by **Guest** » Thu Apr 21, 2005 6:41 pm

I fear that IJ's calculations aren't quite right; the first reason is that 448 is not always right (it is usually just a bit too much), but the more serious reason is that tinfoil's "fundamental problem" isn't accounted for. The first problem makes only a small difference, but I suspect that the second could make a difference of a couple of orders of magnitude, making Josh's reported estimate earlier much more accurate. But IJ's value is an upper bound - the best yet!

To deal with the first problem first, 448 would be the correct answer for filling in blocks with constraints like

*** 123

*** 456

*** 789

123 ???

456 ???

789 ???

(where the * are irrelevant, and we are trying to fill in the ?) in which not only are there four possible squares for any number, but, conversely, every square has four possible numbers going into it. But consider the configuration

*** 147

*** 258

*** 369

123 ???

456 ???

789 ???

It is still true that every number can go into one of four possible squares; however, some squares can be filled by three numbers and others by six. In fact, it's quite easy to show that there are 432 possible ways to fill this in. I think that there are always between 384 and 448 possible ways to fill in these constraints. (I may have this wrong, but I think that there are five basic patterns, which my computer tells me have 448, 432, 400, 392 and 384 possibilities.)

But the second problem is much more serious. As tinfoil originally said, "with only blocks 6 and 8 to fill, there can be only one unique way to fill them", but later noted that in fact, there can easily be no ways at all to fill them. My guess is that this will be the typical case. The sort of thing that can happen when trying to fill in block 1, say, if we already have blocks 2 and 3, and blocks 4 and 7, is that there might be a picture like the following:

??? *** ***

??? 1** 2**

??? 2** 1**

*12

***

***

*21

***

***

The constraints from blocks 2 and 3 force the 1 and the 2 both the occur in the top row of block 1, and blocks 4 and 7 force them both to occur in the left column - so they have to occupy the same square, which is not possible!

Frazer