Imagine we start with a point at (0, 0), and, in general, we apply the following point transformation $$(m, n)\to (m, n+1), (m+1,n)$$ We apply this transformation in exchange for removing the point at (m,n). Thus, for example, if we were to apply the transformation on our initial point (0,0), we would get a point (0,1) and (1,0). There is another important condition to note; the transformation cannot be applied on (m,n) if there is already a point on either (m, n+1) or (m+1,n) or both. To provide another example, say we now apply the transformation on (0,1) to get the points (0,2) and (1,1),along with the point at (1,0) we had before. Now note we cannot apply the transformation on (1,0) because there is a point already at (1,1) from a former transformation and thus we must get rid of this point first.
Now, consider the xth diagonal such that $$m+n=x-1$$
It is relatively trivial to get rid of all points on diagonals x = 1, 2, however beyond this diagonal is where I am having trouble.
The above picture shows what the plane will look like after the first two diagonals are cleared.
My question, then, is how might I go about getting rid of all the points on this red third diagonal, or the fourth diagonal, etc... I have found via experimentation that what makes this process so difficult is the sheer number of points that accumulate in the way of the others.
Any suggestions/help/hints/resources for a question like this would be greatly appreciated,
Thank you for taking the time to read my question and (hopefully) provide me with some insight!