd and even numbers. Beyond that, we have diminishing returns and an uninteresting answer.

  49. ccbowers says:

    petrossa-

    It is true that confirmation bias becomes a problem in science, but I disagree with your conclusion about psychology. Although human behaviors are complex, they are based on human physiology, so we should expect to find some patterns, and we do.

    We just have to be cognizant of the limitations of our observations and be careful not over-extrapolate from them. Psychology gets a bad rap, because the difficulties that arise from that nature of the subject matter that it studies, human behavior. (Economics has a similar problem) Unless you are arguing that human behavior is not amenable to science, I don’t understand your point. To the extent that I do, I disagree with it.

  50. leo100 says:

    BJ7,

    6, 3, 10, 7, 9, 2, 15, 14, 21, 8, 13, 0, 1, 5.

  51. ccbowers says:

    Leo, what is the rule used to get a 5 for that sequence? Using the simplest rule, and excluding negative numbers, I guess a 0.

  52. Ian Wardell says:

    My sequences were:

    1,2,3
    -1,-2,-3
    2343,22981,8934845
    747372,455,123
    -111,-45,299
    -1.22,-0.33,5
    -2,-3,-1
    1,2,-1000000
    1,2,1000000

    I have the propensity to try sequences that are more likely to confirm my hypothesis. Yes there’s no reason why I should do that. I’ll remember that lesson.

  53. Ian Wardell says:

    wfr
    “The people who found the puzzle from this blog are probably wreaking havoc on the statistics (if they’re keeping them). It’s fair to say that fans of critical thinking aren’t typical of the NYT readership demographic”.

    I don’t know if you “skeptics” have any idea how incredibly irritating this sounds. This suggestion that “skeptics” are more rational and intelligent than the rest of us…

  54. BillyJoe7 says:

    Ccbowers,

    “Using the simplest rule, and excluding negative numbers…”

    …and decimals and fractions (;
    But, yes, my question implied that there was only one answer so, yes, no negative numbers, decimals, or fractions.

  55. BillyJoe7 says:

    leo,

    Why is the answer 5?

  56. mumadadd says:

    Ian,

    If you self-identify as a skeptic then you’re at least going to place more value on rationality, even if you’re not actually any good at it. Plus, if you read any of the skeptical literature, including this blog, then it’s pretty unlikely that you won’t have heard of and understand confirmation bias. On top of all that, the fact that the puzzle is the topic of a Neurologica blog post should tip you off that something is going to be not as it seems with it. The puzzle is hinged on critical thinking – this blog is in large part concerned with critical thinking. No special abilities needed – just familiarity with the subject matter.

    Like…duh.

  57. leo100 says:

    BJ7,

    Because that is my bias. Of course their are other answers.

  58. BillyJoe7 says:

    Leo, exluding decimals, fractions, and negative numbers, there is only one answer.
    Please explain how you got 5.

  59. Ian Wardell says:

    mumadadd I did the test last night. I saw it on facebook before seeing it here.

  60. leo100 says:

    BJ7,

    Because 5 wasn’t in the sequence you gave me. I could of went with 4 but I chose 5. All other numbers positive numbers in the following sequence were taken except 4 and 5.

  61. ccbowers says:

    Leo doesn’t seem to understand these types of puzzles.

    Here, I’ll help him out: Leo would be correct if the rule was that every 7th number in the sequence had a “5” in the ones place. Yeah that’s a stretch.

    “I could of went with 4 but I chose 5.” could….of?….went?

  62. BillyJoe7 says:

    leo, what about 11, 12, 16, 17, 18, 19, 20, and all the numbers greater than 21?

    In fact, the answer is 0.
    Can you see why?

  63. Pete A says:

    I don’t know if you “Ian Wardell” has any idea how incredibly irritating you sound. This suggestion that “Ian Wardell” is more rational and intelligent than the rest of us…

  64. Willy says:

    “I could of went with 4…” ROFLMFAO!!!

  65. leo100 says:

    BJ7,

    Yes I do thanks for helping me ccbowers. Pete A, I don’t think he meant that he meant that often materialist’s think they are smarter than anyone who disagrees with them.

  66. Damlowet says:

    BJ7,

    That is interesting I was thinking that zero may be the answer (of course 🙂 ).

    If you take two numbers in order, (grouping each two numbers into group) the second number is always smaller than the fist. With the last group’s first number a ‘1’,(and no negative numbers in the sequence) I though it may have been ‘0’, but that was a pretty big stretch and had a quite loose connection.

    Damien

  67. ccbowers says:

    These exchanges are really painful. Like the opposite of schadenfreude, but not quite empathy.

  68. mumadadd says:

    Man alive. I couldn’t find any connection between those numbers. I’m boxed in by expectation: I only looked for mathematical operations and when I couldn’t find them I was almost certain that BJ7 was pulling our collective plonker. Maybe this is why I’m shit at non linear computer games….

  69. ccbowers says:

    Damlowet. I had indicated the correct answer above, but we have avoided revealing the answer for the sake of Leo100 explaining the rule, or his rule. It has probably gone on long enough, so it is probably good that you revealed your explanation as you are more or less correct.

    The sequence alternates between whole numbers that are smaller, then larger, then smaller, then larger… than the previous number in the sequence. I say whole number to exclude negative numbers and decimals. Since the last number before the ? is 1, and we need a whole number lower than it, there is only “0” left.

  70. This part was interesting:

    “They don’t want to hear the answer “no.” In fact, it may not occur to them to ask a question that may yield a no.”

    I think one of the main things you learn when you become more skeptical and/or scientific in thinking is that you need to SEEK OUT the “no” answers. You can’t know that something is TRUE unless you COULD know that it was false, so we try and prove our hypotheses false so that we can know that they’re true.

    My tests went something like this:

    10, 20, 40 (fits the “doubling” hypothesis)
    1.5, 3, 6 (still fits, even with fractions)
    100, 100, 200 (this is a “no”, still fits doubling)
    120, 200, 400 (RED ALERT! This is a “yes”???)
    200, 200, 400 (but this is is a NO…)
    199, 200, 400 (yes..)
    199, 201, 202 (yes..)
    1, 400, 999 (yes..)
    400, 200, 100 (“no” — it’s not just a sequence in either direction)

    Answer: Each number needs to be higher than the previous one.

    But still, that answer is tentative! Just like science, it’s the best explanation of the data available, and it will change if more data becomes available which contradicts this answer. Interesting experiment on multiple levels.

  71. Pete A says:

    BillyJoe7, I really enjoyed your question: “What is the next number in this sequence:
    6, 3, 10, 7, 9, 2, 15, 14, 21, 8, 13, 0, 1, ?”

    The acronym PRBS means pseudorandom binary sequence. There are an infinite number of both pseudorandom and naturally-occurring stochastic sequence generators (including multi-sided die) in which that sequence will occur.

    The sequence you presented is far too short for the purposes of determining the algorithm.

    What is the next number in this sequence: 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, ?

  72. Willy says:

    Leo: I feel kind of bad for you. You just aren’t getting it, though. Here’s an easy example:

    5, 10, 15, 20, 25, 30,… ?

    What’s next is obvious, right? The examples above aren’t as straight forward, but they have correct answers. In some of the examples, there can be more than one proper answer. Listening to you assure us that you “get it” is quite similar to listening to a Cub Scout assure the Den Mother that he really, really did see a snipe during the snipe hunt or having a child assure you that he saw Santa crawl up the from the chimney and fly away with the reindeer.

    Please do ponder your posts for a while. F’rinstance, materialism and the paranormal aren’t even relevant to this thread.

  73. RickK says:

    A friend from New York once asked me what came next in this sequence:

    86, 79, 72, 66, 59, 42, 34, 28, 23, 18, 14

  74. RickK says:

    Oops .. Extra credit for finding the missing number in that sequence.

  75. tmac57 says:

    Ok, since we’ve completely gone there..

    What is the last number in this sequence?

    3,3,3,4,4,6,4,3,6,7,7,7,7,4,2,3,1

    Hint- Think in terms of rows.

  76. BillyJoe7 says:

    The next number is 96, but it comes at the beginning of the sequence, not the end. The sequence actually stops at 14. The missing number is 50 between 42 and 59. Also there is no 18 or 28. I guess you just misremembered the sequence (:

  77. BillyJoe7 says:

    …that was a reply to Rick

  78. RickK says:

    BJ7 – spot on – technically the answer to my question “what comes next” is “Christopher St. – Sheridan Square”.

    i never said it was a number. 🙂

  79. leo100 says:

    Willy,

    Your perfectly right I should ponder what I am thinking.

  80. Bruce says:

    Just a small note; the original problem is to find a rule using sequences of three numbers. Asking for the next number in a sequence heavily implies that there is a “right” answer and usually only one of. These two are actually different questions:

    a) What is the rule for the following sequence:

    1, 2, 3

    b) What is the next number in the sequence:

    1, 2, 3

    The original problem also allowed for guesses and for people to get immediate feedback on them. This is why it is a much better question to ask.

    Asking for the next number in a sequence, in the media that this commenting section allows, is almost meaningless given the lesson in the original post.

    The ability to test and retest is the key here, not the mental gymnastics required to multiply my dog by the square root of Jupiter. While a lot of science involves complicated maths, I think it is important to remember that critical thinking and the willingness to be wrong is what is key and all those big numbers can actually figure themselves out one way or another (be them in my head or your head or in a calculator or supercomputer).

  81. BillyJoe7 says:

    PeteA,

    Sorry, I misssed your post.

    “What is the next number in this sequence: 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, ?”

    The next number could be 1
    Or it could be 3 if the drawn out sequence is:

    1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 3, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3,

    Or almost any other number depending on the drawn out sequence.
    I assume that was your point.
    On the other hand, if your point is valid, we cannot do number sequences.

  82. BillyJoe7 says:

    tmac,

    Here, I assume, are the rows with the missing number

    3,3,3,
    4,4,6,
    4,3,6,
    7,7,7,
    7,4,2,
    3,1,?

    I’ve no idea what the missing number is.

  83. tmac57 says:

    BillyJoe7- Interesting. I wonder what your process was there, but you’re on the wrong track.
    Think :

    row, row, row.

    (Oh, and a stream of consciousness might help too)

  84. LC says:

    Leo, let me see if I can help, as others don’t seem to be getting through.

    The assignment was to come up with the RULE. Not with a sequence of numbers that would work, or with multiple sequences, or with any numbers at all. From the feedback of yes/no answers to your suggested sequences, you were to derive the RULE. The answer was the RULE — the way of testing any sequence of numbers to determine whether or not it is correct. It wants you to figure out the RULE, not just more numbers that happen to work. It wanted the RULE. I’m repeating this over and over because I want it to sink in that the answer is the RULE.

    Your response: “I got it on my first try. 3,5,7.”

    So you’re saying that “3,5,7” is the rule?

    Because it looks to me and everyone else here is that “3,5,7” is just another sequence of numbers that happen to conform to the rule, and not the rule itself. Which means you didn’t complete the assignment of figuring out the RULE.

    Ah, but then you clarify:

    “By the way 1,3 and 7 is another correct number sequence.”

    …by providing another sequence of numbers, and not the RULE.

    In the following post, you clarify further:

    “HI Mumadadd I get it. Yes the correct answer was 3,5,7.”

    No, Leo, you don’t get it. “3,5,7” was not the correct answer, because the correct answer is the RULE, and not a sequence of numbers that happens to conform to the rule.

    And then, because you’re still not clear and think this is something different than it actually is, you offer this gem:

    “I know that people can cheat. I wouldn’t be surprised if some took this test and hacked into it.”

    I really, really, really, really, really, really doubt that anyone, on the basis of what was being asked for (just to remind you, it was asking for the RULE), found it necessary (or even in the least bit helpful) to hack into this test in order to derive the RULE.

    So then you start defending yourself when others point out that you don’t get it. For example:

    “Of course their is no answer. As there is an infinite number of answers that would be yes and no.”

    This also isn’t correct. There IS an answer! The answer is a RULE. There are also not an infinite number of correct answers (i.e., an infinite number of RULEs, unless you want to try to claim that there are infinite variations in the wording of this rule). Here’s the important distinction that you continue to miss — there are an infinite number of sequences that conform to the RULE, but there is just one RULE, and the puzzle is asking you to figure out the RULE, and not any sequences.

    Still defending yourself, you come back with:

    “My point was to demonstrate that I am smart enough to know how to do number sequences. Because I am sure many of you here think I am probably stupid. That I cannot use critical thinking, which couldn’t be farther from the truth. Plus the ample evidence your talking about is hearsay (anecdotal evidence) which you materialist’s claim must be all bogus when it comes to the so called paranormal.”

    Leo, I have no idea whether you are smart enough to do number sequences, but you have failed in the immediate task of reading through an assignment and correctly understanding what it was asking you to do.

    The rest of it was just getting to be more and more of a non sequitur. For example, I have no idea what “which you materialist’s claim must be all bogus when it comes to the so called paranormal” has to do with being able to follow the instructions in an assignment. (And since I haven’t repeated it in a few paragraphs, I’d better reiterate that the assignment was to figure out the RULE.)

  85. leo100 says:

    LC,

    The point wasn’t to follow the instructions to get to the rule but to show the materialist’s here that I am capable of doing at least simple number sequences.

  86. BillyJoe7 says:

    dream on leo

  87. BillyJoe7 says:

    🙂

  88. Pete A says:

    BillyJoe,

    My sequence 1,2,3, repeated 4 times was indeed an example of seeing a very obvious pattern, but not being sure if we’ve yet gathered enough information to predict the next number. In other words, we’re certain that the sequence is correlated, just not quite sure of the underlying correlation mechanism (the cause).

    Actually, the sequence I presented was just an arbitrary outcome of 12 consecutive rolls of an unbiased 6-sided die. The chance of that particular sequence occurring is only 1 in 6^12, approx. 1 in 2 billion. However, the following two sequences also have that same probability of occurring, but only the last sequence seems to be unpredictable (random):
    2 2 6 2 2 6 2 2 6 2 2 6
    3 5 6 4 1 6 2 1 4 5 3 2

    Humans are very poor at generating and detecting random sequences because we are so hard-wired to think that patterns (our perceived correlations) can result only from causation; never from a sequence of uncorrelated events.

  89. mumadadd says:

    Tmac – 5. 🙂

  90. Willy says:

    Leo: 1) The point was indeed to find the rule, 2) You failed to show the “materialists” that you can do simple sequences, and 3) “Materialism” has nothing to do with this thread. I am stunned that you still don’t get it.

  91. leo100 says:

    How have I failed to do simple sequences. For example 2, 4, 6, 8, 10, 12. I know that materialism has nothing to do with this thread. It’s just that with this thread is seems to show that only materialist’s have critical thinking skills. Which is totally untrue.

  92. BillyJoe7 says:

    mumadadd,

    You’re too late! 🙂
    The answer was supplied in crytic form at 2:12am 😉

  93. tmac57 says:

    BillyJoe7 and mumadadd- Excellent! I especially enjoyed the cryptic answer BJ7.
    I didn’t say anything then, as I wanted to see if anyone else wanted to weight in.
    Nice job!

  94. Damlowet says:

    You’re all bastards! 😉 How the hell is the answer 5?
    I had spent a little time looking up the chords of Row Row Row you boat, nothing there, 3 rows of 7, nothing obvious, 7 rows of 3, nothing there either. What am I missing. I thought I had a little handle on sequences.

    Damien

  95. tmac57 says:

    Damlowet- Well, I can tell you are not a person of letters. 😉

  96. SteveA says:

    leo100: “How have I failed to do simple sequences. For example 2, 4, 6, 8, 10, 12. I know that materialism has nothing to do with this thread. It’s just that with this thread is seems to show that only materialist’s have critical thinking skills. Which is totally untrue.”

    Leo, the question is, what’s the rule behind your sequence and what test could you devise to try and disprove it.

    In your example, the rule seems to be ‘ the next even number’. To try and disprove that I could test it by saying the next number in the sequence is 13. If that turned out to be the right answer, then the rule is disproved and I’d have to try to think of a new rule that accommodates this new information.

  97. Pete A says:

    SteveA, excellent points.

    A sequence of numbers [and all other events] is indistinguishable from an unordered set of those same numbers [or events], unless and until, there is a clearly established causal mechanism (e.g. a rule) that has sufficient power to adequately predict [as in: fit for its purpose] the next event in the sequence.

    Leo has progressed only to the stage of generating basic number sequences from known simple rules. Leo does not yet begin to understand that short sequences, which have a clearly discernible pattern, frequently occur in the long-term sequences produced by highly complex multi-rule systems, chaotic systems, and stochastic systems.

  98. Damlowet says:

    @Bruce

    I don’t think the two questions that you are posing are as unrelated as you suggest. If the question was asked to find the sequence from adding an additional number only to the end of the sequence, the rule could still be figured out (if basic enough). I probably still would have used a similar logic to determine what I did in the original.
    Using your example, if you were asked for the next number to the original sequence, all increasing numbers would have worked, not just one as you suggest, as all increasing numbers above 8 would have been correct using the rule.

    @Leo, 20 odd years ago, IIRC, ‘we’ were presented a few number sequences to figure out in a year 11/12 math class. I seemed to be the only student in that class at the time to get the jist. Try this out, (but don’t feel bad if you can’t solve it). 52,74,48,77……… even with the answer, the sequence isn’t obvious.

    @TMAC57, screw your letters and their obscure usages! :), I will solve it (I hope).

    Damien

  99. TheBlackCat says:

    I tried 30 combinations, including 13 no’s. I first tried any sequence, then the same numbers in any order, then the given numbers in the given place, then monotonically increasing. I seem to have tried a bigger variety of numbers than most others, including (1002,1004,1008), (-2, 0, 4), (6.1, 6.2, 6.3), and (10000, 100000, 1000000). Once I thought I had it I tested every reasonably simple alternative I could think of, like (6, 6, 8), (3, -4, 8), (7, 5, 4), and (-4, 5.4, 3.3).

  100. tmac57 says:

    Damien- I like a person with tenacity! I know you will get it. (then you will throw a brick at me 😉 )

  101. QuiteDragon says:

    # tmac57

    Got it. Took me a while. It was a great (necessary) clue, but I am a little thick.

  102. QuiteDragon says:

    Interestingly, my wife and I have been watching “Gran Hotel” (man, talk about a soap!). The same puzzle (using different input and result than “row”) was on an episode we watched today. Possibly influenced my “getting” the answer to yours, but I like to think nothing happens in my brain without my notice and approval ;^)

  103. ChrisH says:

    This test illustrates why I hate these so called “logic” tests. I was not going to waste one of my ten free looks at the NY Times so early in the month for some ridiculous “test.”

    I am glad I waited because the answer was revealed by Dr. Novella on the SGU. The only rule was that the numbers needed to increase. Woo hoo.

    It is too freaking general, and not at all interesting. It is not a Fibonacci sequence nor a Pascal’s triangle. But at least my favorite three numbers qualify: i, e and pi. Essentially the square root of negative one, the base of the natural logarithm and the ratio of a circle’s circumference to its diameter.

    Well it works if the “rule” also applies to complex numbers. One reason why these silly number sequence puzzles annoys me is that I am an engineer and I deal with applied math where the “number line” is a complex plane. Reality is quite circular, and we are in awe of Euler’s Formula. (my last advanced engineering math exam included lots of drawings of circles… and yes I got a decent grade, it was the probability class that caused me grief).

    In simpler language (humor for some): I used to be a random structural vibration engineer, so all number sequences need to go through a Fast Fourier Transform prior to consideration.

  104. BillyJoe7 says:

    Chris,

    “The only rule was that the numbers needed to increase. Woo hoo”
    That was not the point of the exercise.

    “It is too freaking general, and not at all interesting”
    The interesting part is not the rule but the process by which the rule is discovered.

    “But at least my favorite three numbers qualify: i, e and pi.”
    In fact, these numbers do not qualify. They are not even acceptable.

    “I used to be a random structural vibration engineer, so all number sequences need to go through a Fast Fourier Transform prior to consideration”
    That at least was clever. 🙂

  105. ChrisH says:

    “The interesting part is not the rule but the process by which the rule is discovered”

    Which is tiresome, especially when the solution is so mundane. Put in numbers, hit button, see result. Try other numbers, see results. Try to find a pattern. Blah.

    “In fact, these numbers do not qualify. They are not even acceptable.”

    So only real whole numbers? No transcendental numbers and nothing complex. Boring, and ignores an infinite amount of possibilities. Mathematics is too interesting to be bothered with random made up rules.

    I have always hated puzzles, especially if there is some random trick that has no real logic. But that is just me. It may be part of the reason I stopped reading fiction, since it was often much more mundane than non-fiction. (My latest fave is Glorious Misadventures: Nikolai Rezanov and the Dream of a Russian America by Owen Matthews)

  106. BillyJoe7 says:

    Oh well, perhaps this post wasn’t meant for superior beings such as yourself. 🙂
    Some of us are happy to be reminded that we are all susceptible to bias from time to time. 😉

  107. ChrisH says:

    I also hate baseball, and prefer spoken word to music. It is just my bias.. oh look a squirrel!

    😉

  108. BillyJoe7 says:

    “I…prefer spoken word to music”

    I retract the first sentence of my last post. |:

  109. ChrisH says:

    Okay dokay. I did enjoy my mix of Edgar Allen Poe being read by Basil Rathbone and Vincent Price combined with the songs of “Tales of Mystery and Imagination” by the Alan Parson’s Project. Dear hubby does tell me I am not normal, and I am good with that.

  110. leal says:

    I’m a little late to the party, but I read this blog post when it first came out and knew right away what the rule was. It’s been bugging me ever since, but I finally came across the YouTube video I saw about exactly this about a year ago: https://www.youtube.com/watch?v=vKA4w2O61Xo

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