Achievement "This is a Lock"
Location: Room 7

The solution to the lock is found in the lab at the end of Chapter 5.
(Thanks hexaae, Source[turingtest.fandom.com])

You will see two blue orbs like this, grab them both


Put them in:
Row 2 Spot 3
Row 3 Spot 5

A Polite Conversation
Location: Room 16

You will see a blue barrier that will get disabled from fast movement.
Instead hold the walk button (CTRL on PC) to proceed across without setting off the sensor

The Universe's Machine Epsilon
Location: Room 26

Grab all the boxes from the bottom area


Stack them on top of each other to block the camera

Proceed up the ladder and through the exit

Achievement "Algorithmic Composition"
Location: Room 36

Grab the box from to the left


Drop it on the edge of the steps so that it will roll downwards to the button
THEN hurry up and rush through the door to your right


DO NOT JUMP DOWN TO THE LOWER LEVEL!
Grab the blue orb from this spot and move it to the one on the right to open the door

Now you can jump down and proceed through the two doors

Achievement "The Chinese Room"
Location: Room 46

This one is rather challenging so I'm just going to describe to you the colors you require on each door. Also note there are two doors at the end (both of which you can access and both give you the achievement) for a total of four doors.

First Door = Two Greens
Second Door = One Green and One Purple


Bring One Purple and One Green with you, and grab the Blue (from the Fourth Door)
Third Door = Purple, Green and Blue


To access the Fourth Door (not required for achievement)
First Door = Two Purples
Second Door = Blue and Green

Bring Two Green Orbs with you
Quickly grab the Green from the Second Door while Blue is active
Place all three Greens on the Fourth Door

Achievement "Boolean Logic"
Location: Room 56

Another complicated one so I'll just describe what's required to pass each door. These are Logic Gates which means each symbol means something specific and tells you the order of which you need to open the door.


∧ = AND
∨ = OR
⊕ = BOTH OR ONE
⚑ = NOT

"Thinking Outside of the Box"
Location; Room 66

Optional 7: Chapter 7 Sector 66 (thanks to djcj)

Solution: take the box from the large room and use it to block the door. Take the orb that keeps the large room's door open. The box under the door should keep a gap open. Shoot the orb through the gap to the receiver of the large room.

PS: if you take the box, enter the other room and drop the box, your "gun" will be floating in the air.

Video Guide:
https://www.youtube.com/watch?v=UTPF6BWs5IM