CB(n) or \(\delta(n)\) So, the chess board is a hypothetical Turing machine thing.
Definition
The maximum sum of BB(n)-1’s that can be written (in the finished plane), with BB(n) state and a BB(n) colour halting Turing machine. Note that we do not use a tape but a Plane. As well as that, we allow for movement in 8 directions. Up, down, left, right, up right, up left, down right, down left.
Better Explanation
We have a read and write head moving on the infinite paper. Instead of turning it into a 0 or 1, it turns it into 0,1,2, etc, with n-1 as the maximum number. It reads and writes, until it eventually halts. The final question, is what is the sum of all n-1s?
Augmented CB(n)
Also called \(\Delta(n)\) It should be more powerful.
Instead of asking what is the sum of all n-1s, we ask what is the sum of all numbers that are not 0.
Better Definition
The maximum sum of all nonzero numbers that can be written (in the finished plane), with a BB(n) state and a BB(n) colour halting Turing machine. Note that we do not use a tape but a Plane. As well as that, we allow for movement in 8 directions. Up, down, left, right, up right, up left, down right, down left. BB(n) is the busy beaver function.

Finally, \(\Delta_\Delta(n)). This is defined to be the maximum number that can be computed for an n-state, n colour, n dimentional turing machine to halt.