# Mathematics and chess

• Last updated on 9/27/13, 5:27 AM.

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The number of possible chess positions after White’s first ply move is 20 (16 pawn moves and 4 knight moves).  There are 400 possible chess positions after two ply moves (first ply move for White followed by first ply move for Black).

There are 5,362 possible positions (White’s second ply move) or 8,902 total positions after two ply moves each. There are 71,852 possible positions or 197,742 total positions after four moves. There are 809,896 possible positions or 4,897,256 total positions after 5 moves.There are 9,132,484 total positions after 6 moves. From move 7 the possible positions stabilize as chess lines end, even from move 2 some chess lines end. There are +-10,921,506 total possible positions after 7 moves.

The special draw, the King's draw, should occur a minimum of 32 times. The longest recorded game ended in a draw after 269 moves.

***

There is a built in limit in the logical positions as the average chess game is about 30 moves, 60 moves and above chess games are a rarity. Lots of chess games end between moves 3 and the final move and the pieces decrease as they are captured. In end game situations the material combinations their frequency and the number of moves needed to mate or draw are known and it is in the region of tens of thousand, limiting the logical possible positions in an end game situation to hundreds of thousand.

Phase                         Classification                ~ # of positions          Moves

·         Initial position                     *                                     1                            0

·         Opening                        xxo*oxx                +-    5     x 10^6           1   -  5

·         Opening                  xxxooo*oooxxx           +-  40     x 10^6          6   - 10

·         Middle game       xxxoooo*ooooxxx          +-  45     x 10^6         11  - 15

·         Middle game         xxxooo*oooxxx            +-  40     x 10^6         16  - 20

·         End game                xxxo8*8oxxx              +-    5     x 10^6         21  - 25

·         End game                      xo*8x                   +-    5     x 10^6         26  - 30

·         End game                        o*8                    +-    0.1  x 10^5         31   - Final move

 Logical possible positions                                 +- 140.1   x 10^6  + 1 Possible/playable chess games (Avg game 30 moves)  +- 4,670,033 ~# Of total draw positions @ 7% of playable games    +-    326,933

*=draw, o=winning/lose, x=other, 8=known end game combinations

A guesstimate is that the maximum logical possible positions are somewhere in the region of +-140,100,033, including trans-positional positions, giving the approximation of 4,670,033 maximum logical possible games, thus making chess very playable.

When compared to the numbers available from online databases the actual number of games played so far , for reasonable players, seem to be somewhere in the region of +-2,910,286 which should be taken as a minimum number for the possible logical games.

See Shannon Number for the Upper bound for Random Chess.

Also, see this.

• 8 years ago · Quote · #41
The sad thing is most people may be able to remember several values for Pi, they completely forget what Pi is associated with.
• 8 years ago · Quote · #42
lol well lets not crack chess, lets leave value in the game
• 8 years ago · Quote · #43
the name for the number of distinct combinations is twenty-quattuordecillion
• 8 years ago · Quote · #44

if x - y = 0

then 2x - 2y = 0

so does x - y = 2(x - y)

or 1 = 2.

Sorry but I have nothing useful to contribute...  For some reason I only remember three things from maths A-Level, 1) as above 2)13,983,816 (The chance of winning the lottery) 3) How to sucessfully administer an Axe Kick. (Our teacher was hard)

• 8 years ago · Quote · #45

you divided by zero.

And a better question is how many chess moves are actually good per move.  Usually there is anywhere from 1 to 20

• 8 years ago · Quote · #46

I love those formulas where you can make 1=2, but they are only correct in the context of a graph with a curve where a curve intersects a plane at the 1 and 2 integer.  Of course, this is not considering anything but one plane, but it is fascinating nevertheless.  You just have to see it through the right eyes :)

• 8 years ago · Quote · #47

if x-y = 0

and x-y = 2(x-y)

then 0 = 2 x 0

not 1 = 2

BTW anything divided by 0 = math error

• 8 years ago · Quote · #48

For n=number of digits, isn't

Lim (n->infinity){0.9999...} = 1,

So for extreme large values of 2, don't we have 2 + 2 = 6. If we consider a limiting case.

(Bosco?)

• 8 years ago · Quote · #49

i like maths, but this is going a little bit to far.

:)

• 8 years ago · Quote · #50

proof that 0=1:

------------------------------------------------------

first we prove:

Lemma 1: if Lemma 1 is true then 0=1.

Proof of lemma one: assuming that lemma 1 is true we have to show that 0=1.

a) Now lemma 1 is true - by assumption.

b) hence (by what lemma 1 says) if Lemma 1 is true then 0=1.

from a and b: 0=1.

This proves lemma 1.

--------------------------------------------------------

now proof that 0=1 is:

(c) Lemma  1 is true (we just proved it!)

(d) if Lemma 1 is true then 0=1

(because we have proved lemma 1, and  (d) is what it says!

By (c) and (d),  0=1.

Q.E.D.

• 8 years ago · Quote · #51

lim (2->inf) 1/2 = 0

• 8 years ago · Quote · #52
Now I understand why I'm so confused!
• 8 years ago · Quote · #53

no, simple: lim(x->inf) 1/x=0

substitute 2 for x :)

• 8 years ago · Quote · #54
Being a mathematician and a philosopher, I believe that chess has nothing to do with mathematics.  We can figure out the combinatorics of calculating all the possible moves there is.  But if we cannot know what the best move is by our logic and reasoning, then we have lost the game right there.  It is not the number of combinations we should calculate, but which ones are best to employ in the game.  Sometimes we need to set theory and conventions aside.  We need to enjoy the game for what it is worth.  Leave mathematics out of the chess board and let pure reason battle your way to checkmate.
• 8 years ago · Quote · #55
mathteacher, on a different note [since you are a mathematician and a philosopher - so i suppose you at least partly do mathlogic] , do'nt you wonder why Martin-Lob's theorem is not taught just after Goedel's second? I mean, the proof is short and trivial to understand after Godel 2nd. (Many places they don't even teach Tarski's impredicability of Truth - that one pops out just like that from the proof og Godel's 2nd! [tarski's genius there was in *posing the question*.]
• 8 years ago · Quote · #56

To be honest, I have not studied any of those Theorems in mathematical logic, but I do believe that there should be an order of the presentation of certain proofs in a logic class.  You need to learn truth tables and negations of statements before you can attempt to do proofs by contradiction.  I will have to check out Godel and Tarski sometime.  I like how mathematicians questioned the validity of the axioms sometimes and tried to make amends for the ones they didn't think were completely true.  However, the universal laws in science are expressed with mathematical equations.  But even these equations can be altered once we discover the truth of our universe with far better accuracy and precision.  Only God knows the absolute truth of our universe.  It is up to man to discover it.

• 8 years ago · Quote · #57

aniakovas,

What in hell's bells are you talking about? nxn chess? what is the initial position and what are the rules as n -> inf? A momentary lapse of reason?  P, NP, NP-complete, NP-hard, BPP, exponential, "elementary" ... apply to a *class* of problems parametrised by (at least one) parameter, say  n , which takes infinitely many values [typically we are taking about n -> inf]

Chess is a specific problem [or, in complexity-theory lingo, an *instance*, not a *problem*] - so obviously it is of the simplest time-complexity (ie constant) , ditto for space-complexity :)

PS: please to review the definition of NP-hard --- not from popular mangled expositions, but from a standard text like Aho, Hopcroft and Ullman or Papadimitrou ...

or just look up wikipedia --- this topic is nonpolitical so wikipedia can be trusted on this one.

aniakovas wrote: (there are arguments about whether chess is NP-Hard, it seems to me that is the strongest computers can beat the strongest humans then it is not, but this is my opinion, I am unable to determine if it has been mathematically shown to be NP-Hard)

• 8 years ago · Quote · #58
this formula is called the Game tree
• 8 years ago · Quote · #59

45,111,908,353-45,111,908,364th digits of pi 27182818284, e (the number) without the decimal.

• 8 years ago · Quote · #60

Scoville: "Ever hear about that guy who memorized enough of the 'number pie'..."

I loves me some of that number pie!