Toggle

Current status: Development in progress, mostly completed.

Toggle is a Boggle-like variant (the name is a play on Topological Boggle) that was inspired by the fact that on a regular Boggle board, the dice on the edges of the board (and in particular in the corners of the board) are less "playable," since they have fewer neighbours than the central dice.

I have always wanted to play Boggle on the surface of a torus to treat the outer tiles in the same way as the inner tiles, and hence, the project idea was born. Given that a torus can be modeled as the quotient space a unit square (into which I can embed a board), I figured: why stop there when there are so many other surfaces that can be made from the quotient space of a unit square where the opposite edges are related (or "glued together," like with a Mobius strip) in various ways?

And hence Toggle came to be.

1. The board can be:

   1. Embeddable in the plane, as per usual Boggle;
   2. Embeddable on a horizontal or vertical cylinder;
   3. Embeddable on a torus;
   4. Embeddable on a Mobius strip;
   5. Embeddable on a Klein bottle; or
   6. Embeddable on a projective plane.

2. The game can be timed or untimed.

3. I plan on ranking game boards (i.e. assigning them a unique number) so that players can play simultaneously - or with the same board and rules at different times - without requiring any kind of connection.

This is also intended to be a personal exercise in playing with tries, trie-packing, learning JavaFX, and playing more with Java 9 and 10.

I would additionally like to use this as an opportunity to investigate information compression and value by using tries and packed tries to evaluate the inherent value in glyphs and words in different languages.

Other projects and ideas will probably emerge as the project continues.

Here is an example of a board (board #9223372036854775807:839283) that is on the surface of a torus, i.e. you can loop around from top to bottom and left to right, and how you can use that to spell the word ENSEALING, with the initial E circled in green, the final G circled in blue, and the lines connecting dice shown in red.

Here is an image of a board embedded on the surface of a Klein bottle. The white squares indicate the dice, and the grey squares indicate their adjacencies that are non-obvious.

Here are the calculations worked out for the non-obvious adjacencies of a 4x4 Boggle board with valid positions indexed (i, j) for 0 <= i, j < 4, where the columns mean:

• The grid board is omitted, since this is evident.
• CX is the horizontal cylinder board.
• CY is the vertical cylinder board.
• MX is the horizontal mobius strip board.
• MY is the vertical mobius strip board.
• KX is the horizontal Klein bottle board (note KX = CY + MX).
• KY is the vertical Klein bottle board (note KY = CX + MY).
• P is the projective plane board (note P = MX + MY).

Updates

2018-11-02

Currently, the project has:

• Tries, trie-packing, and statistical comparisons between them.
• Board geometries as described above.