I don't like this math XD
How Many 4PC Custom Starting Positions are Possible?
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lol
im too lazy to read all...
Dec 8, 2020
0
#5
if the position contains no pawns, then it will basically be the same as other positions but with a different orientation. Does this count?
If you are not good at figuring out real world examples check out Wolfram|Alpha
It is an AI Search engine (it is not very good at ai but brilliant at calculations just enter the number and it will compare it for you(it found me pi with 300 digits))
Dec 9, 2020
0
#7
I'm sure grable knows about WolframAlpha. But I think it was something like an answer to this theme.
According to the grable's conclusion we have infinite possibilities, i.e. number of new custom positions we can create has no limit. Yes, grable missed a lot, in particular the one that Box described. But the sense remains the same, we can create an infinite number of new positions.
So it is very important to remember that creating a new position only is not enough, it should be fair, balanced and interesting. Otherwise why not to take another position from infinite list of possible positions? Also I'd remind that making positions symmetric is just the simplest way of making position more fair. I.e. there are many other ways of making position fair and balanced. And it needs very hard work and very long tests. Therefore some words like "I don't want to waste my time testing my new position" are just devoid of common sense. Creating a new custom position is not just a filling the board with pieces and walls, it is much a much more. You have to create a logic, strategies and goal of the game and only after that to fill the board with pieces which will help to embody your idea. We undoubtedly appreciate your work in creating new positions, but I often see that some positions are posted as semi-finished, some are even without a sense - just a board filled with pieces, walls and rules.
So in conclusion, I would like to emphasize once again that filling the board with pieces and walls is only a small part of creating a new variant.
I'm also sure that everyone here is smart enough not to quote my whole little rush of inspiration and write "ok" or whatever else.
Dec 9, 2020
0
#8
qilp wrote:
I'm sure grable knows about WolframAlpha. But I think it was something like an answer to this theme.
According to the grable's conclusion we have infinite possibilities, i.e. number of new custom positions we can create has no limit. Yes, grable missed a lot, in particular the one that Box described. But the sense remains the same, we can create an infinite number of new positions.
So it is very important to remember that creating a new position only is not enough, it should be fair, balanced and interesting. Otherwise why not to take another position from infinite list of possible positions? Also I'd remind that making positions symmetric is just the simplest way of making position more fair. I.e. there are many other ways of making position fair and balanced. And it needs very hard work and very long tests. Therefore some words like "I don't want to waste my time testing my new position" are just devoid of common sense. Creating a new custom position is not just a filling the board with pieces and walls, it is much a much more. You have to create a logic, strategies and goal of the game and only after that to fill the board with pieces which will help to embody your idea. We undoubtedly appreciate your work in creating new positions, but I often see that some positions are posted as semi-finished, some are even without a sense - just a board filled with pieces, walls and rules.
So in conclusion, I would like to emphasize once again that filling the board with pieces and walls is only a small part of creating a new variant.
I'm also sure that everyone here is smart enough not to quote my whole little rush of inspiration and write "ok" or whatever else.
ok.
Dec 9, 2020
0
#9
grable wrote:
qilp wrote:
I'm sure grable knows about WolframAlpha. But I think it was something like an answer to this theme.
According to the grable's conclusion we have infinite possibilities, i.e. number of new custom positions we can create has no limit. Yes, grable missed a lot, in particular the one that Box described. But the sense remains the same, we can create an infinite number of new positions.
So it is very important to remember that creating a new position only is not enough, it should be fair, balanced and interesting. Otherwise why not to take another position from infinite list of possible positions? Also I'd remind that making positions symmetric is just the simplest way of making position more fair. I.e. there are many other ways of making position fair and balanced. And it needs very hard work and very long tests. Therefore some words like "I don't want to waste my time testing my new position" are just devoid of common sense. Creating a new custom position is not just a filling the board with pieces and walls, it is much a much more. You have to create a logic, strategies and goal of the game and only after that to fill the board with pieces which will help to embody your idea. We undoubtedly appreciate your work in creating new positions, but I often see that some positions are posted as semi-finished, some are even without a sense - just a board filled with pieces, walls and rules.
So in conclusion, I would like to emphasize once again that filling the board with pieces and walls is only a small part of creating a new variant.
I'm also sure that everyone here is smart enough not to quote my whole little rush of inspiration and write "ok" or whatever else.
ok.
omg lol grable. ok.
Dec 9, 2020
0
#10
qilp wrote:
Yes, grable missed a lot
I wonder why grable decided to quote this and say "ok"
Dec 9, 2020
0
#11
Obviously to set a good example to others.
Dear @soggydollar. We are glad to inform you that you have officially been promoted to the title of @grable's apprentice.
lol
grable wrote:
Let's do this.
Each square can be occupied by one of 7 classic pieces or 21 fairy pieces, in one of the 4 possible colours, or colour-less dead piece. Squares can also be left empty, or filled with a wall.
That gives us 5*(7+21)+2 or 142 options for each square. There are 160 squares on a 4PC board, so there are 142^160 possible arrangement of pieces, or about 2.3*10^344. That's our starting point.
However, we also must consider that there must be pieces of at least 2 different colours, as 4PC starting positions must have at least 2 players, so let's back up a step and say that of those 2.3*10^344 possible arrangements, there are 30^160 in which the pieces are all Red, 30^160 which are all Blue, 30^160 which are all Yellow, and 30^160 in which they're all Green, so let's subtract those, and add back in an amount which is valid.
142^160 - 4*(30^160)+4*(??)
What are the corrected amounts though? I'd say that in these positions, 159 of the squares can be any of the 30 single-colour options, but the 160th square must be one of the 3*(7+21) other-colour pieces. So if I'm correct here, that's 30^159*84. So, that means our total has been decreased a bit from 2.3*10^344 to:
142^160-4*(30^160)+4*(30^159*84) or about 2.3*10^344 (yes, it's exactly the same when you're looking at this order of magnitude, but it's about 1.57*10^237 less).
You could start to calculate how many permutations there are of the various gamerules, but I think this is sufficient. 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. Anything I missed?
TL;DR: about 2.3*10^344. That's a two, then a three, followed by 343 zeroes.
Because it's the trendy thing to do, I'll compare the number 2.3*10^344 to something we can all imagine. The Indianoplis Motor Speedway in Indiana, United States, has a seating capacity of 250,000 people. You'd need to fill that about 9.3*10^338 times. Sorry, I'm not creative enough to figure out a "Real-World" example of a number this terrifyingly huge.
XD math
grable wrote:
qilp wrote:
I'm sure grable knows about WolframAlpha. But I think it was something like an answer to this theme.
According to the grable's conclusion we have infinite possibilities, i.e. number of new custom positions we can create has no limit. Yes, grable missed a lot, in particular the one that Box described. But the sense remains the same, we can create an infinite number of new positions.
So it is very important to remember that creating a new position only is not enough, it should be fair, balanced and interesting. Otherwise why not to take another position from infinite list of possible positions? Also I'd remind that making positions symmetric is just the simplest way of making position more fair. I.e. there are many other ways of making position fair and balanced. And it needs very hard work and very long tests. Therefore some words like "I don't want to waste my time testing my new position" are just devoid of common sense. Creating a new custom position is not just a filling the board with pieces and walls, it is much a much more. You have to create a logic, strategies and goal of the game and only after that to fill the board with pieces which will help to embody your idea. We undoubtedly appreciate your work in creating new positions, but I often see that some positions are posted as semi-finished, some are even without a sense - just a board filled with pieces, walls and rules.
So in conclusion, I would like to emphasize once again that filling the board with pieces and walls is only a small part of creating a new variant.
I'm also sure that everyone here is smart enough not to quote my whole little rush of inspiration and write "ok" or whatever else.
ok.
XD @grable you did the opposite of what @qilp told you
Dec 9, 2020
0
#15
Very important information. You are the only one who noticed that.
Dear @Pika_Pokemon. We are glad to inform you that you have officially been promoted to the title of @grable's Chief Secretary.
Dec 9, 2020
0
#16
BoxJellyfishChess wrote:
if the position contains no pawns, then it will basically be the same as other positions but with a different orientation. Does this count?
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!?
Dec 9, 2020
0
#17
ChessDemon732 wrote:
If you are not good at figuring out real world examples check out Wolfram|Alpha
Okay, here's what W|A tells me:
2.3*10^344 is 2.8*10^-27 times smaller than Skewes number. Oh yes, that's much better for me to imagine now! Thanks!
Dec 9, 2020
0
#18
There are also positions you can create with insufficient material. Or positions which are over (someone is in checkmate or stalemate).
Dec 9, 2020
0
#19
qilp wrote:
qilp wrote:
Yes, grable missed a lot
There are also positions you can create with insufficient material. Or positions which are over (someone is in checkmate or stalemate).
Which is why I didn't try to calculate the number of playable custom positions. Even if the game ends on move #1, it's still being counted here. I think it's a much much tougher task to calculate the playable positions.
Let's do this.
Each square can be occupied by one of 7 classic pieces or 21 fairy pieces, in one of the 4 possible colours, or colour-less dead piece. Squares can also be left empty, or filled with a wall.
That gives us 5*(7+21)+2 or 142 options for each square. There are 160 squares on a 4PC board, so there are 142^160 possible arrangement of pieces, or about 2.3*10^344. That's our starting point.
However, we also must consider that there must be pieces of at least 2 different colours, as 4PC starting positions must have at least 2 players, so let's back up a step and say that of those 2.3*10^344 possible arrangements, there are 30^160 in which the pieces are all Red, 30^160 which are all Blue, 30^160 which are all Yellow, and 30^160 in which they're all Green, so let's subtract those, and add back in an amount which is valid.
142^160 - 4*(30^160)+4*(??)
What are the corrected amounts though? I'd say that in these positions, 159 of the squares can be any of the 30 single-colour options, but the 160th square must be one of the 3*(7+21) other-colour pieces. So if I'm correct here, that's 30^159*84. So, that means our total has been decreased a bit from 2.3*10^344 to:
142^160-4*(30^160)+4*(30^159*84) or about 2.3*10^344 (yes, it's exactly the same when you're looking at this order of magnitude, but it's about 1.57*10^237 less).
You could start to calculate how many permutations there are of the various gamerules, but I think this is sufficient. 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. Anything I missed?
TL;DR: about 2.3*10^344. That's a two, then a three, followed by 343 zeroes.
Because it's the trendy thing to do, I'll compare the number 2.3*10^344 to something we can all imagine. The Indianoplis Motor Speedway in Indiana, United States, has a seating capacity of 250,000 people. You'd need to fill that about 9.3*10^338 times. Sorry, I'm not creative enough to figure out a "Real-World" example of a number this terrifyingly huge.