# Possible positions after 40 moves

When clearing out my loft, I came across an old Usborne Childrens guide to chess (clearly didn't help much). Whilst flicking through I found this fact which i remember even then fried my brain.

'The total number of different games lasting 40 moves each is greater than the number of atoms in the universe'

It always amazed me, because lets face it, theres quite a lot of atoms in the universe. Just wondering if anyone can verify or rubbish this statistic for me, I'd look into it myself but I'm a little bit too, erm, busy.

i;ve heard that number very many times....in many places...if that counts for anything!

There are 400 different positions after each player makes one move apiece. There are 72,084 positions after two moves apiece. There are 9+ million positions after three moves apiece. There are 288+ billion different possible positions after four moves apiece. It goes on, by 40 moves each I won't be surprised if there were more than the atoms in 20 universes.

cheater_1 wrote:

Just as you CANNOT put 20 ounces of water in a 16 ounce glass, You cannot have have something larger than the universe contained in the universe. It's a physical impossibility.

but that statistic is not saying that the game is bigger than the universe, just that the number of possibilities in chess is greater than the number of atoms in the universe, not that i agree with it...

the estimated number of possible positions in an entire game of chess (never mind after 40 moves) is actually 1050 according to victor allis (A computer scientist that has made huge advances in game AI), FAR less then the 4x1079 and 1081 atoms in the known universe.

However, the game tree complexity is estimated at 10120 By claude shannon, an information therorist.

Referances (you probably don't want to read these):

Shannons report on programming a computer to play chess:

http://archive.computerhistory.org/projects/chess/related_materials/text/2-0%20and%202-1.Programming_a_computer_for_playing_chess.shannon/2-0%20and%202-1.Programming_a_computer_for_playing_chess.shannon.062303002.pdf

A thesis by Victor allis:

http://fragrieu.free.fr/SearchingForSolutions.pdf

I don't have any referances for the atoms in the known universe, sorry.

Numbers....Too....Big....Brain....Going....To....Explode

after some brief calculations, and a little homework, i got an answer...i think.

number of atoms in the universe- 3  x 10 to the 79 power

Number of possiable chess postions-  10 to the 120 power

Sorry Cheater_1 your wrong. It isn't even close.

Take one piece. There are 65 possible locations (64 on the board, plus 1 more - "off the board") where it can be located. Take a second piece. There are 64 possible locations for it; total possible combinations? 4160.

Add a third piece, and you've got 262080 possible combinations.

By the time you've placed all 32 pieces, you're talking about 9.5 x (10 to the 53rd power) - 95 followed by 52 zeros. That's a big number - but nowhere near the estimated 4x10^79 (four followed by 79 zeros) hydrogen atoms in the universe.

AND not all of these are legitimate or possible.  For example, bishops are limited to half the squares. Pawns can never be on the back rank, and can't legitimately reach but about half the remaining squares. Wherever one king gets placed, that's between three and eight squares the other king can't be on. So even that theoretical number gets reduced by quite a bit.

But that's not quite the same thing as the number of possible SEQUENCES of 40 moves by each player. If each player has a choice of 16 pieces to move on each turn, then the total comes to 16^80, or about 2.14x10^96 - 2 followed by 96 zeros. (And of course, each player has multiple choices as to how he will move a piece, once he's selected it - multiple different squares he could move it to.) But of course, that's not accurate; each player is limited to 10 pieces for his first move, many of the move sequences lead to checkmate before 40 moves, so there would be no continuation, and so on.

But let's see if there's an easier way to reach a meaningful answer. The requirement is that the game has to last 40 moves. How many "decent" moves does each player have on average for each turn? There may be a lot of POSSIBLE moves on each turn, but many of these lead to immediate loss of material or bad positions, are illegal under the rules of chess, and so on. Let's just guess and say that on average, each player has a choice of five "decent" moves. That drops the number of possible sequences to about 8.27x10^55 (if we assume six possible decent moves, the number becomes 1.8x10^62). Or, to put it another way, we have to assume that each player has an AVERAGE of 10 decent moves on each turn, in order to pass that 4x10^79 figure.

Conclusion? Nah - there are NOT more possible 40+ move games of chess, than there are atoms in the universe.

cheater_1:  How much bigger do you think 10^120 is compared to 10^11?  Well if it was 10^121, it would be 11 times the power.

That's:

10^11 x 10^11 x 10^11 x 10^11 x 10^11 x 10^11 x 10^11 x 10^11 x 10^11 x 10^11 x 10^11

Lets say 10^11m is atoms in a drop, then imagine the drop as another atom and so on.  You have to zoom out 11 times to reach 121.  The universe just isnt that big.  Where do you go to make your next jump.  A quick google throws up 10^50 atoms in the world.  that means in order for there to be 10^120 atoms in the universe, there would have to be 10^70 worlds!  I really think you need to rethink just how big 120 zeroes after a number is!  Each three zeroes is not just another thousand.  It's another thousand times everything you have done already!

hmm, I can assure you, it is more than possible to play a bad move in a game of chess...

why are we still debating? just count them. The Opening post never said "good postions"

There are 168024 possible legal positions with just a King and pawn vs. king endgame

I suppose the guy didn't take the universe to be endless, did he?

the universe isn't endless....which makes this problem much easier,lol

Since we have no idea how BIG the Universe is, we can't say that it is a FACT.

That is why I believe chess is a sport, because if there were only so many moves available then I would only consider it a game. The choices become expotential as more moves are played. I read somewhere also that the starting position of a chess game is pregnant with more possibilities than the universe. But that doesnt mean that much, its an abstract notion.

lol, you missed MY point.  you will not ever get close to 120, even with ridiculous overestimates, like assuming there are 1,000,000,000 ocean's worth of atoms on earth, and 1,000,000 planet's worth of atoms in the solar system, and so on, you will still not get anywhere near ^120 atoms in the universe.  In fact, you could have more than 10^11 universes and still not get close.  Clear enough yet?  In fact, if we weren't being extremely generous in estimates, and were taking more reasonable numbers, if we had an entire universe for every atom of the earth, you would have ~ 10^129 atoms, so the number of chess games is only a billionth of that.

Why are we bring water into this?

The  best possible estimate number of atoms in the Observable universe- 3 x 10 to the 79

The number of possible chess psotions after 80 moves- 10 to the 120 power.

No math required. The second number is bigger! No way around it!

Get My Point?

Not only bigger, but given those estimates 10^41 times bigger.  ~10^41 universes of atoms.  That's 1 followed by 41 zeroes!

cheater_1 wrote:

Just as you CANNOT put 20 ounces of water in a 16 ounce glass, You cannot have have something larger than the universe contained in the universe. It's a physical impossibility.

I would agree if the number of possible positions after 40 moves was a physical thing like atoms of hydrogen and oxygen, but it is not -- it is an abstract concept.

Similarly, the number of possible arrangements of any number of the atoms in the universe up to and including all of them is an abstract concept that vastly outnumbers the total number of atoms in the universe -- by your reasoning, is this an invalid concept?