To reopen an old wound, I think

ravel mentioned this method earlier. Help to verify my figures and estimations would be appreciated !

If there really are 10^16 puzzles - A "real" puzzle distribution

might look like this !

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`18 0`

19 40

20 2,000

21 500,000

22 ? 10,000,000,000

23 ? 500,000,000,000,000

24 ? 3,000,000,000,000,000

25 ? 3,000,000,000,000,000

26 ? 3,000,000,000,000,000

27 ? 500,000,000,000,000

28 ? 50,000,000,000

29 ? 1,000,000,000

30 ? 100,000,000

31 ? 10,000,000

32 ? 1,000,000

33 ? 10,000

34 ? 1000

35 4

36 0

Lets see if I can estimate the number of 25 clue puzzles in a grid

How many ways are there to put 25 clues at random into a single valid grid

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`81! / 56!*25! = 525652003943603702568 = 5.25 * 10^20`

how many will be invalid ?

Only a very small proportion will have absent clues in two rows in the same band, and only a small proportion will fail to have at least 8 clue numbers [less than 1% total]

I estimated how often 25 clues will [minimally] solve a given grid by making a random mask and solving it over a large number of "random" grids

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`Out of 800,000 puzzles`

mask minimal puzzles

1 18

2 7

3 5

average 10 puzzles, therefore = 1 in 80000

Out of 4,900,000 puzzles

mask minimal puzzles

4 23

therefore = 1 in 200,000

This small study may give an average of

1 in 100000 randomly picked masks of 25 clues will be minimal and valid.

I appreciate that this does not tally with ravels work - and that the effect that is shown here is that some masks are much better than others....

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` [5.2 * 10^20] `

Estim number of 25 puzzles in a grid = --------------------------------- = 5.1*10^15

100000 [average]

only 3/10s of puzzles have 25 clues......

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`Estim number of puzzles in a grid = 5.2*10^15 * 10/3 = 1.7*10^16 `

C