afj wrote:We've had two different heuristic methods both suggesting 6 x 10^21 as an approximate answer.

Frazer

Wow, what an interesting problem. I thought I would take a fresh stab at it and see where I got. Well, I thought I hit upon a good technique, but ....

I made the assumption that the 1st, 5th and 9th grids could all be filled in independently. There are (9!)^3 possible ways to do this. Then I figured you could brute force how many ways there were to fill in the rest of the grid for for a few of those combinations. I was hoping that any of the (9!)^3 initial start positions would have the same number of ways to complete. Failing that, an average would point to the total number of combinations.

I first tried this start point for my brute force approach:

123 --- ---

456 --- ---

789 --- ---

--- 123 ---

--- 456 ---

--- 789 ---

--- --- 123

--- --- 456

--- --- 789

My algorithm churned out 283 576 different ways of legally filling in the rest of the grid.

Then I changed grids 5 and 6 to:

123

468

579

and

258

713

496

Result: 174937 combinations. Oh well

Then grid 5 to:

468

123

579

Result: 174937 combinations. Interesting that the same number appeared twice.

Anyway, if I take the average of my three tests, I get a figure of 10.1 * 10^22 as an approximation to the total number of possible solutions. A little higher than 6 * 10^21, but in the same ballpark.