On Apr 12, 11:08*pm, Rich Hutnik wrote:

But, if one is working with Chess Variants, then the issue does
arise that if the number of variants is finite, then you can have a
classification system in place that could capture them all, and even
simplify, and perhaps bridge them.

I think that one can always go 'outside the system' and come up with a
reasonable new Chess variant that is not included in any
classification system, even if that system embraces an infinite number
of variants.

Yet, the fact that people can only handle games up to a certain finite
level of complexity means that the number of Chess variants is finite.

A large, but poorly-defined finite set, therefore, can behave for
practical purposes as if it had properties that, in an exact
mathematical sense, can only apply to a set with at least aleph-one
elements. This doesn't defy any law of mathematics (and, indeed, due
to the subject matter, I've pulled sci.math back in, since it's
relevant now).

John Savard

On Apr 15, 3:53 pm, Quadibloc wrote:
On Apr 12, 11:08 pm, Rich Hutnik wrote:

But, if one is working with Chess Variants, then the issue does
arise that if the number of variants is finite, then you can have a
classification system in place that could capture them all, and even
simplify, and perhaps bridge them.

I think that one can always go 'outside the system' and come up with a
reasonable new Chess variant that is not included in any
classification system, even if that system embraces an infinite number
of variants.

Yet, the fact that people can only handle games up to a certain finite
level of complexity means that the number of Chess variants is finite.

A large, but poorly-defined finite set, therefore, can behave for
practical purposes as if it had properties that, in an exact
mathematical sense, can only apply to a set with at least aleph-one
elements. This doesn't defy any law of mathematics (and, indeed, due
to the subject matter, I've pulled sci.math back in, since it's
relevant now).

John Savard

So, then, to make this more mathematical, are the number of rules
variants for a game like chess an Aleph of any sort? I will this
topic to have it ask that. Maybe someone else who is more math(y) in
their knowledge could frame this in a more mathematically proper form.

- Rich

On Apr 16, 4:58 pm, Rich Hutnik wrote:
On Apr 15, 3:53 pm, Quadibloc wrote:



On Apr 12, 11:08 pm, Rich Hutnik wrote:

But, if one is working with Chess Variants, then the issue does
arise that if the number of variants is finite, then you can have a
classification system in place that could capture them all, and even
simplify, and perhaps bridge them.

I think that one can always go 'outside the system' and come up with a
reasonable new Chess variant that is not included in any
classification system, even if that system embraces an infinite number
of variants.

Yet, the fact that people can only handle games up to a certain finite
level of complexity means that the number of Chess variants is finite.

A large, but poorly-defined finite set, therefore, can behave for
practical purposes as if it had properties that, in an exact
mathematical sense, can only apply to a set with at least aleph-one
elements. This doesn't defy any law of mathematics (and, indeed, due
to the subject matter, I've pulled sci.math back in, since it's
relevant now).

John Savard

So, then, to make this more mathematical, are the number of rules
variants for a game like chess an Aleph of any sort? I will this
topic to have it ask that. Maybe someone else who is more math(y) in
their knowledge could frame this in a more mathematically proper form.

- Rich

It is relatively simple to come up with an infinite number of
variants.
Consider that in standard chess the king can be captured by a single
attack. Consider a variant where the king can only be captured by
two attacks. This generalizes to 3,4,...,n,... attacks.

(Actually, Simon Smith's argument above falls a bit short. It is
not enough to show there are an infinite number of descriptions
of variants of Calvinball chess (after all for each variant there
are an infinite number of descriptions))

- William Hughes

Rich Hutnik wrote:

So, then, to make this more mathematical, are the number of rules
variants for a game like chess an Aleph of any sort? I will this
topic to have it ask that. Maybe someone else who is more math(y) in
their knowledge could frame this in a more mathematically proper form.

Consider the following variants of chess:
Variant 1: standard set of men, 8x4 board.
Variant 2: standard set of men, 8x5 board.
Variant 3: standard set of men, 8x6 board.
Variant 4: standard set of men, 8x7 board.
Variant 5: standard set of men, 8x8 board. --standard chess
Variant 6: standard set of men, 8x9 board.
Variant 7: standard set of men, 8x10 board.
Variant 8: standard set of men, 8x11 board.
Variant 9: standard set of men, 8x12 board.
....

The above set of variants is clearly infinite
and maps to the set of integers.

It even offers interesting play; at, say, 8x32,
do you try to launch an attack on the opponent
right away with your long range men (QBR), or
do you keep them behind a wall of pawns that you
slowly march toward the opponent? And if both
sides start marching pawns, what is the best
pawn structure to have when they meet? Diagonal
line? Arrowhead? V? zig-zag? straight across?
And what is the best knight and king placement?

Now consider these variants of chess:

Variant 3: standard set of men, 8x6 board. -- same as in list above
Variant 3.1: standard set of men, 9x6 board.
Variant 3.2: standard set of men, 10x6 board.
Variant 3.3: standard set of men, 11x6 board.
Variant 3.4: standard set of men, 12x6 board.
Variant 3.5: standard set of men, 13x6 board.
Variant 3.6: standard set of men, 14x6 board.
....

The above set of variants is also clearly infinite,
larger than the previous infinite set, and maps
to the set of fractions. Eventually you get to
Variant 3.141592653589793238462643383: standard
set of men, 141415926535897932384626433811x6 board
and on to any other fraction you choose.

The question of *meaningful* differences is more
interesting. I don't see any meaningful difference
between playing on an 8x1000 board and playing on an
8x1002 board. But the loss of meaningfulness is
gradual; where exactly does it reach zero?


--
Guy Macon
http://www.guymacon.com/

On Apr 16, 4:48 pm, Guy Macon http://www.guymacon.com/ wrote:

The question of *meaningful* differences is more
interesting. I don't see any meaningful difference
between playing on an 8x1000 board and playing on an
8x1002 board. But the loss of meaningfulness is
gradual; where exactly does it reach zero?

But what about the variant on an 8 x 3,698,201,443,...,828,216 board,
where the ... stands for a number of digits which, if printed in 4
point type, in the pages of a thick telephone directory, would require
enough of those volumes to cover Manhattan Island to a height of one
mile?

The number of practical variants of chess that real humans can play is
strictly finite - yet people keep coming up with unexpected new ideas
for variants.

John Savard

On Wed, 16 Apr 2008 18:58:13 -0700 (PDT), Quadibloc
wrote, in part:

On Apr 16, 4:48 pm, Guy Macon http://www.guymacon.com/ wrote:

The question of *meaningful* differences is more
interesting. I don't see any meaningful difference
between playing on an 8x1000 board and playing on an
8x1002 board. But the loss of meaningfulness is
gradual; where exactly does it reach zero?

But what about the variant on an 8 x 3,698,201,443,...,828,216 board,
where the ... stands for a number of digits which, if printed in 4
point type, in the pages of a thick telephone directory, would require
enough of those volumes to cover Manhattan Island to a height of one
mile?

The number of practical variants of chess that real humans can play is
strictly finite - yet people keep coming up with unexpected new ideas
for variants.

Here is a more practical variant of Chess which belongs to a class of
variants with a very large number of members.

(Best seen with fixed-pitch font.)

R Q K R
B N N B
P P P P
P P P P
R P - . - . P R
N P . - . - P N
B P - . - . P B
K P . - . - P K
Q P - . - . P Q
B P . - . - P B
N P - . - . P N
R P . - . - P R
P P P P
P P P P
B N N B
R Q K R

The two players have their narrow arrays at the top and bottom of the
board. The pieces in arrays on the side of the board may not be captured
by either of the two players, but they can capture the players' pieces.

These pieces move once after every three ply. That is, the sequence
of moves is:

White, Black, White, Left Pieces, Black, White, Black, Right Pieces

The left pieces, as White, and the black pieces, as Black, when they
move simply replay the moves of the Immortal Game between Anderssen and
Kieseritzky.

If one's piece happens to be standing on a square which was empty in the
game to which a piece moves, then it is captured.

Replace the Immortal by the Evergreen, and you get another variant.

So here is a very practical chess variant belonging to a very large
class of chess variants - as many variants of chess as there are *games*
of chess!

John Savard
http://www.quadibloc.com/index.html

Guy Macon http://www.guymacon.com/ wrote:
Consider the following variants of chess:
[for i3, Variant i: standard set of men, 8xi board.]

The above set of variants is clearly infinite and maps to the set of
integers. [...]

Now consider these variants of chess:
[for i3, j7, Variant i.j: standard set of men, jxi board.]

The above set of variants is also clearly infinite, larger than the
previous infinite set, and maps to the set of fractions.

These are properly called the positive rational numbers (i.e., the set
of numbers that can be written as i/j for positive integers i and j).
The set of positive rationals is *not* larger than the set of integers:
it has the same cardinality.

Proof. (Writing N for the positive integers, Q' for the positive
rationals and |S| for the cardinality of the set S.) Every positive
integer n can be written n/1, so is a positive rational. Therefore,
|N|=|Q'|. Any positive rational m/n can be coded unambiguously by
the positive integer 2^m x 3^n, so |Q'|=|N|. QED.


Dave.

--
David Richerby Solar-Powered Expensive Book (TM):
www.chiark.greenend.org.uk/~davidr/ it's like a romantic novel but it'll
break the bank and it doesn't work in
the dark!

Quadibloc wrote:
The number of practical variants of chess that real humans can play
is strictly finite

Trivially: we only have a finite universe in which to store the rules.


Dave.

--
David Richerby Homicidal Umbrella (TM): it's like an
www.chiark.greenend.org.uk/~davidr/ umbrella but it wants to kill you!

On Apr 16, 6:48 pm, Guy Macon http://www.guymacon.com/ wrote:
Now consider these variants of chess:

Variant 3: standard set of men, 8x6 board. -- same as in list above
Variant 3.1: standard set of men, 9x6 board.
Variant 3.2: standard set of men, 10x6 board.
Variant 3.3: standard set of men, 11x6 board.
Variant 3.4: standard set of men, 12x6 board.
Variant 3.5: standard set of men, 13x6 board.
Variant 3.6: standard set of men, 14x6 board.
...

The above set of variants is also clearly infinite,
larger than the previous infinite set, and maps
to the set of fractions. Eventually you get to
Variant 3.141592653589793238462643383: standard
set of men, 141415926535897932384626433811x6 board
and on to any other fraction you choose.

The question of *meaningful* differences is more
interesting. I don't see any meaningful difference
between playing on an 8x1000 board and playing on an
8x1002 board. But the loss of meaningfulness is
gradual; where exactly does it reach zero?

Meaningful differences is very important. One could argue that the
look of pieces, or their names, could be considered changes, and that
could be about infinite. But it has no effect on gameplay.

What I do see so far is several things that would lend to chess
variants be unbounded:
1. Time control, being infinite. One could do an infinite range of
time delays for a Bronstein clock. Not practical.
2. Size of board, being infinite. This then means an infinite number
of shapes. Unless the size of the board is infinite, then the number
of boards is finite.
3. Recursion. Here is an example. Say you can Calvinball the rules.
Let's say to implement a new rule, like Gipf introduces a new pieces,
one has to win another game to do this. If the number of possible
games one can play is infinite, then there is an infinite number of
varieties of chess. If then someone else where to go about wanting to
change the rules for the new game to see about the old game, they have
to play yet another game, then it is possible to cause an infinite
recursive set of action in place. This is not practical, but is
arguably meaningful to the game experience as a whole.

Pretty much here, either the parameters of a rule are unbound, or the
number of rules is unbound, or there is an infinite recursive rule
that can take place. If so, then such would be unlimited. The first
two be Heraclitian, and the last one being Calvinball. If anyone can
find any others, please say so.

Now, whether or not there is an infinite number of piece types, or
rules, that is another issue that would need to be considered here.
Anyone have evidence that there is an infinite number of rules that
can come in existence for a game like chess?

- Rich

Ed Murphy wrote:
David Richerby wrote:
Quadibloc wrote:
The number of practical variants of chess that real humans can
play is strictly finite

Trivially: we only have a finite universe in which to store the
rules.

Counterargument: infinite sets of variants can be encoded in a
finite description (e.g. arbitrary board length, as mentioned
earlier).

Yes but, in order to play a game, we need to know which of those
variants we are playing. We only have a finite universe in which to
store either an explicit presentation of the rules or the parameters
that generate them from your finite description.


Dave.

--
David Richerby Carnivorous Portable Composer (TM):
www.chiark.greenend.org.uk/~davidr/ it's like a pupil of Beethoven but
you can take it anywhere and it
eats flesh!