Among n identical-looking coins, one is fake and lighter than the others. The algorithm uses a balance scale to compare sets of coins and determine which set contains the fake in O(log n) comparisons. It works by repeatedly splitting the coins into two piles, comparing the piles on the scale, and discarding the heavier pile until the fake coin is found. If n is odd, one coin is kept aside and the rest split into two piles. If even, all n coins are split into two piles.

1. Fake-Coin Problem
Decrease and Conquer Technique

2. Definition
Best illustrates the decrease-by-a-constant-factor
strategy
Among n identical-looking coins, one is fake.
Among n identical-looking coins, one is fake.

3. Definition
Among n identical-looking coins, one is fake.
Problem: to design an efficient algorithm for
detecting the fake coin
detecting the fake coin
Assumption: fake coin is known to be lighter than the
genuine one

4. How to compare?
With a balance scale,
any two sets of coins
any two sets of coins
can be compared

5. How to compare?
 Based on tipping to the left, to
the right, or staying even, the
Balance scale will tell whether
Balance scale will tell whether
the sets weigh the same or
which of the sets is heavier than
the other but not by how much.

6. How to compare?
Staying even,
No Fake Coin
No Fake Coin

7. How to compare?
Tipping to the left,
Fake Coin -
Right
Fake Coin -
Right

8. How to compare?
Tipping to the right,
Fake Coin -
Left
Fake Coin -
Left

9. Basic Idea
 If n is odd (n−1) is even
 Keep one extra coin aside
 Split (n−1) coins into two piles and put each half on the balance scale
 If the piles weigh the same, the coin put aside must be fake
 If the balance is not even, then we choose the lightest pile of coins to
be the one containing the fake.
 Proceed in the same manner with the lighter pile - one with the
fake coin

10. Basic Idea
 If n is even
 Split n coins into two piles and put each half on the balance scale
 If the piles weigh the same, No fake coin
 If the balance is not even, then we choose the lightest pile of
coins to be the one containing the fake.
 Proceed in the same manner with the lighter pile - one with
the fake coin