The Odder One Out

My recent entry, where I asked you to choose the odd one out among these images

Odd One Out

was extremely popular. It was republished all around the world and brought my blog as much traffic in one day as I used to get in a month. Not only did I read the many comments I received, I also followed up on other peoples’ blogs who reprinted my puzzle — at least those that were in either Russian or English. I also got private emails and had many conversations in person about it. The diversity of answers surprised me, so I would like to share them with you.

As I’ve said before, I do not think there is a correct answer to this type of question, but I was disappointed by some of the answers. For example, those who simply said, “The green one is the odd one out,” made me feel that either they hadn’t read the question or hadn’t thought about it very much. It’s a shame that these people spent more time sharing their opinion with the world than thinking about the problem in the first place.

I wouldn’t mind someone arguing that the green one is the odd one out, but in this case an explanation is in order. Many people did offer explanations. Some told me that we perceive the color difference stronger than all other parameters I used, and the green figure pops out of the picture more than anything else. In fact, I personally perceive color difference the strongest among all the parameters, but since there are people who are color blind, I would disregard my feelings for color as being subjective.

You can create a whole research project out of this puzzle. For example, you can run an experiment: Ask the question, but flash the images above very fast, so there is no time for analysis — only time to guess. This allows us to check which figure is the first one that people perceive as different. Or you can vary the width of the frame and see how the perception changes.

Color was not the only parameter among those I chose — shape, color, size and the existence of a frame — that people thought was more prominent. My readers weighed these parameters unequally, so each argued the primary importance of the parameter they most emphasized. For example, one of my friends argued that:

The second figure should be the odd one out as, first, it is the only one without a frame, and, second, it is the only one comprised of one color rather than two. So it differs by two features, as others differ only by one feature.

A figure having one color is the consequence of not having a frame, so this particular friend of mine inflated the importance of not having a frame.

However, I can interpret any feature as two features. For example, I can say that the circle is the odd one out because not only is it a different figure, but it also doesn’t have any angles. Similarly, the last one is the smallest one and the border width is in a different proportion to its diameter.

On a lighter side, there were many funny answers to the puzzle:

  • The one that says I am special.
  • The right one because it is right.
  • The fourth one, because four is the only composite index.
  • The one that says I am not special.

For the which-is-the-odd-one-out questions, the designer of the question is usually expecting a particular answer. So here’s the answer I expected:

There is only one green figure. Wait a minute, there is only one circle. Hmm, there is only one without a frame and there’s only one small figure. I see! The first one is the only figure that is not the odd one, that doesn’t have a special property, so the first must be the odd one out. This is cool!

And the majority of the answers were exactly as I expected.

Since this is a philosophical problem, some of the responses took it to a different level. One interesting answer went like this:

All right, the last four figures have special features; the first figure is special because it is normal. Hence, every figure is special and there are no odd ones here.

I like this answer as the author of it equated regular features with a meta-feature, and it is a valid choice. This answer prompted me to write another blog entry with a picture where I purposefully tried to not have an odd one out:

Find Odd One Out

Though I wrote that the purpose of this second set of images is to show an example where there is no odd one out, my commentators still argued about which one was the odd one out here.

Finally, I would like to quote Will’s comment to my first set of images:

The prevailing opinion is that the first is least unique and is therefore the oddest. But it is the mean and the others are one deviation from it. Can the mean be the statistical anomaly?

And Cedric replied to Will:

Yes, I think the mean can be a statistical anomaly. The average person has roughly one testicle and one ovary. But a person with these characteristics would certainly be an anomaly.

17 Comments

  1. Leo:

    A possible answer to the second puzzle is “the invisible one”.

  2. Qiaochu Yuan:

    The mean can certainly be the statistical anomaly depending on what you mean by “the mean.” A person who has a single average trait is average. But a person all of whose traits are average is exceptional. For example, a person all of whose facial proportions are average would be extremely beautiful; every feature would be in just the right place.

  3. ecjs:

    I was going to say that : “There is only one green figure. Wait a minute, there is only one circle. Hmm, there is only one without a frame and there’s only one small figure. I see! The first one is the only figure that is not the odd one, that doesn’t have a special property, so the first must be the odd one out. This is cool!”.

    That’s not cool it was expected. :'(

  4. Mike Theodore:

    Great post. Thanks.

  5. Merlijn van Veen:

    You can choose the first form, because it only differs one feature from the others, while the others differ two features from each other.

    Or you can say that it is the first because this form has no unique feature on its own.
    This is what I saw.

    But if you talk about the one that is ‘most unlike’ the others, then it is definitely the green one – it jumps at you the moment you see the question.
    But this way of looking at it is not analytical.

    This question has an obvious analytical pattern here however: it is deliberately created in order to make the one that is not odd stand out with a gameplay of size, colour and shape.
    That is the essential meaning of this question, and only one answer addresses that.

    Kinship in forms, as if they evolved from each other, is another perception.
    Here also the first form takes central stage: there is a dwarf ‘subspecies’, a green and a round variety, and the variety without a shell / exoskeleton.
    In that case it is not usual to say that the central, holotype, form is the odd one out.
    It is the one that STANDS out from the others because its form is typical for ‘the species’.

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  7. Marco Tocornal:

    What about saying: “the question is wrong: there’s no odder one”, or “‘odd’ is not well defined”.
    In the second set of figures, all have two diferences with the other ones so I think that there’s no odder one in that one. IN the first set, it’s not so easy.
    What about:
    In a set of figures, we can create a partition of it such as in all its components’ elements are exactly equal, we have some sub-sets then such that in every one there’s different figure, but every element of a sub-set is different to any element of any other sub-set, therefore we say A is relatively-odd to B if A and B belong to different sub-sets.
    In our case, all are relatively odd to the other ones.
    We can also use a Venn diagram where every circle has a lable as “square”, “presence of frame”… Then the first figure of the first set whould be it’s intersection, therefore it’s the less odd, but the other ones are equally odd, so the answer of the questions whould be “all except from the first one”. In the second set, there’s no intersection and then there are no odd one.

  8. Erik van Bergen:

    From a evolutionary point of view the left figure is the ancestral one. All other figures can derive from it in just one step…

    In the second puzzle you see that you will always need two steps the get from one figure to the other.

  9. Annelyse:

    I’m going to use this one on my blog. Great!!!

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  13. Doug jenkins:

    I’m the author of the Newton Excel Bach blog links above, and I’ve been giving this some more thought.

    It seems to me that both the puzzle and the Monty Python scene (see link) are examples or Russell’s Paradox, which asks if a set which is not a member of itself is a member of the set of all sets that are not members of themselves. If it is, it isn’t, and if it isn’t then it is.

    Similarly in the Monty Python scene the crowd on being told that they are “all different” look at one another and proudly respond in unison “yes we are all different”, all except for one who has to report that he isn’t. As soon as he does so he becomes different, as the only one who isn’t different, but then immediately becomes one of the crowd who are all different, and so on …

    With the shape puzzle (and “odd man out” puzzles in general) we are looking for some set of features that all the shapes share, except for one. So, on the first pass, we see that common features, shared by all but one are: colour (red), shape (square), size (big), and border (present). But now we have a problem, there are four odd ones, not one, so we need to look for another feature shared by all but one of the shapes, so we add a fith feature, to match the “standard” the shape must have exactly one “non-standard” feature. Now (counting from left to right) shapes 2 to 5 can all proudly proclaim, yes we are all different, but poor old shape number 1, with its boring large square shape, plebian red colour, and common as muck thick black border, must nervously half raise its hand and say “I’m not!” But being different is part of the standard, so it has one non-standard feature, so it doesn’t, so it does, and so on …

    But wait, if it is not different, it has exactly one feature that is not part of the standard, so it does match the standard, but if it does match the standard it is not different to the other four shapes, so does not have one differing feature, so it does not match the standard, and so on …

    So the question is, if a set is defined as being those objects that meet a certain standard except in one respect

  14. Merlin:

    @Doug Jenkins

    But what is the standard?
    The standard is not being different at all cost, being different is the question not the standard.
    The standard concerns on what basis a shape is different from another.
    So the question becomes: on what basis does one shape stand out?
    If I say: now it is unique because it lacks a unique feature other shapes do have, I cannot say that this uniqueness
    is ex- or interchangable with the unique features of the other shapes. I cannot exchange the basis.
    One deals with a state of being, the other with having a well defined property.
    And in that state of being, lacking the unique property, it stands out. To be or to have, that is the question.

  15. Edward Welbourne:

    Qiaochu Yuan: indeed, while everyone is used to the standard one-dimensional Gaussian’s strong bias for being near the middle, if you look at the n-dimensional Gaussian, the probability of being a given radius from the centre rapidly (as n increases) comes to fit a pattern where being *near* the mean is anomalous. The distribution of distance from the mean is a gamma distribution, whose order parameter grows with n. Much the same happens with any distribution, not just the gaussian, essentially because the set of values within h of radius r grows in proportion to r^n.h and, for small r, this dominates over the density, regardless of (continuous) distribution. Thus, whenever several (more or less) independent random variates are considered together, being near the mean *on all of them* becomes exceptional.

    Doug Jenkins: I, too, enjoyed the liar paradox element – the first is unique in being the only one that’s not unique; c.f. the standard proof that every natural is interesting (if not, which is the least non-interesting natural ? but surely, being the first such renders it interesting). All the same, one can readily escape the paradox in this case, by the very fact that each of the others has a unique actual geometric property, specifiable without regard to the company in which we find it, where the one with the majority value for each property is only unique in a second-order sense, specifiable only in terms of which other figures it is considered alongside.

  16. Nate:

    Interesting, but the frame attributes are color and proportion, I think. The frame colors are black and white but white frame doesn’t show on a white background so that’s the one with the unique attribute.

  17. Julia:

    Wonderful concept. I really like the way that you point out how this can be adapted to puzzle based learning.