Crack the Code: Can You Solve the Demon Dance Party Riddle?

Imagine a gathering of the world's sharpest minds, descending into the desert for an annual meeting of logicians. At the heart of this event lies an exclusive club, hosting a legendary rave under the full moon. But entry isn't easy; it's guarded by the Demon of Reason, who challenges each hopeful entrant with a perplexing riddle. Are you clever enough to join the party?

The Demon's Challenge

You arrive late, finding your 23 friends already inside. To gain entry, you must face the demon alone and solve his intricate puzzle:

Your friends were each given a mask, and forbidden from communicating. They stood in a circle, each able to see the masks of others, but not their own. The demon declared that, using logic alone, each person could eventually deduce the color of their mask. Every two minutes, a bell would ring, signaling those who knew their mask color to step forward and be admitted.

Here's what transpired:

• At the first bell, four logicians entered.
• At the second bell, a number of logicians wearing red masks entered.
• The third bell saw no one enter.
• At the fourth bell, logicians wearing at least two different colors entered.
• All 23 of your friends, employing perfect logic, eventually made it inside.

The demon's question: How many people entered when the fifth bell rang?

Unraveling the Logic

At first glance, deducing your mask color based solely on what others are wearing seems impossible. However, a crucial realization dawns on everyone even before the first bell.

Consider a logician with a unique mask color. Seeing no one else with that color, they could never logically conclude that they possess it. This violates the rules, implying that there must be at least two masks of each color.

Now, imagine exactly two people wearing the same color mask. Each sees only one other mask of that color. Knowing they can't be the only one, they instantly realize their own mask color.

This explains the first bell: two pairs of logicians each recognized their mask colors upon seeing a unique color in the room.

Inductive Reasoning

What if there are three people with the same color mask? Each sees two others with that color. From one's perspective, the other two should behave like the pairs, leaving at the first bell. When that doesn't happen, each of the three realizes they are the third person with that color, and all three leave at the next bell.

This is what occurred with the red masks, meaning there were three of them.

We've now established a basis for inductive reasoning. By solving the simplest case and identifying a pattern, we can apply the same logic to successively larger sets.

The pattern: everyone will know their group as soon as the previously sized group has the opportunity to leave.

Solving the Puzzle

After the second bell, 16 people remained. The lack of movement at the third bell indicated that there were no groups of four.

Multiple groups, necessarily of five, left at the fourth bell. Three such groups would leave a single mask wearer, which is impossible. Therefore, there were two groups of five.

This leaves six logicians outside when the fifth bell rings. That's the answer to the demon's riddle!

With the puzzle solved, there's nothing left to do but join your friends and dance the night away.