Version 3 ( Case 2 and 3)

Version 3 ( Case 2)	Use A, B ,C and D to represent groups of three coins	Rational
First weighing	Place A and B on either side of the scale.	If the scale does not balance, try by replacing A with C.
Second weighing	Replace A with C	If it is also unbalanced, counterfeit is in B.
Third weighing	Place any 2 coins, each on either side of the scale	If it does not balance, replace 1 coin with the last coin.
Fourth weighing	Replace 1 coin with the last coin.	1) If the scale balances, counterfeit is the coin that was replaced. 2)If the scale becomes unbalanced(meaning that the other two are equal in weight), counterfeit coin will have a different weight compared to the other 2.

Version 3 ( Case 3)	Use A, B ,C and D to represent groups of three coins	Rationale
First weighing	Place A and B on either side of the scale.	If the scale does not balance, try by replacing B with C.
Second weighing	Replace B with C	If it is also unbalanced, counterfeit is in A.
Third weighing	Place any 2 coins, each on either side of the scale	If it does not balance, replace 1 coin with the last coin.
Fourth weighing	Replace 1 coin with the last coin.	1) If the scale balances, counterfeit is the coin that was replaced. 2)If the scale becomes unbalanced(meaning that the other two are equal in weight), counterfeit coin will have a different weight compared to the other 2.

This is my investigative work

Done by: Jin Yuan 3E1 (41)

Version 2 case 2 and version 3 case 1

Version 2 (Case 2)		Rationale
First weighing	Put any 3 coins on either side of the scale.	If the scale is not balanced, the counterfeit is in the heavier side.
Second weighing	Place any 2 coins from the heavier group , each on either side of the scale.	1) If the scale balances, the counterfeit is the last coin. 2) If they do not balance, the heavier coin is the counterfeit.

Version 3 ( Case 1)	Use A, B ,C and D to represent groups of three coins	Rationale
First weighing	Place A and B on either side of the scale.	If the scale balances, try by replacing B with C.
Second weighing	Replace B with C	If it also balances, counterfeit is in D
Third weighing	Place any 2 coins, each on either side of the scale	If it does not balance, replace 1 coin with the last coin.
Fourth weighing	Replace 1 coin with the last coin.	1) If the scale balances, counterfeit is the coin that was replaced. 2)If the scale becomes unbalanced(meaning that the other two are equal in weight), counterfeit coin will have a different weight compared to the other 2.

Version 2 case 1

Version 2 (Case 1)		Rationale
First weighing	Put any 3 coins on either side of the scale.	If the scale balances, the counterfeit is in the 3 remaining coins.
Second weighing	Place 2 of the remaining coins, each on either side of the scale.	1) If the scale balances, the counterfeit is the last coin 2)If the scale is unbalanced, the heavier side contains the counterfeit coin

Investigative work ( version 1)

Version 1 (Case 1)		Rationale
First weighing	Put any 3 coins on either side of the scale, leaving you with 2 coins.	If the scale balances, the counterfeit is in one of the two remaining coins.
Second weighing	Place the remaining 2 coins, each on either side of the scale.	The lighter coin is the counterfeit

Version 1 (Case 2)		Rationale
First weighing	Place any three coins on either side of the scale.	If the scale is unbalanced, the counterfeit is in the lighter group of three coins.
Second weighing	Place any 2 coins from the lighter group, each on either side of the scale.	1) If the scale balances, the counterfeit is the coin that was not weighed. 2) If the scale is not balanced, the lighter coin is counterfeit.