A book claims there are $10^{120}$ board positions on a chess board. How would one prove this?

Question

I have recently bought an exercise copy and there, in the cover page I got an amazing fact about chess board, "There are total $10^{120}$ board positions in a chess board."

But I was just thinking how to prove this.

My work:

I think to first find the no of board positions including all pawns(yet don't know how to do that) and then get a motivation excluding some no of pawns. Can anyone give some idea?

Answer 1

$10^{120}$ is the Shannon number, a lower bound estimate of the number of possible chess games.

For number of positions, this is on the Shannon number wiki:

"John Tromp estimated the number of legal chess positions with a 95% confidence level at $ 4.5\times 10^{44} \pm 0.37\times 10^{44}$ based on an efficiently computable bijection between integers and chess positions."

https://en.wikipedia.org/wiki/Shannon_number#cite_note-5 https://github.com/tromp/ChessPositionRanking

Answer 2

I think that's a rough estimate of how many games you can create using legal moves. That is a very strange estimate given by $20^{80}$ catering to the fact that in each position, there are an average of $20$ moves you can play, and an average chess game lasts for $40$ moves. I found this information in a Numberphile video. I guess, it's called Shanon number.

But, if you want to have an elementary estimate of the number of (legal) board positions, you can go like-

The two kings has to be there. So, place them on any two squares in $\binom{64}2$ ways.

Now, the rest $30$ pieces may or may not be included in $2^{30}$ ways. Arrange them in the other $62$ squares in $\binom {62}{30}$ ways.

So, the total number of ways is $2^{30}\times \binom{64}2\times \binom {62}{30}\approx 9.7\times10^{29}$

Note that this estimate doesn't eliminate the chance of two kings being on adjacent squares which is illegal. Also, this estimate may put two bishops in the same coloured square (which is half illegal since considering pawn promotions make it legal). That reminds us, we also haven't taken pawn promotions into account.