I don't understand, are your questions serious or joky (or rhetoric), but anyway I consider them seriously.

dobrichev wrote:Serg wrote:So, you can see that rows pair with MI = 7 (no two-row UA) is the most probable event!

Is this phenomenon similar to this observation for sudoku grids in Minlex Form: Min and Max Lists / Chaining?

Yes, variant of two-row configuration for Sudoku solution grid when that configuration doesn't contain any UA sets is the most probable too (as for 9 x 9 Latin Squares). Michael Derevin's article contain data necessary for calculation of probability. This case (no UA sets in two-row combination) takes place for Ranks: 3, 9, 11, 12 - totally for 144+432+1296+1296 = 3168 (variants of the 2-nd row). Probability to observe such case is 3168/12096 = 26.2 %. The next nearest (frequent) case is U4+U14 case - probability to observe it is 21.4 %.

dobrichev wrote:Michael A. Deverin wrote:In just over 70% of the cases, the lowest ranking pair was of rank = 4. This is especially fortuitous, since there is only one way to convert the row pair to minlex form.

It is possible to estimate probability to get the lowest ranking pair of rank = 4 for random Sudoku solution grid.

- Code: Select all
`Rank Frequency Probability to observe (approx.)`

1 72 0.00595

2 216 0.01786

3 144 0.01190

4 1296 0.10714

...

In total 12096 r2 row form when r1 = 123456789

Probability not to get Rank1 in one two-row (two-column) pair is 1 - 0.00595 = 0.99405 . Probability not to get Rank1 in 18 two-row/column pairs is 0.8981. Probability not to get Rank2 in 18 two-row/column pairs is 0.7230. Probability not to get Rank3 in 18 two-row/column pairs is 0.8062. Probability to get Rank4 at least once in 18 two-row/column pairs is 0.8700. Multiplication of all these probabilities gives 0.455 or 45.5 %. This is substaintially less, than 70%. Probability of 70% is observed during random Sudoku solution grids simulation. It seems different row/column pairs in Sudoku solution grids may not considered as independent (especially rows/columns in the same band/stack). So, this model doesn't work well for Sudoku, but works rather good for 9 x 9 LS.

Serg