The 100 Coin Puzzle: A Classic Brain Teaser

The 100 coin puzzle is a classic brain teaser that has been around for decades. It’s a simple puzzle to understand, but it can be surprisingly tricky to solve. In this blog post, we’ll explore the puzzle, its solution, and the logic behind it.

The Puzzle

Imagine you have 100 coins. Some of them are heads up, and some are tails up. You can’t see the coins, but you can flip them over. Your goal is to determine the number of heads-up coins after a specific sequence of flips.

Here’s the catch: you can only flip the coins in groups. You can flip over 1 coin, 2 coins, 3 coins, and so on, up to 100 coins. After each flip, you’re allowed to count the number of heads-up coins.

The Solution

The key to solving the 100 coin puzzle lies in the fact that you can use the number of heads-up coins after each flip to deduce the number of heads-up coins in the original arrangement.

Here’s how it works:

1. Flip all 100 coins. Count the number of heads-up coins. Let’s say you get 50 heads-up coins.
2. Flip every other coin (2, 4, 6, 8…). Count the number of heads-up coins again. Let’s say you get 25 heads-up coins.
3. Flip every third coin (3, 6, 9, 12…). Count the number of heads-up coins. Let’s say you get 16 heads-up coins.
4. Continue this process, flipping every fourth coin, every fifth coin, and so on, until you flip every 100th coin.

The number of heads-up coins you count after each flip will give you the clues you need to determine the original number of heads-up coins.

The Logic

The logic behind the solution is based on the concept of parity. Parity refers to whether a number is even or odd.

When you flip every other coin, you’re changing the parity of the number of heads-up coins. If the original number of heads-up coins was even, it will become odd, and vice versa.

Similarly, when you flip every third coin, you’re changing the parity of the number of heads-up coins that are multiples of 3. And so on.

By carefully observing the parity changes after each flip, you can deduce the original number of heads-up coins.

Example

Let’s say you get the following counts after each flip:

Flip	Heads-up Coins
1	50
2	25
3	16
4	12
5	10
6	9
7	8
8	7
9	6
10	6

Notice that the number of heads-up coins changes parity after each flip. This means that the original number of heads-up coins must be odd.

To find the exact number, we can look at the differences between the counts after each flip. For example, the difference between the counts after flip 1 and flip 2 is 25. This means that there were 25 coins that were flipped twice (once in flip 1 and once in flip 2).

Continuing this pattern, we can deduce that the original number of heads-up coins was 37.

Conclusion

The 100 coin puzzle is a fun and challenging brain teaser that demonstrates the power of logic and deduction. By understanding the concept of parity and carefully analyzing the number of heads-up coins after each flip, you can solve this classic puzzle and impress your friends with your problem-solving skills.