You are blindfolded, and 10 coins are placed on a table in front of you. You can touch and move the coins, but you cannot determine whether a coin is heads or tails by feeling it. You are told that exactly 5 coins are heads up and 5 coins are tails up, but you do not know which ones.
Your task is to divide the coins into two groups such that both groups have the same number of heads. You can flip the coins any number of times.
Check if you were right - full answer with solution below.
Solution:
Divide the 10 coins into two groups of 5 coins each. Then, choose any one of the groups and flip all the coins in that group.
The key idea is based on the total number of heads, which is given as 5.
When you divide the coins into two equal groups, suppose the first group contains x heads. Since there are only 5 heads in total, the second group must contain (5 − x) heads.
Now, flip all the coins in the first group. After flipping:
- The X heads in the first group become tails.
- The remaining coins in that group (which were tails) become heads.
So, the first group now has (5 − x) heads, and the second group was not changed, so it already has (5 − x) heads.
Example: Consider one possible arrangement:
Pile 1: H H T T T
Pile 2: H H H T TNow flip all coins in Pile 1:
Pile 1 becomes: T T H H H
Pile 2 remains: H H H T TNow both piles have 3 heads each.
Both groups end up with the same number of heads, even without knowing the position of any individual coin.