The first floor is a simple map with only 1 trainer in it. after defeating the trainer you receive a hint to one of the later puzzles

On this floor you walk forward and fall down the hole (the holes aren't visible until after you step on it).

Falling down the hole leads you to this floor under the first floor. you need to hit the switches in a certain order so that the rocks all go up so you can walk through (this puzzle is similar to the original games where you hit the switch and the doors open and close).after getting through both puzzles you have a double battle. defeat them to move on and receive another hint to the next puzzle

Now you are back to the second floor (on the top left side of the room), defeat the trainer (who gives you another hint) and continue to the door on the right.

This is a different area of the gym but is still located in the first floor. To get through this area you have to walk through the pillars in a certain pattern so you don't fall down the holes (the holes aren't visible until you step on them, also falling down takes you back to the basement where you have to walk all the way back again and a trainer awaits you to battle). The first trainer gives you the hint to solving this puzzle. He tells you that the only way to get through is to view the room from the statues perspective. so pressing the A button on either one of the statues asks you if you want to see from the statues view. If you say "yes" then this map appears:
You get a view of the same room but from a different perspective, so from this view you will notice the flags hanging on the pillars which show you the path you need to take (it will show you this view only once so you need to memorize it). Exit out of the map and navigate downwards until you reach the stairs at the bottom

This room is also located in a different part of the gym but is still considered as the 2nd floor. defeat the trainer to receive the last hint to the final puzzle and continue up the stairs.

This is the last puzzle of the gym and basically all you have to do is enter each door in the correct order. The last 4 trainers you faced all had hints as to which door you need to enter and at which order you should enter them. the hints consist of math problems where the answer is the number of times you need to count the doors and from which door to start and what direction you need to count them in. for example if the first answer was 10 and the trainer told you to start on the second door from the right and count 10 doors left or right or back and forth (example: count 3 doors to your left, then 7 to your right). Then you enter the next 3 answers and count the doors in the correct direction to take you to the next room. This puzzle is a bit difficult to explain so I hope that everyone was able to understand it.

The gym leader is located on the fourth floor and is blocking you from finding out the location of the Pokemon league. defeat him to receive his badge and enter the door behind him.

on the fifth floor you encounter your rival who tells you that he has found the location of the Pokemon league and will be waiting for you there for one final battle.

This is the highest floor in the gym, the reason the gym is so big and why you need to climb to the top is because the Pokemon league is on an island lost at sea and the only way to find its location is from the highest building and at that particular spot (where the gym is located). look through the binoculars to find out where you need to go next.