Two soldiers, William and Ethan, are assigned to guard a bridge, which connects the West and East sides of the Great Kingdom. Each soldier is ordered to stand at an end of the bridge to make sure no criminals cross.
On one side of the bridge stands William, watching over the West side of the kingdom, and making sure no shady characters try to cross the bridge.
Ethan stands on the other side of the bridge, facing the East side of the kingdom with his rifle at the ready in case any criminals try to pass across.
"Any criminals today?" William asks.
Ethan rolls his eyes. "What do you think?" he asks.
"You roll your eyes too much," William says.
How could William tell that Ethan was rolling his eyes?
William is on the east side of the bridge, facing the West side of the kingdom, while Ethan is on the west side of the bridge, facing the East side of the kingdom. So William and Ethan are facing each other, and can see each other's faces.
A farmer challenges an engineer, a physicist, and a mathematician to fence off the largest amount of area using the least amount of fence. The engineer made his fence in a circle and said it was the most efficient. The physicist made a long line and said that the length was infinite. Then he said that fencing half of the Earth was the best. The mathematician laughed at the others and with his design, beat the others. What did he do?
The mathematician made a small circular fence around himself and declared himself to be on the outside.
100 men are in a room, each wearing either a white or black hat. Nobody knows the color of his own hat, although everyone can see everyone else's hat. The men are not allowed to communicate with each other at all (and thus nobody will ever be able to figure out the color of his own hat).
The men need to line up against the wall such that all the men with black hats are next to each other, and all the men with white hats are next to each other. How can they do this without communicating? You can assume they came up with a shared strategy before coming into the room.
The men go to stand agains the wall one at a time. If a man goes to stand against the wall and all of the men already against the wall have the same color hat, then he just goes and stands at either end of the line. However, if a man goes to stand against the wall and there are men with both black and white hats already against the wall, he goes and stands between the two men with different colored hats. This will maintain the state that the line contains men with one colored hats on one side, and men with the other colored hats on the other side, and when the last man goes and stands against the wall, we'll still have the desired outcome.