Your First Fugueal Protolock

Some of these puzzles are 'easy', but for those that need speed, there's actually a 'helper addon', that works on all 3 kinds of puzzles (Cantaric, Mezzonic and Fugueal): Zereth Mortis Puzzle Helper available on curse.

Don't thank ME, thank httpsx by supporting him and his addon if possible :D

Edit: I found another kind of puzzle, similar to the untying ones from Nazjatar, but those I found are easy enough.

As I tried to figure this out, I came to this conclusion:
Think of your 3 clicks as either:
SAME + SAME + SAME or DIFFERENT + DIFFERENT + DIFFERENT

SAME border shapes can be:
circle + circle + circle
square + square + square
hexagon + hexagon + hexagon

SAME core shapes can be:
cube + cube + cube
diamond + diamond + diamond
round + round + round

SAME core numbers can be:
one number + one number + one number
two numbers + two numbers + two numbers
three numbers + three numbers + three numbers
For example: one cube + one round + one diamond
For example: two cubes + two rounds + two diamonds

SAME border numbers can be:
one border + one border + one border
two borders + two borders + two borders
three borders + three borders + three borders
For example: A circle with one border + a square with one border + a hexagon with one border

DIFFERENT border shapes can be:
circle + square + hexagon

DIFFERENT core shapes can be:
cube + diamond + round

DIFFERENT core numbers can be:
one number + two numbers + three numbers
For example: one round + two diamonds + three cubes
For example: one cube + two cubes + three cubes

DIFFERENT border numbers can be:
one border + two borders + three borders
For example: a square with one border + a circle with two borders + a hexagon with three borders
For example: a square with one border + a square with two borders + a square with three borders

All 4 criteria (border shape, core shape, border numbers, core numbers) have to all be either
SAME + SAME + SAME or DIFFERENT + DIFFERENT + DIFFERENT

Example:

A square with one border and two cubes in the core.
A circle with one border and two diamonds in the core.
A hexagon with one border and two rounds in the core.

Border shapes are DIFFERENT + DIFFERENT + DIFFERENT (square, circle, hex)
Border numbers are SAME + SAME + SAME (one border, one border, one border)
Core shapes are DIFFERENT + DIFFERENT + DIFFERENT (cubes, diamonds, rounds)
Core numbers are SAME + SAME + SAME (two cores, two cores, two cores)

Hope this helps.

Update:

After reading some more comments in this post and solving some more of these puzzles I want to add some more info to provide you with additional help.

Observations:
1] These puzzles get easier to solve the more you do them.
2] I have not yet come accross an "unsolvable puzzle".
3] Do you see the patterns ... the fractals? (my favorite [npc=177958] quote).

Observation 1 needs no explanation. Like most things, the more you do it the better you get.

Observation 2 requires a bit of explaining. The themes of the comments are the following:
"Are you sure each puzzle actually has a solution?"
"I clicked on 3 pieces and they dissapeared, but there is no way to remove the other 6 pieces"
"I clicked on 6 pieces and they dissapeared, but there is no way to remove the final 3 pieces"

My answer is two fold.
A] It is my guess that every puzzle is generated by a computer program which means two things.
1] Every puzzle has a solution unless the program has a flaw ... which I doubt.
2] There is only a finite number of puzzles, but due to probability and statistics that's a huge number.

B] A puzzle might seem like it is unsolvable only because you clicked on 3 pieces that have a "valid" criteria.
Those 3 pieces dissapear and you are left with 6 pieces. Of the remaining 6 pieces, there is no valid criteria
to remove any more pieces. That means you chose the "wrong" 3 pieces. It means that one or more of those 3
pieces belongs to a different set of criteria. To phrase it in another way "You have chosen 3 pieces that are valid,
however those 3 pieces do not lend itself to the overall puzzle solution".

Here you will click "restart the puzzle" and this time do not choose those same 3. They won't work for the overall.

Observation 3 Do you see the patterns ... the fractals?
Here is the fun part, but this is too long, so don't read it. [TL;DR]

I'm looking at a puzzle and I'm only looking at the border shapes.
I see 3 circles, 3 squares, and 3 hexes. This is telling me something. What is it telling me?
It's telling me that I can choose:
SAME + SAME + SAME circle circle circle
SAME + SAME + SAME square square square
SAME + SAME + SAME hex hex hex
or I can choose:
DIFFERENT + DIFFERENT + DIFFERENT circle square hex
DIFFERENT + DIFFERENT + DIFFERENT circle square hex
DIFFERENT + DIFFERENT + DIFFERENT circle square hex

I'm looking at a puzzle and I'm only looking at the border shapes.
I see 2 circles, 2 squares, and 5 hexes. This is telling me something. What is it telling me?
It's telling me that there is no way to have SAME + SAME + SAME with the circles because there are only 2 of them.
It's telling me that there is no way to have SAME + SAME + SAME with the squares because there are only 2 of them.
It tells me that I will need 1 DIFFERENT + DIFFERENT + DIFFERENT circle square hex
It tells me that I will need 2 DIFFERENT + DIFFERENT + DIFFERENT circle square hex
It tells me that I will need 3 SAME + SAME + SAME hex hex hex

Do you see the patterns ... the fractals?
There is always more to learn.

I will not go any deeper than this. I think you understand how to better solve these puzzles.

Final Note: I have solved many puzzles in my life and have never seen this one before. The creator of this
puzzle deserves much respect. Thank you Blizzard developer for this wonderful puzzle.

After solving this puzzle more I am afraid I have to go deeper.
Why? People keep saying there are unsolvable puzzles. So far, never true.
If you come accross an unsolvable puzzle, take a screen shot of it and I will show you the solution.

I am now going to do some math to explain more.

I will start with the border shapes ... ONLY the border shapes.
All of the rules will pertain to the other three ...
Number of Borders, Number of Cores, Shape of the Cores

I am going to lay out every single BORDER SHAPE possibility starting with one square border and stopping
there so there so I do not have to go through every possibility. You just need to see the patterns to understand.

There are 9 puzzle pieces, each border is either a circle, square, or hex.
Here is every possibility starting with only 1 square border.

Choices for 1 square border:
One - 1 square, 0 circles, 8 hexes
Two - 1 square, 1 circle, 7 hexes
Three - 1 square, 2 circles, 6 hexes
Four - 1 square, 3 circles, 5 hexes
Five - 1 square, 4 circles, 4 hexes
Six - 1 square, 5 circles, 3 hexes
Seven - 1 square, 6 circles, 2 hexes
Eight - 1 square, 7 circles, 1 hex
Nine - 1 square, 8 circles, 0 hexes

Out of these 9 possibilities, some of them will be impossible to solve. You'll never see them.

One - 1 square and 8 hexes
Never can the sqaure be used becasue there is no circle to be different.

Three - 1 square, 2 circles, 6 hexes
You would have to have the square + circle + hex to be d+d+d
but then you'd be left with only 1 circle an 5 hexes
and there would be no way to deal with that one circle because there's no more cicles to be s+s+s
and not enough squares to be d+d+d

Four - 1 square, 3 circles, 5 hexes
The square would have to be used with 1 circle and 1 hex to be d+d+d
but then you will be left with 2 circles and 4 hexes
No way can you have s+s+s on the circles because there is only 2 of them remaining.
No way could you have d+d+d on the circles because there's no more squares.

Six - 1 square, 5 circles, 3 hexes
The square would have to be used with 1 circle and 1 hex to be d+d+d
but then you would be left with 4 circles and 2 hexes
No way could you have s+s+s on the hexes becasue there is only 2 of them remaining.
No way could you have d+d+d on the hexes because there's no more squares.

Seven - 1 square, 6 circles, 2 hexes
The square would have to be used with 1 circle and 1 hex to be d+d+d
but then you would be left with 5 circles and 1 hex
No way could you have s+s+s or d+d+d for only 1 hex remaining

Nine - 1 square, 8 circles, 0 hexes
Never can the square be used because there is no hex to be different

That only leaves the following possibilities:
Two - 1 square, 1 circle, 7 hexes
Five - 1 square, 4 circles, 4 hexes
Eight - 1 square, 7 circles, 1 hex

These 3 have possible solutions because you could have d+d+d and s+s+s
Two = square + circle + hex (d+d+d) hex + hex + hex (s+s+s) hex + hex + hex (s+s+s)
Five = 1 square + 1 circle + 1 hex (d+d+d) 1 circle + 1 circle + 1 circle (s+s+s) and same with hex
Eight = 1 square + 1 circle + 1 hex (d+d+d) 1c + 1c + 1c (s+s+s) and 1c + 1c + 1c (s+s+s)

I could continue this mathematical argument and go to two square borders. Then to three square borders
4, 5 ,6, 7, 8 and 9. Then I could go to circle borders and do the same thing. Then hex borders. I could
keep going to show the same thing is true for the number of borders and the number of cores and the
core shapes ... but what you need to see is the patterns.

Seriously blizzard, what a horrible quest.