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• As your country's top spy,

• you must infiltrate the headquarters of the evil syndicate,

• find the secret control panel,

• and deactivate their death ray.

• But all you have to go on is the following information

• picked up by your surveillance team.

• The headquarters is a massive pyramid with a single room at the top level,

• two rooms on the next,

• and so on.

• The control panel is hidden behind a painting

• on the highest floor that can satisfy the following conditions:

• Each room has exactly three doors to other rooms on that floor,

• except the control panel room,

• which connects to only one,

• there are no hallways,

• and you can ignore stairs.

• Unfortunately, you don't have a floor plan,

• and you'll only have enough time to search a single floor

• before the alarm system reactivates.

• Can you figure out which floor the control room is on?

• Pause now to solve the riddle yourself.

• To solve this problem, we need to visualize it.

• For starters, we know that on the correct floor

• there's one room,

• let's call it room A,

• with one door to the control panel room,

• plus one door to room B,

• and one to C.

• So there must be at least four rooms,

• which we can represent as circles,

• drawing lines between them for the doorways.

• But once we connect rooms B and C,

• there are no other connections possible,

• so the fourth floor down from the top is out.

• We know the control panel has to be as high up as possible,

• so let's make our way down the pyramid.

• The fifth highest floor doesn't work either.

• We can figure that out by drawing it,

• but to be sure we haven't missed any possibilities,

• here's another way.

• Every door corresponds to a line in our graph

• that makes two rooms into neighbors.

• So in the end, there have to be an even number of neighbors

• no matter how many connections we make.

• On the fifth highest floor, to fulfill our starting conditions,

• we'd need four rooms with three neighbors each,

• plus the control panel room with one neighbor,

• which makes 13 total neighbors.

• Since that's an odd number, it's not possible,

• and, in fact, this also rules out every floor that has an odd number of rooms.

• So let's go one more floor down.

• When we draw out the rooms,

• low and behold, we can find an arrangement that works like this.

• Incidentally, the study of such visual models

• that show the connections and relationships between different objects

• is known as graph theory.

• In a basic graph, the circles representing the objects are known as nodes,

• while the connecting lines are called edges.

• Researchers studying such graphs ask questions like,

• "How far is this node from that one?"

• "How many edges does the most popular node have?"

• "Is there a route between these two nodes, and if so, how long is it?"

• Graphs like this are often used to map communication networks,

• but they can represent almost any kind of network,

• from transport connections within a city

• and social relationships among people,

• to chemical interactions between proteins

• or the spread of an epidemic through different locations.

• So, armed with these techniques, back to the pyramid.

• You avoid the guards and security cameras,

• infiltrate the sixth floor from the top,

• find the hidden panel,

• pull some conspicuous levers,

• and send the death ray crashing into the ocean.

• Now, time to solve the mystery

• of why your surveillance team always gives you cryptic information.

• Hi everybody.

• If you liked this riddle, try solving these two.