# Reality Conditions

## Monday, March 06, 2006

### Weirdest Google Search leading someone to this blog (so far)

Someone got here some hours ago from the Google search page:

does a queen have more entropy than rook and bishop

Now, in case you have come back still looking for an answer: I'm not sure what you think "entropy" means and I can't imagine why you would like to know this, but according to the standard physical meaning of entropy as "logaritm of the number of possible states" a queen has less entropy. During a game of chess it can be in any of 64 squares, so its entropy is log (64) = 4.158..... For a rook and a bishop the calculation is more complicated because it is not clear if the bishop is any bishop, or one in particular of the two bishops each player has (the light-squared one or the dark-squared one). If the first, then the entropy is log (64 x 63) = 8.302..... , and if the second, then it is log (32 x 63) = 7.608..... You see that in each case the queen's entropy is smaller. (I have used units in which Boltzmann's constant equals 1).

Or maybe you were not asking for the abstract, "informational" entropy of the pieces, but for the concrete thermodynamical entropy the wooden chess pieces have? Then again the rook and bishop have more entropy, simply because combined they have much more atoms than the queen, and entropy is an extensive magnitude. Assuming, of course, all the pieces are kept at the same temperature and pressure.

Hope that was useful...

#### 8 Comments:

• But the queen has more (thermodynamical) entropy than either rook or bishop one-on-one (that is how I understood the query). One-on-two, she has less.

• Thanks for commenting, coturnix! I'm a long-time reader of your blog and it is a nice surprise to see you drop by here.

I actually think the most plausible interpretation of the query is that the searcher thought "entropy" to mean something like "power" or "number of available moves". This makes it an fairly interesting question to ask of a queen contrasted with a rook and bishop put together (as the queen combines their possible moves, but is in practice a more powerful piece than the conjunction of the other two).

With either of the interpretations I suggested (a bit toungue-in-cheek) in the post, or the one you are suggesting, there is no point in asking for a queen, rook or bishop (instead of any other chess pieces, or any small objects in fact).

• I wonder if, tongue-in-cheek, of course, once can put a number on entropy of each chess piece, including pawn, king and knight, then make a cute visual that can be spread around the blogosphere...

• Hum, intriguing idea... but with which interpretation of "entropy"? I guess it would have to be redefining it as number of possible moves or something similar. Because for the actual thermodynamical entropy results would depend on the size of the pieces, which varies from chess set to chess set; for informational entropy they would be the same for all pieces with the exception of pawns (which cannot be in the first line) and perhaps bishops (which can only be on one colour) and maybe even kings (because there is no valid position with the two kings in contiguous squares)...

A different project: to define a notion of chess entropy as a property of chess positions, instead of pieces. This would be a closer analogue to entropy in physics as a property of macrostates. For example, maybe the entropy of a position could be defined as (the logaritm of) the number of possible ways that position can be reached from the initial one with legal play.

Hey, the second law of thermodynamics seems to hold! There will normally be more such possible paths to a position for later positions, so entropy grows in time. Pawn movements and captures are irreversible transformations; other movements are reversible and leave entropy constant...

OK, this is clearly material for another post. If tomorrow when I wake up all this seems to make sense and not be just a random expression of nerdiness, which is most likely.

• Thank you for the great story.

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• I'm fairly certain the searcher was looking for information-theoretic entropy of the given piece(s) on an empty board, i.e. the average number of bits needed to represent each subsequent position assuming a uniform starting position (and probably a uniformly selected subsequent position).

That way the problem is harder because the bishop can get in the way of the rook (or vice-versa) so that there are not as many possible subsequent positions as there would be with a single piece alone on an empty board.

By  Anonymous, at 3:34 AM, March 14, 2011

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• Let's suppose that the game of Chess was created by some Higher Minds and thru they supraconscious link with the ecosystems using the chess pieces are able to tune in into its entropy and produce amazing results.Wizard apprentice.

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