yes that one is required to open that gate.
that 8 square 3x3 one is the simplest one in the world. later on in the square above that will be a 15 square 4x4 one that is a repeatable quest, and is optional.
but why pay any BB's? I didn't think that the option to pay to open gates existed before this Christmas event.

as for those puzzles, there are any number of online solvers that you can set with your particular random orientation and they tell you what to move.
http://www.brian-borowski.com/Software/Puzzle/ is the best one and can even do the one that's so simple 3x3.

Yes, one 'solved' position is as follows, and you can actively work to that one.
1 2 3 4
5 6 7 8
9 10 11 0
13 12 14 15

I have checked every possible one with the first 2 rows correct and the last 2 rows in all possible orders, and this is the only one that slips its way through the buggy coding that the developers have for determining if it's 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0
such simple coding but they got it wrong.

Ah, I didn't see it was a door, stupid me.
Is it supposed to be 8 or 15 pieces?

Also, if you get the big puzzle in under 100 moves, you win an achievement. Should that be win an achievement, achieve an achievement or completed an achievement? The world may never know!

Another identified incorrectly as solved is close to the one listed above.
1 2 3 4
5 6 7 8
9 10 11 0
12 13 15 14

not sure why I didn't catch this one the last time I experimented.

Yes you do. You get 10 Achievement points for any of the 'repeatable' achievements. That also includes getting to Max First.
Not sure what else it includes, perhaps anything Achievement related that gives you a popup that you can publish.

And I figured out how I missed that 'solution' last time. I was checking all the orientations of such and such a character. I was not necessarily doing them in order but had all of them on paper. I crossed off them as I did them but didn't proceed any farther when I discovered the 'one' incorrect solution. Didn't think that there could be more.

Achievement points are meaningless. They do not factor into the Ranking Score. They do not in and of themselves give you XP when you are getting XP. They do nothing for you.

You can't even sort the rankings by that field. I'd love to know who has max of that, who has done the most repeatable quests, who has won the most Diamonds,... and any number of other things.

As for farming this, I've got 101 of the 117 Achievements. I've done this one in under 100 perhaps a few dozen times. I've maxed out first at least 5 times, think the first time was the Far Away Lands in June. My achievement points is 2030 so does that mean I've done 102 things again and again? Who knows. Who cares? Does it matter? no
I don't even bother to try to get this one in under 100 moves, I try for exactly 99 or 101 or just do it in whatever when I'm not doing this current experiment.

I checked many combinations, 360 times about 25 effectively.
There are no new ones other then the real one and those 3 unreal ones that the game thinks of as solved. But of course this is a far cry from the billions of combinations that can exist.

I discovered something very interesting though.
I can put all the pieces into the following position and have the game not think it's solved.
1 2 3 4
5 6 7 8
9 10 11 0
13 12 14 15

It seems that if you move the 0 up from the bottom right square (and the 15 down) it will flag a 'solved' condition.
But if you get that 0 there from the left it is not solved until you move the 0 down then up again. That makes absolutely no sense. It seems like they programmed the solved check as an ongoing thing, which one can't really do for a puzzle like this, instead of doing a full check after every move. Perhaps as a time saving thing, but a check of 16 equalities takes no time at all.