jomapil wrote:

I also agree that modulo is the hardest and bitwise the easiest. To gain more adherents of these special puzzles it would be valuable anyone explain, with more details and examples, these puzzles.

As a curiosity, the total number of different possibilities (for 8x8) are

Bitwise 2-cells -----28

Bitwise 3-cells -----112

Mod 2-cells ---------56

As there are many puzzlers that haven't mathematical formation it would be helpful to make available the respective tables.

I hope, hopefully, that Patrick don't remember to do module 3-cells

Okay, I'm curious, why is this any different than just 8*7 (Assuming a fixed cell shape) in the case of 2 cell bitwise cages by the fundamental counting theorem? They have the same number of possibilities, it's just that a bitwise cage gives you more information to resolving it.

As for any 3 (Except division and subtraction), assuming it's L shaped, it's 8*1*7+8*7*6, which is the number in which the two branches are equal, and the nexus different, plus the number with all three different. This is 392 for an L shaped, and for 3 in a row, it's 6*7*8, which is 336. All of this, mind you, is still the fundamental counting theorem for the sake of proof.

But the thing is that the rate at which this can be reduced is drastically different in a bitwise cage.

1| doesn't exist.

2| doesn't either.

3| contains only 1's and 3's, so there are 2 combinations in a 2 cell cage, and 2 in a 3 cell cage.

4| doesn't exist.

5| contains only 1's, 4's, and 5's, making 6 combinations for a 2 cell cage, and 12 combinations for a 3 cell cage.

6| contains only 2's, 4's, and 6's, making 6 combinations for a 2 cell cage, and 12 combinations for a 3 cell cage.

7| can contain absolutely anything < 8. But it's still dramatically reduced because it must contain at least 1 number > 3. There are 24 possible 2 cell solutions, and a lot more 3 cell than I care to calculate, enough to the point at which you won't get much of anywhere by trying to eliminate possibilities on an l shaped 3 cell 7 bitwise or.

8| doesn't exist.

9|-15| in an 8x8 is just 8 and the difference in 8 and the cage number. So 13| is 8 and 5 in two cells, and in 3 is 8 + any of the options for 5| for two cells from earlier, etc.

Basically, even with the difficulty set lower as it apparently was, bitwise or is still fundamentally easier than any of the other functions at the size we're doing it (Personally, I'll take easier bitwise puzzles in exchange for not having to do a 16x16 Modulo puzzle. At that point, it wouldn't matter which function was harder, since I don't think I'd have the patience to solve any of them.).