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1000000 random P-sudokus were checked - has it at least one "Afghan" pair (i.e. is it possible to buld "Afghan grid" from checked P-sudoku and its pair)? I got 1729 "Afghan-ready" P-sudokus (ratio 1:600 approx.). Then I analyzed those 1729 "Afghan-ready" P-sudokus.

It is known, that any band (stack) of sudoku solution grid may contain "repeating triples" or not. So, there are 2 possible types of any band (stack). I counted bands and stacks containing "repeating triples". For example, MC grid has 3 bands containing "repeating triples" and 3 stacks containing "repeating triples". So, MC grid would have such count equal to 6.

Some statistics concerning "repeating triples" of 1729 "Afghan-ready" P-sudokus.

- Code: Select all
`Number of bands and stacks with triples Grids`

0 67

1 289

2 534

3 532

4 247

5 53

6 7

P-sudokus without "repeating triples" are more rare, than one would expect (67 grids), but P-sudokus having all bands and stack with "repeating triples" (like MC grid) are more frequent, than one would expect (7 grids).

If a band is randomly generated, than probability of "repeating triples" appearance is 1/4. Probability of having 3 bands and 3 stacks containing "repeating triples" - (1/4)^6 = 0.000244, that corresponds to 0.4 grids among 1729 grids (20 time less, than observed). On the other hand, probability of having all bands and stacks without "repeating triples" - (3/4)^6 = 0.178, that corresponds to 307.7 grids among 1729 grids (5 times higher than observed).

So, more bands/stacks with "repeating triples" are there, higher probability of random P-sudoku to be "Afghan ready" is.

Serg