by **Guest** » Sat Mar 26, 2005 12:32 pm

Interesting. How about this...

All the rows in insolation give you 9 * 9! possibilities

Now, considering columns can only reduce this number, so col 1 at the moment has 9^9 possibilities (column one will be 999999999), but only 9! are valid, so that gives you...

(9* 9!) - (9^9-9!) which rationalises to

(10*9!)- (9^9)

Column two has 8^9 possibilities so we need to take off 8^9 - 9!...

(11*9!) - (9^9) - (8^9)

So, I think rows and columns alone would give you

(18*9!)-(9^9)-(8^9)-(7^9)-(6^9)-(5^9)-(4^9)-(3^9)-(2^9)-9

Oh bugger, that's negative too!

OK, how about this, start with this grid:

987654321

876543219

765432198

654321987

543219876

432198765

321987654

219876543

198765432

This is consistent for rows and columns, so you only need to worry about boxes:

So how many possibilities are there in the top box: at the moment 9*8^2*7^3*6^2*5, and how many can actually be valid? 9!, of course, which is true for all the boxes. So that doesn't go anywhere either.

Actually, I don't see any sensible answer other than 9*9!. I'm beginning to think this is a self-cancelling problem. By that I mean that although two adjacent boxes, rows or columns with possibilities of 9! seems contradictory, in fact it may not be - consider the first three rows, that might look like this:

987987987

543543543

321321321

Although the first row is over egged (this is way over the actual 9! possibilities) the third line is under-egged. Maybe this balance achieves the right answer.

What do you think?