I believe that this achievement can only be obtained after you beat the main story line and have turned on all of the lasers. However, you do not need to have beaten all of the random puzzles spread out across the island.
The top part of this solution is all about beating the challenge and any hints that I can give. The bottom part of this solution is telling everyone how to get to the challenge room.
Beating the Challenge:
This is not an easy achievement. There is no way to cheat the system, and there are no online solutions because the puzzles change every time.
Once you are in the cave, go to the very back and start the record. That is accomplished by moving the cursor to the far left instead of going up, Going up will stop the record.
As soon as you start the record, run up to the wall and solve the first three puzzles. These are simple puzzles. The first two are mazes and the last one is black and white squares.
When you are doing these, make sure to hold down the button that speeds up your cursor (I believe it is LT) and push RB after you solve the puzzle to quickly switch to the next puzzle.
After this, run to the next puzzle which is a maze. In a little bit you will need to walk a full size version of this same maze, so pay attention not only to the solution of the maze but also to where the two dots are in the maze. In the full size maze, these two dots are the locations of two triangle puzzles that you must solve. I took a picture of the maze with my phone before I solved it so I would remember the location of those two puzzles.
Once you are done with the maze (which shouldn't take too long) run to the next section. Here there are four puzzles spread out around a pillar. Two puzzles are on the top part, and two are on the bottom. Only one puzzle will be lit up at a time, and the order is always random, so be quick about checking all 4 puzzles to know which one you can solve.
The puzzles rotate between these 4 locations. There will be a simple maze, a tetris with suns puzzle, a black and white squares puzzle, and a symmetry puzzle with multicolored dots.
The last puzzle I mentioned, the symmetry puzzle with multicolored dots, means that there will be three different colored dots on the symmetry puzzle. Orange dots, blue dots, and black dots. The color of line that that you control will be blue, and the color line you don't control will be orange (that may be backwards. I don't remember). The orange line must go through the orange dots, and the blue line must go through the blue dots. Any color line can go through the black dots.
If you fail any of these puzzles, you must go back and re-solve the last puzzle you did. So if one of them are too hard for you, just purposefully fail it to get a new puzzle of that same type.
Another thing that I started doing with the tetris puzzle was counting the amount of squares that the tetris designs filled up, and then just looking for that number of squares I could section off.
After you solve all 4 of those puzzles, run across the bridge to the next set of puzzles. In this section, there are three black and white square puzzles, but only one of them is possible to complete. The other two are impossible. Once you solve the one you can beat, go to the next set of three. These are multi-colored squares, and once again, only one of them is possible to complete.
When you are looking at the puzzles trying to figure out which of them is possible, look for quick hints. One thing that is nice about all of these puzzles is that because you are timed, none of the puzzles can be too difficult. So if one of the puzzles looks impossible right from the beginning, then it probably is. Look for what you believe is the easiest and try that one first.
Things to look for in the puzzles that be beat include corner sections of one color that can be separated from the other colors easily, and clumps of the same color going straight down the middle.
If you see a puzzle with three different color touching each other and none of them are touching a side of the puzzle, that puzzle is almost certainly impossible.
Another thing that I did was I attempted this challenge when a friend was watching, just so I could have a second set of eyes and try to determine which puzzle was possible faster.
After you solve the two possible puzzles, run into the full size maze. Pull up the picture on your phone and run to those locations in the maze. Then spin around. The puzzle will be on one of the screens close to your location. Solve the first one and then run to the next location and solve the second one. These puzzles are triangle puzzles. If you don't know how they work, scroll down in this solution. I talk about them farther down.
EDIT: Christoph 5782 has linked a website that will solve these puzzle triangles for you. I am posting that link here!http://witness.webcrow.jp/
Again, I had someone else watching when I solved these puzzles to quickly count the sides of each box to make sure that the lines touching the box equaled the number of triangles in the box. This is really important at this step because if you screw up here you must re-solve the puzzle from before, which requires you getting out of the maze.
After you solve the two triangle puzzles in the maze, leave out the exit and come to the last two puzzles. They are pillar puzzles. They also aren't terribly difficult. However, because they are pillar puzzles, there is really no way to double check and make sure that you have the right answer before you finish the puzzle.
There will be two pillar puzzles. One is a black and white puzzle, the other is a symmetry puzzle with black squares. If you are having difficulty here, I recommend going to the end of the game right before the elevator and working on this pillars.
Those are the last two puzzles, so once you beat those, you will have beaten the challenge.
The entire challenge is timed with the record that you started at the beginning. The record plays two classical songs and once the songs are finished you are out of time.
If you are stuck on the same puzzle for too long, either fail it on purpose so you can get a new puzzle that is the same type, or run back to the record and restart it.
There is no way to pause this puzzle and solve it "offline" like there is on the ps4. On the ps4 you can enter sleep mode and solve the puzzle, and then turn the ps4 back on.
If you turn off the Xbox One, the puzzle will restart. And if you pause the game, the timer will continue, even if you open a new app.
I hope that this solution helps. Good luck on getting this achievement!
Getting to the Challenge Room:
So once you have done completed the game and started all of the lasers, go up to the top of the mountain where you started the final quest to finish the game.
(Side note. If you have already completed the game, just load an old file. Try and save before you get in the elevator to finish the game)
At the top of the mountain you can see a statue holding a box. If you get down beneath the box by walking down the stairs, you can see that there is another solution for the box.
If you are confused, look at this photo.
Once you re-solve the box, go back down into mountain until you reach the statue staring at a puzzle on the ground. This puzzle will be right before the room with all of the pillar puzzles that you needed to solve.
The solution to this puzzle is as follows.
Left, Up, Right, Up 4, Left, Down 3, Left, Up, Left, Down, Left, Down 2, Left, Up 4, Right, Up, Left
This puzzle was hard for me to see just because it was hard to get close enough to it.
If you don't understand this puzzle, the concept is pretty simple, There are several boxes that contain yellow triangles. For every one triangle in a box, that is how many times you must draw a line that touches the box that the triangle is in. So if a box has one triangle, one side of the box (And only one side) must have a line going through it. So if a box has three triangles in it, you must cover up three sides of the box with lines.
Remember this puzzle, it shows up in the finale.
Again, here is the link for the triangle puzzles that Christoph 5782 posted in the commentshttp://witness.webcrow.jp/
Once you solve this puzzle, it will open up a secret door behind you, the direction that the statue is looking. The door closes after a couple seconds, so don't waste to much time getting there.
Once you do that, you must solve a black and white square puzzle
Solution for that puzzle
Again, from IGN.
After that, another door will open. Back here there are lots and lots of puzzles to solve. However, only two of them need to be solved. Wander through this area a bit until you make your way to a pillar puzzle. The solution to this puzzle is as follows.
Right, Up 2, Right, Down 2, Right 3 , Up, Right, Up, Left 2, Down, Left, Up 3, Right 2, Up 2 Left, Down, Left 2, Down, Left, Down, Left, Up 2,Right, Up, Left, and then finish it.
After that, you will need to solve one more puzzle just opposite of the pillar puzzle. It is a sun and tetris puzzle.
The solution is as follow.
Up 5, Right, Down 4, Right 4, Up 2, Left 2, Up 2, Right 2
Now you have made it into the challenge room.