The following argument intends to prove that the game of Torus is an
equal goals game. If it's not a tremendously rigorous proof, then at
the very least it strongly suggests that conclusion.
1. A hexagonally tessellated rhombus and a hexagonally tessellated
square can both be used to generate the exact same torus.
2. Winning helices for NS are mirror images of winning helices for
EW. NOTE: Permutations which contain helices will not be considered
in this proof.
3. Every EW loop on a rhombus oriented as shown below has exactly one
corresponding EW loop on a square, both of which generate the exact
same EW loop on a torus.
I'll give you a couple of examples of EW loops on a rhombus and their
corresponding EW loops on a square. You basically just generate a
cycle and sample it pi radians away from the original sample with a
square template.
o . . o . o o . . o o . . o o .
. o o o o o . o o o . o o o . o
. . . . . . . . . . . . . . . .
.. . . . . . . . . . . . . . . .
. o o . o . . o o . . o o . . o
. o . o . o . o . o . o . o . o
. o . o . o . o . o . o o . o .
o o . . . . o o . . o o . o o .
4. On a rhombus half of the total number of possible permutations
contain EW loops only and half contain NS loops only.
5. A square has the same number of points and therefore the same
number of permutations as a rhombus.
6. Since half of the permutations on a rhombus contain EW loops only,
and since there is a one to one correspondence between EW loops on a
rhombus and EW loops on a square, and since there are the same number
of permutations on a square as on a rhombus, half of the permutations
on a square contain EW loops only.
7. Since half of the permutations on a square contain EW loops only,
then the other half contain NS loops only. Therefore Torus is an
equal goals game.
