How Many 4PC Custom Starting Positions are Possible?

So we subtract all the positions without pawns, and add it back, divided by 4 - one for each rotation? In other words, find how many there are with pawns, then multiply it by 3/4 and subtract that quantity.

PositionsWithoutPawns = 122^160-4*(25^160)+4*(30^159*69)

So let's get our final final number:

142^160-4*(30^160)+4*(30^159*84) - 3*[122^160-4*(25^160)+4*(30^159*69)]/4, or about... wait for it...

2.3*10^344, yep, still the same with this order of magnitude.

And obviously, I'm using W|A for this, you think I'm doing this by hand!?

i'm a math whiz, so 2.3*10^344 is about...

Think about just 2.3*10^10. That's 23000000000. Or 23 billion.

I will make this easier to figure out myself even though I'm breaking the rules of math and scientific expressions. 23*10^344 is...

230000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

Or, two hundred thirty tredecicentillion. That's just crazy!

What about positions in which all are dead pieces

you are correct, grable didn't factor this in correctly I think. He subtracted 30^160 for each group of positions with only a color (4 times), but it's not actually 30. you have 29 piece types, and they can be either dead or a single color. So 58 + 2 (wall or blank) = 60; for 160 squares you would subtract 60^160. Am I wrong?

you have 29 piece types, and they can be either dead or a single color. So 58 + 2 (wall or blank) = 60; for 160 squares you would subtract 60^160. Am I wrong?

this is why I like math so much

idk and i dont care lol

There are 28 piece types (7 classic + 21 fairy).
There are 5 colors (RBYG + dead).
28 * 5 = 140 different pieces.
140 + 2 (wall or blank) = 142 different states for each square.
142^160 ≈ 2,323459e+344 different states for 160 squares or for 4PC board.

So revised you would be subtracting 4 * 58^160 to remove the positions in which there is only one player.

I didn't try to calculate the number of playable custom positions.

You could technically change the starting position by which pieces are royals, but then you'd also have to confirm you're only including legal placements of royals. Also consider the N-Check gamerule: you could theoretically assign any number of lives to any royal, which would give infinite possibilities, so for that reason alone, I think I'll stop my calculations here.

so he kinda went halfway...