## Thursday, February 27, 2014

### True or false?

Approximately 988 million positions that can be reached after just 4 moves for white and 4 for black! True or false?

Anonymous said...

I can't count more than 64.

Anonymous said...

No, seems it should be more like 15 billion.
(25 avg moves per position raised to the eighth power), i.e. 8-ply
(4 moves white and 4 moves black).

Anonymous said...

That seems about right to me. There are not 25 avg moves during the first four moves. The rooks, bishops, etc. are constrained. The pawns, knights, are 10 pieces, and a couple minor pieces will be able to move, so call it 13 or so moves on avg. 13.3^8 = 1 billion, which matches the claim.

Anonymous said...

True. If you don't believe me, then count it yourself.

Prof.S.G.Bhat said...

TRULSE

Peter said...

Should be a lot more if you count all possible moves. There are 20 possible moves to start (8 pawns x 2 moves plus2 knights x 2 moves), and it goes up from there. I followed one line of the London System for 4 moves and counted the possibilities in each position: (20x20) x (27x27) x (27x29) x (39x32) and get 2.85 x 10^11

Luc Despontin said...

I think MORE. +-500Billions

Anonymous said...

Definitely seems false to me as 988 million seems way too little. However 988 million is a very specific number. Surely that would have been rounded off to approximately 1 billion if it isnt an exact calculation.

So lets try to do a bit of rough calculation (after all thats what chess players do all the time).

There are 20 initial moves for white so if we make a conservative estimate 20^8 that would be 25.6 billion.
That does not take into account the additional moves that would be possible once white has moved (for example 1.e3 gives white 29 possible options for move 2).

So the actual number has to be greater than 25.6 billion. There will be a small subtraction as the question asks about positions so transposed move orders would count only as a single position.

So from the rough calculation and the fact that the question gave the specific number "988", I would guess that the real answer is actually 988 billion positions

Anonymous said...

Ok I did some research after guessing that the real answer is 988 billion and it seems like 988 million is actually the correct answer. This however seems rather counter intuitive. The only explanation has to be that there are far more transposed positions after 8 half moves than I had initially considered. There are also illegal positions where one side plays a move that does not get out of check. And of course there are many possible positions where one side gets checkmated before 8 half moves.