by Sami Casanova
Answer: WAY
Problem:

The first thing to notice is that the colors of the boxes are all unique and not obvious choices for standing apart from each other, and thus are not there just for decoration. If you view the source code of the page, you can get the color of each box expressed as a hexadecimal number.

ClueColor
There is an exit to the east, leading to a pronic room. There is also a staircase leading up.001FC0
There is a staircase leading down to an admirable room. There is also a staircase leading up.015555
There are exits to the east and west, and a staircase leading up.020AE1
There is an exit to the south, leading to an automorphic room. There is also a staircase leading down.11EEE1
There is a staircase leading up to a room that is amicable with this one. There is also a staircase leading down to an Ulam room.15F507
There is an exit to the east, leading to a Catalan room. There is also an exit to the west, leading to a harmonic room.1D5AB8
There is an exit to the south, leading to the smaller room of a betrothed pair. There is also an exit to the north.287F43
There is a staircase leading up to an Eulerian room. There is also an exit to the west.28CF08
There is a staircase leading down to a Carmichael room. There is also a staircase leading up.35C7E2
There is a staircase leading up to a Saint-Exupery room. There is also a staircase leading down.3BADC8
There is a staircase leading down.50BAF4
There is a staircase leading up to a square room. There is also a staircase leading down.590A6F
There is a staircase leading up to a magnanimous room. There is also an exit to the east.5C56AC
There is a staircase leading down to a prime room. There is also a staircase leading up.63BCBE
There is an exit to the south, leading to a Moran room. There is also an exit to the north.6C7B00
There is a staircase leading down to a perfect room. There is also a staircase leading up.6FFCBF
There is a staircase leading up from here.7029A3
There is an exit leading south to this room’s fiancee. There is also an exit to the north.71D4A4
There is a staircase leading up to a house room. There is also a staircase leading down.79757E
There is a staircase leading up to a Cullen room, and also a staircase leading down.94016A
There is a staircase leading up to a Perrin room, and also a staircase leading down.980001
There is an exit to the south, leading to a tau room. There is also an exit to the north.998BA5
There is a staircase leading down to an astonishing room. There is also an exit to the north.9D8000
There is an exit to the east, leading to a Curzon room. There is also an exit to the west, leading to a Rhonda room.9F89E3
There is a staircase leading down to a Fibonacci room. There is also an exit to the north.A6B75B
There is an exit to the south, leading to a cube room. There is also a staircase leading down.A761A5
There is an exit to the east, leading to an anti-perfect room. There is also an exit to the west.AA79D2
There is an exit to the south, leading to a Lucas room. There is also an exit to the north.B73E1D
There is a staircase leading down.B8CA42
There is a staircase leading up to a lucky room. There is also an exit to the west.BC09A2
There is an exit to the south, leading to a brilliant room. There is also an exit to the north.C294AB
There is a staircase leading down to a Smith room. There is also a staircase leading up.CB7B25
There is a staircase leading down to a hungry room. There is also a staircase leading up.D0FC21
There is a staircase leading up to a primorial room, and also a staircase leading down.EFEED8
There is a staircase leading up to a Zuckerman room. There is also a staircase leading down.F13840

Each box’s number has one or more properties (such as being prime, or a perfect cube, or an amicable number). For example, the last box on the puzzle page has the color #F13840, which is 15,808,576 in base 10—and a perfect square. The flavortext indicates that you don’t know where you are, which hints that you need a map. The room exits that describe the property of the room they lead to, provide enough information to create the map. All of the rooms are contained within a 5 × 5 × 5 cubical structure. Any two rooms that share a wall (or a floor/ceiling) are connected and have a door (or staircase) between them.

First (Bottom) Floor

 Color: 7029A3 Property: Exit: U Color: 5C56AC Property: Exits: E, U Color: 1D5AB8 Property:  (not used) Exits: E, W Color: 28CF08 Property: Exits: W, U Color: 001FC0 Property:  (also Ulam, and Harmonic) Exits: E, U Color: AA79D2 Property: Exits: E, W Color: 020AE1 Properties: (also Lucky) Exits: E, W, U Color: 9F89E3 Property:  (not used) Exits: E, W Color: BC09A2 Property: Exits: W, U

Second Floor

 Color: 63BCBE Property: Exits: U, D Color: 79757E Property: Exits: U, D Color: EFEED8 Property: Exits: U, D Color: 6FFCBF Property: Exits: U, D Color: CB7B25 Property: Exits: U, D Color: 590A6F Property: Exits: U, D

Third Floor

 Color: 015555 Property:  (also Curzon) Exits: U, D Color: A761A5 Property: Exits: S, D Color: 94016A Property: Exits: U, D Color: B73E1D Property: Exits: N, S Color: C294AB Property: Exits: N, S Color: 998BA5 Property: Exits: N, S Color: D0FC21 Property: Exits: U, D Color: 9D8000 Property: Exits: N, D Color: F13840 Property: Exits: U, D

Fourth Floor

 Color: 15F507 Property:  with 1,175,265 Exits: U, D Color: 980001 Property: Exits: U, D Color: 35C7E2 Property: Exits: U, D Color: 3BADC8 Property: Exits: U, D

Fifth (Top) Floor

 Color: 11EEE1 Property:  with 1,438,983 Exits: S, D Color: B8CA42 Property: Exit: D Color: 6C7B00 Property: Exits: N, S Color: 287F43 Property: Exits: N, S Color: 71D4A4 Property:  to 10,925,915 Exits: N, S Color: A6B75B Property:  to 7,460,004 Exits: N, D Color: 50BAF4 Property: Exit: D

Once you have mapped the cube, you need to look at the shadows it would cast if a light were shone on the entire cube from the outside; first to the north, then to the east, then down, as clued by the flavortext. If the occupied rooms (colored maroon in the image below) cast a shadow and the unoccupied rooms do not, the shadows in those directions form the letters W, A and Y, respectively—and the answer to the puzzle is WAY.