Mathimagics wrote: a, (a+3) mod 3 and (a+6) mod 3
mod 9 of course
Mathimagics wrote: a, (a+3) mod 3 and (a+6) mod 3
Mathimagics wrote:.
Fake news! I did not write that at all!
hkociemba1 wrote:Are there any known transformations which preserve X but do not necessarily preserve P ?
For a given P-grid created with your method it easy to generate a different valid grid by just swapping N pairs of cells. In case of N=3 you can take for example the rows a, (a+3) mod 3 and (a+6) mod 3 and swap in each row the elements of two columns in the same stack, for example column 7 and 8. But this surely is nothing new for you. This operation always destroys the X-property so this way round it is no problem.
Thanks, I also noticed this. But except from destroying part of the grid and searching for another solution there does not seem to be any obvious transformation which gets rid of the only 3 (or N in general ) minirow types.blue wrote:"Swap rows 4&6, swap columns 4&6", is one.
Mathimagics wrote:An intriguing question arises. What happens to the relative population of PWX's as we increase the box size?
For example, we have 5.47 billion ED Sudoku grids (for N=3), but only 2922 ED PWX grids (and a much smaller set of isomorphic transformations).
What will happen with N=4, N=5 etc? Will the ability to form compatible intersections tend to grow? Diminish? Do we in fact have any prospect of ever knowing?
It will also be very interesting to find out (for N=3) whether these 2922 are in fact connected or not by Pittenger-moves. The fact that they are otherwise comparable to finding marbles in inter-planetary space does not absolutely exclude this possibility … that's one question that is at least accessible!