The next version of Tic-tac-toe Collection will include support for playing on different topologies. Other games have done this before but probably not with so many options. To be specific, the options available will be:
- Horizontal cylinder
- Vertical cylinder
- Horizontal Möbius strip
- Vertical Möbius strip
- Horizontal Klein bottle
- Vertical Klein bottle
- Real projective plane
The ones that have horizontal and vertical versions are basically the same, but it does make a difference for how Drop Four is played.
A cylinder is formed by repeating the play space in one direction, and a torus is formed by repeating it in both directions.
A Möbius strip is like a cylinder, but each repeat is flipped (so a horizontal Möbius strip repeats horizontally, but each copy of the board is flipped vertically relative to its neighbours).
A Klein bottle is like Möbius strip, but also repeats (normally) in the opposite direction.
The real projective plane is like a Möbius strip in both directions. That is, vertical copies flip horizontally and horizontal copies flip vertically.
That last item, the real projective plane is interesting and led to a new game option being added. One of its odd features is it makes all the corner squares diagonally adjacent to themselves. This means, for instance, you can get three-in-a-row with just two moves. This seemed odd to me so I did a bit of investigating. The corners are not really adjacent to themselves, but just appear that way because trying to render the real projective plane on a flat surface is very misleading. The end result, is a new game option:
“Allow cell reuse” determines whether the same cell can be used more than once in the same line. For example, imagine playing on a 3×3 torus with a winline size of 4. If you get three-in-row horizontally, it would appear you have also have four-in-row (and five-in-a-row and six-in-a-row etc.) if you allow cells to be reused. Whether allowing it or not makes for better gameplay or not, I don’t know yet. But the option is there.