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 re: 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/>
|
