Any such set with this quality is a "cover". Still, it's nice to construct such sets with some order in mind. My set of 416 representatives are kept in row-minimal-form. That is, no spin-and-relabel operation will produce a smaller 27-digit number (after row concatenation).

The 416 cover representatives fall into 44 equivalence classes, where the class is defined by the number of ways the grid can be completed to form a legal finished Soduko matrix. Reference: "Enumerating possible Sudoku grids", by Felgenhauer and Jarvis; June 20, 2005.

I am attempting to construct a 4-row cover, and have reached the stage where the veracity of the results are in doubt. I would like to describe my methodology and present some results, hoping that some of you smart people

can either confirm or refute the findings. Here it goes ...

For counting purposes, there are ordering restrictions placed on digits in column-1 of rows four thru nine. They are r4 < r5 < r6 and r7 < r8 < r9 and r4 < r7.

For each 3-row representative, I tacked-on all possible fourth rows and then counted (semi-brute force) the number of completions to a full grid. There were 834148 such representatives. Regarding the number of completions, each

representative fell into one of 7739 equivalence classes.

- Code: Select all
`AAAAAA BBBB A :: 4-row representative count`

4 432 B :: Number of configurations

6 864

18 1296

288 2592

504 3888

833328 7776

======

834148

I then considered that the 834148 representatives might not be mutually exclusive, regarding spin and relabel operations. So I "spun" each one into its row-minimal-form and found that their were indeed many duplicates.

After removing duplicates, my representative set was reduced to 790171 4-row entries. There were still 7739 equivalence classes.

- Code: Select all
`AAAAAA BBBB A :: 4-row representative count`

4 432 B :: Number of configurations

4 864

12 1296

185 2592

387 3888

789579 7776

======

790171

All seemed to be going well until my final check. My experience has been that the cross product sum of configurations times completions for each representative should be a constant.

- Code: Select all
`Constant = Sum { configurations[i] * completions[i] }`

= 255,322,533,620,736

However, for my set of 790171 4-row representatives, the cross product sum is 299,540,420,207,424 (about 17% bigger).

Here follows a small table comparing 1-row, 2-row, 3-row, and my probably faulted 4-row cover.

- Code: Select all
`Legend:`

A = Sum{ configurations[] }

B = Sum{ completions[] }

C = Sum{ configurations[] * completions[] }

Set AAAAAAAAA BBBBBBBBBBBBBBB CCCCCCCCCCCCCCC

1 1 255322533620736 255322533620736

2 12096 317247725328 255322533620736

3 2612736 40715927744 255322533620736

4 6141771216 38540330792 299540420207424 Wrong!

Total Number of Grids

= 6670903752021072936960

= (9!) * 72 * 255322533620736

I would like a little help. Can some of you smart people either confirm or refute these findings?