According to this walkthrough:
These small items allow you to make your mark on the world by painting your own QR code on the walls. The messages you leave can be seen by any other player that is on your friends list, so make things interesting by hiding them in obscure areas. Of note, the paint bucket is the only item in the world that can pass through the purple puzzle gates.
So its purpose is to leave messages around for your friends to find.
There are also 2 achievements related to paint buckets:
Leave Your Mark: Unlocked after using a paint bucket to write a QR message for the first time.
Split Personality: Find the paintbuck in a level that has a recorder puzzle. Bring the paintbucket into the puzzle and start the recording. During this time, paint a QR code and then stop the recording. During playback, read the ghost QR code to obtain the achievement.
First of all. There are 30 stars at The Talos Principle as far as I know:
- There are 27 stars in the halls (A,B,C).
- There are 2 stars in the main world - the one with the tower.
- There is 1 star in the messenger places.
Then, second, please, don't think that knowing exact locations of each star will save you from exploring, often you still need to search for instruments (jammers, cubes, etc.) to get the star and it is essentially the same task. Even more, I think this game is about exploring. Even during the sigil puzzles you need to find a jammer, you need to find a position for connector, you need to find the sigil at the end, etc.
But anyway, in the following I give hints, which are aimed to reduce: a) process big area exploration, b) "looking for I don't know what", c) trials and errors when you trying to jump over a wall. After
you read all hints you will face a usual sigil-type puzzle. But be
carefull, some of this hints makes the rest of the star-puzzle
trivial.
When you read the hints note that:
- "Open world" is all world excluding the puzzle areas, sometimes it is inner
area (surrounded by puzzles), sometimes outer area. sometimes both.
- Right/left/front/back are given respectively to your position when you enter a puzzle or teleport to a world.
- "Hidden area" usually means an area completely surrounded by walls/fences. Often you can look inside if you jump.
The hints:
A1. Exact location:
Open world. Forward from the teleport to another puzzle area. Twice to the left to L-shaped puzzle area. It is in hidden area behind the fence to the right from the puzzle door.
Crucial secret:
There is a switch somewhere.
A2. Exact location:
Open world near the teleport. Behind a small wall in bushes to the right.
Crucial secret:
You can take a key outside from a puzzle.
Crucial secret2:
There is a door in the backwards direction from the teleport.
A3.1. Exact location:
"A Bit Tied Up". Right front corner, near the sigil.
A3.2. Exact location:
Open space, under ground near the teleport.
Crucial secret:
You can change time on the clock.
Crucial secret 2:
You will need to know what is HEX and what is ASCII.
A4. Exact location:
"Push it further", in the corner. Look for a narrow path near a wall.
Crucial secret:
There is a hidden connector on the top of a column (or a tree in older version of the game), that is nearby.
A5.1. Exact location:
"Friendly crossfire". Right front corner, where the black ball was.
A5.2. Exact location:
"Things to do with 2 boxes". Behind a wall at hidden area at back left corner.
Crucial secret:
There is a switch somewhere.
A6. Exact location:
Open world in hidden area to the left, between 2 puzzle areas.
Crucial secret:
There is an object in the open world.
A7. Exact location:
"Two Pesky Little Buzzers" Left back corner.
Crucial secret:
You can shoot laser though the violet barrier.
B1. Exact location:
"Something about a star". Right in the center.
B2. Exact location:
"The Tomb", behind right statue, you can see a path from outside.
Crucial secret:
You can come in from the inside of the puzzle.
B3. Exact location:
"Blown Away". In a hidden area right near of the door (to the left).
Crucial secret:
You can level up a bit one of the connectors at another puzzle.
B4.1. Exact location:
"Right angle", hidden area near the front wall, you can see fan.
B4.2. Exact location:
Open world, in the Sphinx.
Crucial secret:
There is a hidden source and receiver in the open world.
Crucial secret 2:
You can get one of the puzzle-connectors outside.
B5. Exact location:
In a middle of the open world, on the top of the spire.
Crucial secret:
You can take things outside from puzzles.
B7.1. Exact location:
Open world, at the top of the big arch structure in front of the big pyramid.
Crucial secret:
You can take things outside from puzzles.
B7.2. Exact location:
"Bouncing side by side". Front left corner.
C1. Exact location:
Open world, at ruins behind the puzzle areas, behind a barrier. You can see it easily if you explore the ruins a little bit.
Crucial secret:
You can find a connector in the open world.
C2. Exact location:
"A Ditch and a fence", at the left front corner, behind a wall.
C3. Exact location:
"Weathertop", at the left front corner, you can see it when you solve the puzzle.
C4.1. Exact location:
"Outbliette", behind the wall on the right.
C4.2. Exact location:
"Throne room", near the front wall.
C5.1. Exact location:
"Time flights", hidden area in front of the recorder.
C5.2. Exact location:
"Close and jammed", hidden area right after first barrier on the left.
C5.3. Exact location:
"Dumbwaiter", behind the front wall.
Crucial secret:
There are two solutions. At the simple you can use locations of 3 puzzles. At the hard you can take a connector inside to the open world.
C6. Exact location:
"Seven doors of the recording", at the right back corner, just follow the barriers.
C7. Exact location:
"Dead Man's switch", around a corner, near the front wall.
MW1. Exact location:
A small building behind the tower.
Crucial secret:
You can take connectors outside from the tower. But you can't solve it until you reach the hall C.
MW2. Exact location:
Tower's 3rd floor, on the left from the door.
Crucial secret:
There is a hole in the roof.
MP1. Exact location:
Messenger place at C hall, on a floating iland.
Crucial secret:
Everything you need is right in front of you. Go for it.
I don't put here exact solutions intentionally, I think a player should solve puzzles by himself. But if you really want full solutions you can see all of them, for example, in this 33 minutes guide on youtube.
Best Answer
I almost completed the game and now I have quite good estimation of number of puzzle it has. There are:
In total this would be 138 puzzles + 39 eggs, may be few more eggs.
P.S. New DLC: "Road to Gehenna" additionally provides 24 very good sigil-like puzzles + 16 star-like puzzles + several eggs.