How many different chess positions are there?

And how many will ever be seen, on a board or only in the memory of a computer?

Does a position that will never be seen exist?

Trees, forest, falling and sound?

A ton of sources say a ton of different things. Kasparov once said "It's like 10 with forty five zeroes after it, or something like that" (not even remotely kidding, that's exactly what he said lol). But I've read that it is actually much higher, something like 10 to the sixtieth or even hundred and twentieth power. Either way, it's something akin to the number of electrons in the observable universe.

What proportion of them would you like to explore? 

10^120 was an early estimate, 10^45 is pretty decent

Also, you cannot use direct statistics maths for this, since most possible positions are not obtainable

https://en.wikipedia.org/wiki/Shannon_number

The 10^45 you cited comes with this caveat at the bottom of the proof ;)...so I would stick with the base Shannon number of 10^120.

I have a significantly more complex program that proves an upper bound of 7728772977965919677164873487685453137329736522 (about 10^45.888 or ~ 2^152.437) on the number of positions, but, like the bound of ~10^46.25 published by Shirish Chinchalkar in "An Upper Bound for the Number of Reachable Positions", ICCA Journal, Vol. 19, No. 3, pp. 181-183, 1996, it requires much better documentation to be considered verifiable

4...there are 4 possible positions.

Why is sequences of moves any different from positions?

This does not change anything. One sequence of moves could be e4 e5. This sequence of moves has a corresponding position. Another sequence of moves could be d4 nf3. This also has a corresponding position. That is 2 sequences of moves and 2 positions. In my mind, every sequence of moves has a corresponding position, and every obtainable position has a corresponding sequence of moves. Now as im writing, im realising that there may very well be multiple sequences of moves leading to the same corresponding position, is this what you mean would be the difference?

There are four positions after two moves in which White is in checkmate. There are eight sequences of moves that lead to these four positions. All of them have 2...Qh4#. All of them have White advancing the g-pawn two moves, but this can occur on either the first or the second move.

The number of move sequences dwarfs the number of positions.

Please show me a game that ends 2...Qxe4#.

3.Qxe5# is possible, as is 3...Qxe4#. There are perhaps 300 sequences leading to checkmate on White's third move. I don't have the exact number in my head and don't feel like looking it up this morning. 292 might be correct. The number of checkmate sequences on Black's third move is vastly higher of course.

I get it now, my mathematical intuition has failed me, god dammit

The problem by looking at the pawn structure as you described is that after 5 moves, the knight cannot be on any square without a pawn or a piece. Since each piece has its own specific way to move, you need to implement rules into your formula. This is not done so easily, and if its possible, it is going to create a huge formula which contains all those rules, described mathematically.

And even if you would assume that the opponent's knight just jumped back and forth, this is only one example, and there are many others, especially considering the pawns can only move forward and attack diagonally, this will be very hard to calculate.

I believe you would have to iterate som kind of formula to find the number of positions, which would take a very long time. I dont think you can use one formula to calculate the number of obtainable chess positions.

I saw that it is 1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000