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• Once each year, thousands of logicians descend into the desert for Learning Man,

• a week-long event they attend to share their ideas,

• think through tough problems... and mostly to party.

• And at the center of that gathering is the world's most exclusive club,

• where under the full moon, the annual logician's rave takes place.

• The entry is guarded by the Demon of Reason,

• and the only way to get in is to solve one of his dastardly challenges.

• You're attending with 23 of your closest logician friends,

• but you got lost on the way to the rave and arrived late.

• They're already inside, so you must face down the demon alone.

• He poses you the following question:

• When your friends arrived, the demon put masks on their faces

• and forbade them from communicating in any way.

• No one at any point could see their own masks,

• but they stood in a circle where they could see everyone else's.

• The demon told the logicians that he distributed the masks in such a way

• that each person would eventually be able to figure out their mask's color

• using logic alone.

• Then, once every two minutes, he rang a bell.

• At that point, anyone who could come to him

• and tell him the color of their mask would be admitted.

• Here's what happened:

• Four logicians got in at the first bell.

• Some number of logicians, all in red masks, got in at the second bell.

• Nobody got in when the third bell rang.

• Logicians wearing at least two different colors got in at the fourth bell.

• All 23 of your friends played the game perfectly logically

• and eventually got inside.

• Your challenge, the demon explains,

• is to tell him how many people gained entry when the fifth bell rang.

• Can you get into the rave?

• Pause here to figure it out yourself.

• It's initially difficult to imagine how anyone could,

• using just logic and the colors they see on the other masks,

• deduce their own mask color.

• But even before the first bell, everyone will realize something critical.

• Let's imagine a single logician with a silver mask.

• When she looks around, she'd see multiple colors, but no silver.

• So she couldn't ever know that silver is an option,

• making it impossible for her to logically deduce that she must be silver.

• That contradicts rule five, so there must be at least two masks of each color.

• Now, let's think about what happens

• when there are exactly two people wearing the same color mask.

• Each of them sees only one mask of that color.

• But because they already know that it can't be the only one,

• they immediately know that their own mask is the other.

• This must be what happened before the first bell:

• two pairs of logicians each realized their own mask colors

• when they saw a unique color in the room.

• What happens if there are three people wearing the same color?

• Each of them—A, B and C— sees two people with that color.

• From A's perspective, B and C would be expected to behave the same way

• that the orange and purple pairs did, leaving at the first bell.

• When that doesn't happen,

• each of the three realizes that they are the third person with that color,

• and all three leave at the next bell.

• That was what the people with red masks did

• so there must have been three of them.

• We've now established a basis for inductive reasoning.

• Induction is where we can solve the simplest case,

• then find a pattern that will allow the same reasoning

• to apply to successively larger sets.

• The pattern here is that everyone will know what group they're in

• as soon as the previously sized group has the opportunity to leave.

• After the second bell, there were 16 people.

• No one left on the third bell,

• so everyone then knew there weren't any groups of four.

• Multiple groups, which must have been of five,

• left on the fourth bell.

• Three groups would leave a solitary mask wearer,

• which isn't possible, so it must've been two groups.

• And that leaves six logicians outside when the fifth bell rings:

• the answer to the demon's riddle.

• Nothing left to do but join your friends and dance.

Once each year, thousands of logicians descend into the desert for Learning Man,

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B1 TED-Ed bell mask demon rave silver

# Can you solve the logician’s rave riddle? - Edwin Meyer

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林宜悉 posted on 2021/02/23
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