The 416 is now understood - excellent.

I note also the confusion in the gang elements in the 44 was much worse than 1/44. That also explains why G coluldnt understand me when I was refering to it.

From Red Eds

Box2 in the 44 is equivilent/the same 25 times out of 44

Box 3 has 33? different combinations. - for what its worth.

When we are combining more than 3 boxes it gets complicated - and this is what defeated normal algebraic attempts at working out the total number of solutions.

It is slightly more complicated because when you have two boxes interacting at right angles there is always a different number of solutions possible to fill the box.

Eg taking B2 B4 independantly there are 5 possibles solution TOTALS for B5.

These are 432,384,392,400 & 448. The most common by far is the 400. It depends on the way the boxes interact see Frazer's post. This took me ages to comprehend by the way - but it is true !

I think I am right with these numbers

If there are 9! ways of expressing B2/B4 [B2 is fixed]

The way it works out can be expressed:

432 -- 1*1296 = 1296

384 -- 27*1296 = 34992

392 -- 54*1296 = 69984

400 --162*1296 =209952

448 -- 36*1296 = 46656

-------------

----------------------362880 = 9!

The rare 432 is eguiv to a gangster1/44 and this has 6^4 ways = 1296

The 3^3 ways of swapping the boxes which give a 384,392,400 - explains the 27 in the 27,54 and 162.

you need to study it to understand it

To answer G , well we dont know why a grid turns up 17 solutions with perhaps much more regularity than usual - if we did we perhaps could find a 16.

I have analysed Gs 17er with multiple 17s in it and it doesnt look special as regards its box combinations - note now 36 not 18. But I think it is special.

It is

400 - 26

392 - 6

384 - 1

448 - 3

432 - 0

Interestly in trying to construct grids with 384s - thats when pairs start appearing turning up - so I am now going for 392s and 400s - avoiding 448 and the dreaded number pairs. That might be the way.

In its favour of this analysis - grids constructed with more of the lower combinations definitely have have less total solutions. That might make all the difference. We need another breakthrough !

Regards