What I've heard called 'six-dimensional' Sudoku has mathematical extensions on rows, columns, and boxes, so I was wondering what the mathematical extensions of diagonals would be. It might be something to look into... depends how you think a diagonal is defined.

For those of you who aren't familiar with the extensions of rows and columns, we can assign every cell a number in ternary (base 3). For each cell, let's define four variables (which go from 0 to 2):

- x (the horizontal position inside the cell's box)

- y (the vertical position inside the cell's box)

- z (the horizontal position the cell's box is in)

- w (the vertical position the cell's box is in)

So cell R7C2 would have x = 1, y = 0, z = 0, w = 2.

Note that row is w*3+y+1 and column is z*3+x+1.

Anyway, we can create an intuitive ordering by calling each cell wyzx (with those being digits, not things multiplied together), so that R7C2 = 2001 (in base 3, of course). In this way, R1C1 = 0, R1C2 = 1, R1C3 = 2, R1C4 = 3, ..., R2C1 = 9, etc. A 'row' is the set of cells when you set the y and w coordinates, a 'column' is where you set the x and z coordinates, and a box is where you set the z and w coordinates. This can be extended, so, for example, you could have another set of cells where the x and y coordinates are set (this is usually called 'disjoint groups'), or x and w, or y and z.

I think - but I'm not sure, I'll have to see what it looks like - that you can call the forward diagonal the set of cells where x = y and z = w, and the backward diagonal the set of cells where x+y=2 and z+w=2. What would this look like with other variables, I wonder?