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### Meta

The first one is not like the others.

The second one has no boundary, the third one has a different shape, the fourth one has a different color, the fifth one has a different size.

Only the first one is not different from the others in any specific aspect. That’s what makes it very different.

Nice puzzle!

Oh sorry, should have followed the link before posting … there are already lots of answers.

When I explained the answer to a friend, I ended up giving paradoxical description of the answer. It’s the only one that’s not unique.

The puzzle is paradoxical.

The first shape is the only shape which satisfies the property “there is no property which the other four shapes satisfy but this shape doesn’t.” But then the property of not having such a property is itself such a property. [There is probably a much better way of writing that.]

So: the fact that it is not special makes it special… which makes it not special again. And so on.

Brandon: It does have a paradoxical feel to it, doesn’t it? One way to get around the paradox is by putting properties into a hierarchy of classes. For example, we could say that a “Level 1 property” is a physical/visual property. Then “Level 2 properties” are allowed to refer to Level 1 properties (but not other Level 2 properties), “Level 3 properties” are allowed to refer to Level 1 or 2 properties, and so on. Then properties like “being red” and “having a border” are Level 1 properties. Then the first shape then satisfies the Level 2 property that “There is no Level 1 property which the other four shapes satisfy but this shape doesn’t”. But now this property is not allowed to refer to itself, since it refers to Level 1 properties and it is Level 2.

Coming up with an interesting Level 3 property and/or a puzzle whose solution involves such a property is left as an exercise for the reader!

Your hierarchy of properties sounds rather like Russell’s Theory of Types! http://en.wikipedia.org/wiki/Type_theory

Pingback: One of These is not Like the Others « Random Walks

Robert: Indeed! You didn’t think I just came up with the idea on my own, did you? ;-) Russell’s original Ramified Theory of Types has fallen out of favor, but there are still other type/logic systems which use a similar hierarchy of universes (some of which come up in my research on dependent type theory).

This puzzle was too simple for me to solve.

When a puzzle has a simple solution I always fail to see it.

But when a puzzle is sufficiently subtle or complex the solution becomes immediatley obvious to me.

Naturally, I have yet to see a sufficiently subtle or complex puzzle.