Hat tip to Tanya Khovanova.

- algorithm Apollonian approximation art bar beauty binary binomial coefficients birthday book book review cards carnival Carnival of Mathematics Cassini chocolate combinatorics complex cookies counting decadic decimal diagrams elements expansion factorization fibonacci formula fractal game games gasket graph groups Haskell hyperbinary idempotent identity integers interactive irrational Ivan Niven Lagrange lehmer lucas MaBloWriMo making Mersenne nim number numbers objects omega order pi prime problem programming proof puzzle rectangles repunit review sequence square squares strategy subgroups tessellation test triangular units video visualization X
### Blogroll

### Fun

### Reference

### Categories

- algebra (43)
- arithmetic (58)
- books (28)
- calculus (6)
- challenges (51)
- combinatorics (8)
- complex numbers (5)
- computation (42)
- convergence (9)
- counting (31)
- famous numbers (48)
- fibonacci (18)
- fractals (12)
- games (24)
- geometry (45)
- golden ratio (8)
- group theory (26)
- humor (6)
- induction (7)
- infinity (17)
- iteration (23)
- links (73)
- logic (6)
- meta (40)
- modular arithmetic (24)
- number theory (67)
- open problems (11)
- paradox (1)
- pascal's triangle (8)
- pattern (79)
- people (20)
- pictures (51)
- posts without words (13)
- primes (33)
- probability (6)
- programming (17)
- proof (61)
- puzzles (11)
- recursion (12)
- review (19)
- sequences (28)
- solutions (28)
- teaching (13)
- trig (3)
- Uncategorized (4)
- video (19)

### Archives

- September 2016 (2)
- August 2016 (5)
- July 2016 (2)
- June 2016 (4)
- May 2016 (4)
- April 2016 (2)
- March 2016 (3)
- February 2016 (9)
- January 2016 (8)
- December 2015 (5)
- November 2015 (29)
- August 2015 (3)
- June 2015 (2)
- April 2015 (1)
- May 2014 (1)
- December 2013 (1)
- October 2013 (1)
- July 2013 (1)
- June 2013 (1)
- May 2013 (1)
- April 2013 (3)
- March 2013 (3)
- February 2013 (2)
- January 2013 (5)
- December 2012 (3)
- November 2012 (4)
- October 2012 (5)
- September 2012 (1)
- August 2012 (4)
- July 2012 (1)
- June 2012 (6)
- May 2012 (2)
- April 2012 (3)
- March 2012 (1)
- February 2012 (4)
- January 2012 (5)
- December 2011 (1)
- November 2011 (7)
- October 2011 (4)
- September 2011 (6)
- July 2011 (2)
- June 2011 (4)
- May 2011 (5)
- April 2011 (2)
- March 2011 (4)
- February 2011 (1)
- January 2011 (1)
- December 2010 (1)
- November 2010 (4)
- October 2010 (2)
- September 2010 (1)
- August 2010 (1)
- July 2010 (1)
- June 2010 (2)
- May 2010 (3)
- April 2010 (1)
- February 2010 (6)
- January 2010 (3)
- December 2009 (8)
- November 2009 (7)
- October 2009 (3)
- September 2009 (3)
- August 2009 (1)
- June 2009 (4)
- May 2009 (5)
- April 2009 (4)
- March 2009 (2)
- February 2009 (1)
- January 2009 (7)
- December 2008 (1)
- October 2008 (2)
- September 2008 (7)
- August 2008 (1)
- July 2008 (1)
- June 2008 (1)
- April 2008 (5)
- February 2008 (4)
- January 2008 (4)
- December 2007 (3)
- November 2007 (12)
- October 2007 (2)
- September 2007 (4)
- August 2007 (3)
- July 2007 (1)
- June 2007 (3)
- May 2007 (1)
- April 2007 (4)
- March 2007 (3)
- February 2007 (7)
- January 2007 (1)
- December 2006 (2)
- October 2006 (2)
- September 2006 (6)
- July 2006 (4)
- June 2006 (2)
- May 2006 (6)
- April 2006 (3)
- March 2006 (6)

### 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.