How to solve a counterfeit coin problem?

You have 12 coins.1 is counterfeit and is lighter/heavier.Weight using balance weighing scale at most 4 times.Find the fake coin.
Solution:Split 12 coins into 4 groups.Group A(3 coins),B(3 coins),C(3 coins) and D(3 coins)
Weight A against B then C,D
If any balance,no fake coin in them.
If fake coin in D(example), and D and A(confirm all real) is unbalance,weight of fake coin is found.
-weigh any 2 coins, -if balance,third coin is counterfeit.
-if unbalance(e.g.lighter),then lighter coin is counterfeit.

How to solve a counterfeit coin problem?

Second problem:
You have 9 coins.1 is counterfeit and is heavier.Weight using balance weighing scale at most twice.Find the fake coin.
Solution:Split 9 coins into 3 groups.Group A(3 coins),B(3 coins) and C(3 coins).
1. weight A against B
a)If equal, Fake coin is in C.
-balance any 2 coins in C.
-if balance,third coin is fake
-if not balance,heavier coin is counterfeit.
b)If A or B is heavier,
-balance any 2 coins in heavier group.
-if balance,third coin is fake
-if not balance,heavier coin is counterfeit.

How to solve a counterfeit coin problom?

First problem:
You have 8 coins.1 is lighter and is fake.Weight using balance weighing scale at most twice.Find the fake coin.
Answer:Split 8 coins into 3 groups.Group A(3 coins),B(3 coins) and C(2 coins).
1. weight A against B
a)If equal, Fake coin is in C.
-balance the 2 coins in C and lighter coin is counterfeit.
b)If A or B is lighter,take 2 random coin from lighter group and weigh.
-balance,third coin is fake
-if lighter,then it is fake