Problem Setup

• Annual Learning Man desert gathering for logicians
• World's most exclusive club rave guarded by Demon of Reason
• Entry requires solving logic challenge
• 23 friends already inside, protagonist arrived late and must solve alone

The Mask Challenge Rules

• Demon placed masks on 23 logicians' faces
• No communication allowed between participants
• No one can see their own mask color
• Standing in circle where everyone sees others' masks
• Demon distributed masks so each person can deduce their color using logic alone
• Bell rings every 2 minutes
• Anyone knowing their mask color can tell demon and gain entry

Game Results Timeline

• First bell: 4 logicians gained entry
• Second bell: Some number of logicians in red masks gained entry
• Third bell: Nobody gained entry
• Fourth bell: Logicians wearing at least 2 different colors gained entry
• All 23 friends played perfectly logically and eventually got inside

Logic Solution Process

• Key insight: Must be at least 2 masks of each color
  • Single mask wearer couldn't deduce their color (would see no matching masks)
  • Contradicts rule that everyone can figure out their color
• Two-mask color groups:
  • Each person sees only 1 mask of their color
  • Knowing minimum is 2, they deduce they're the second
  • 4 people left at first bell = 2 pairs of different colors
• Three-mask color groups:
  • Each sees 2 others with same color
  • Expects those 2 to leave at first bell (like pairs did)
  • When they don't leave, all 3 realize they're the third person
  • Red mask group left at second bell = 3 people

Inductive Reasoning Pattern

• Groups of size N leave when groups of size N-1 had opportunity to leave
• After second bell: 16 people remaining (23 - 4 - 3 = 16)
• Third bell: No one left = no groups of 4
• Fourth bell: Multiple groups left = groups of 5
• Cannot be 3 groups of 5 (would leave 1 solitary person, impossible)
• Must be 2 groups of 5 = 10 people left at fourth bell
• Remaining: 6 logicians for fifth bell

Final Answer

• 6 people gained entry when fifth bell rang