The Dark Coin Riddle: Can You Solve It?

Imagine yourself as a daring adventurer who has finally located a legendary dungeon filled with ancient coins. The wizard who owns the castle is willing to let you have the treasure, but there's a catch. You must solve his intricate puzzle to escape with your newfound wealth.

The Wizard's Challenge

The wizard presents you with a collection of coins. Each coin has two faces: one side bears a silver scorpion crest, and the other a gold one. Your task is to divide the coins into two piles, ensuring that each pile has the same number of coins facing silver side up.

Just as you're about to begin, the torches in the dungeon extinguish, plunging you into complete darkness. You can no longer distinguish between the silver and gold sides of the coins by sight. However, you remember that there were exactly 20 coins facing silver side up before the lights went out. How can you solve the puzzle and escape the dungeon?

The Solution

The solution is surprisingly simple:

1. In the darkness, select any 20 coins from the pile.
2. Flip each of these 20 coins over.

That's it! This seemingly simple action guarantees that both piles will have the same number of coins facing silver side up.

Why Does It Work?

The key to understanding this solution lies in the concept of complementary events. Let's break it down:

• You start with a total of 20 silver-facing coins.
• When you select 20 coins at random, some of them will be silver-facing, and some will be gold-facing.
• The number of silver-facing coins you pick is unknown, but let's say you pick 7 silver-facing coins. This means the remaining 13 coins in your selection are gold-facing.
• In the original pile, there are now 13 silver-facing coins left (20 total - 7 picked).
• When you flip all 20 coins in your selected pile, the 7 silver-facing coins become gold-facing, and the 13 gold-facing coins become silver-facing.
• Now, both piles have 13 silver-facing coins!

This works regardless of how many silver-facing coins you initially pick. The act of flipping the selected coins ensures that the number of silver-facing coins in both piles will always be equal.

The Algebra Behind the Magic

We can express this mathematically:

• Let x be the number of silver-facing coins you move to the new pile.
• The number of silver-facing coins remaining in the original pile is 20 - x.
• The number of gold-facing coins in the new pile is also 20 - x.
• When you flip the coins in the new pile, the 20 - x gold-facing coins become silver-facing.

Therefore, both piles now have 20 - x silver-facing coins.

Escape and Adventure

With the puzzle solved, the wizard releases you, and you escape the dungeon with your hard-earned treasure. As you stand at the crossroads, you flip one of your Stygian coins to determine the path to your next adventure.

This coin riddle demonstrates the power of logical thinking and problem-solving. By understanding the underlying principles, you can overcome seemingly impossible challenges and emerge victorious, just like our intrepid adventurer.