Thank-you Red Ed for doing that - and for confirming the lowest B159.

I must admit that I did not search all of the "237083 essentially-different B1B5B9", I only searched 5000 or so of the essentially different -

- Code: Select all
`+---+---+---+ `

|1xx|...|...|

|xxx|...|...|

|xxx|...|...|

+---+---+---+

|...|123|...|

|...|456|...|

|...|789|...|

+---+---+---+

|...|...|134|

|...|...|287|

|...|...|596|

+---+---+---+

I have been a bit fortunate - my next best around was 104000.

The way I have completed the B5B9 means that The B6 and B8 boxes both have only

392 completions each.

Thee are only 5 different ways

384,392,400,432 & 448 to complete these cross-combinations, as described by Frazer

Frazer wrote:In response to coloin, it takes a little thought to persuade yourself that there are 5 classes of constraints; then a computer determines all of the values 384,...,448. Let's assume that we are trying to count the number of ways to complete

??? ***

??? ***

??? ***

123

456

789

where we are given all the *, and want the ?. The situations are:

(1) The three rows of * have the same numbers as the three columns (in some order); for example, 147/258/369, or 825/417/693 [432 ways to fill in the ?].

(2) One of the three rows has the same numbers as a column, but the other two have just two; e.g., 147/259/368 [384 ways].

(3) All three rows have two numbers from a column; e.g., 148/259/367 [392 ways].

(4) Two rows have two numbers from a column, but the other has just one; e.g., 143/256/789 [400 ways].

(5) All three rows have one number from each column; e.g., 123/456/789 [448 ways].

It takes a little thought to convince yourself that there are no other possibilities.

There are 18 such pairs in a grid. The 400 option is by far the most common.

Edit - or are there 36 such pairs ?

C