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(13/06) Cecil Artavion : Start from the designated area. From there follow the directions on the arrows. I have the final answers for each circled. The only one that is unable to be determined to be heavier or lighter is number 20 shuriken. If you are unable to read the attachment tell what is wrong with either how I sent it or the document itself and I will try and fix it or tell you verbally the answers.
Answer
Shika: Congrats, you're the 1st to get it right again. Though there is a much simpler way to explain without drawing diagrams... Your efforts are commendable though. I would be too lazy to drae diagrams...

(13/06) Warbringer : I am really sorry for this, but I forgot something in the answer I sent previously, and I thought I'd correct that. Although it looks obvious, I just wanted to make sure that everything is perfectly clear.
In Case 1.1, I treated the case where 1,5,6 and 2,7,8 don't balance, and 1,5,6 is heavier. I forgot about the case where 2,7,8 is heavier. It's actually the same logic : the fake shuriken would be 7, 8, or 1. If it is either 7 or 8, it would be heavier than the others, else it would be lighter. We weight 7 and 8 (Move 4). If they balance, the fake shuriken is 1, else it is the heavier of 7 and 8.
Shika: I get what you mean, you're right. This is getting really surreal as once again, Cecil beat you to it. There is however a simpler way also involving piles of 6, 6 & 8. I'll reveal the solution soon after a nap.

(13/06) Warbringer : Awright, this will be pretty long. I'll use numbers for the shurikens to avoid confusions, and copypaste to avoid trouble.
First, We make 3 piles : P1 (1,2,3,4,5,6), P2 (7,8,9,10,11,12), P3
(13,14,15,16,17,18,19,20).
We use the scale to check P1 and P2 (Move 1) :
Case 1: They don't balance, the fake shuriken is either in P1 or P2.
Case 2: They balance, the fake shuriken is in P3.
Begin Case 1
We make 3 piles : C1P1 (1,2,3,4), C1P2 (5,6,7,8), C1P3 (9,10,11,12).
We use the scale to check C1P1 and C1P2 (Move 2) :
Case 1.1: They don't balance, the fake shuriken is either in C1P1 or C1P2.
Case 1.2: They balance, the fake shuriken is in C1P3.
Begin Case 1.1
If C1P2 is heavier, we weight 1,5,6 and 2,7,8 (Move 3).
If they balance, then the fake shuriken is either 3 or 4. We weigh 4 and 9 (Move 4). If they balance then the fake shuriken is 3, otherwise it is 4. If they don't balance, and 1,5,6 is heavier, then the fake shuriken is 5, 6, or 2. If it is either 5 or 6, it would be heavier than the others, else it would be lighter. We weight 5 and 6 (Move 4). If they balance, the fake shuriken is 2, else it is the heavier of 5 and 6.
If C1P1 is heavier, we weight 5,1,2 and 6,3,4.
We can then determin which one is the fake using the same reasoning we used above. I'm not going to write it again, it's too troublesome.
End Case 1.1
Begin Case 1.2
We weight 9,10 and 1,2 (Move 3). If they balance, the fake shuriken is either 11 or 12. We weight 11 and 1 (Move 4). If they balance, the fake shuriken is 12, else it it 11. If 9,10 and 1,2 don't balance, the fake shuriken is either 9 or 10. We weight 9 and 1 (Move 4). If they balance, the fake shuriken is 10, else it it 9.
End Case 1.2
Begin Case 2
We make 2 piles : C2P1 (13,14,15,16), C2P2 (17,18,19,20).
We use the scale to check C2P1 and C2P2 (Move 2).
We are in front of a situation similar to Case 1.1. We follow the same logic to reveal the fake shuriken. Again, it would be too troublesome to write it once more.
End Case 2
This concludes the problem. We could get the fake one in always no more than 4 moves. I hope I didn't screw up at some place, that sure was tough.
Shika: will comment later.

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