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Problems and Puzzles

Many more coming soon…

The Four Lions

Four Lions are released into a quadratic arena with sidelength 100m. The lions start in the four corners of the arena, and every lion moves directly towards the next lion clockwise at a constant speed of 1m/s.

Will the lions every meet, and if they do, how long will it take?

Bonus: What about five lions released into a regular pentagon or n lions released into a regular n-gon?

Party Time

I am hosting a small dinner party for my friends. What is the least number of guests I have to invite, to guarantee that either three of them are mutual strangers, or three of them are mutual friends?

Note that any two guests are either strangers or friends, and their relationship is symmetric.

Bonus: How many guests do I need to invite to guarantee four (or five) mutual strangers or friends?

Crossing the Desert

A truck has to cross a desert which is 800 miles wide. An unlimited amount of fuel is available at one end of the desert, but the truck can only carry enough fuel to travel 500 miles (’one load’).

The truck is allowed to leave any amount of the fuel it is carrying by setting up refueling stations anywhere in the desert.

How many loads of fuel are required to cross the desert?

Bonus: Given unlimited fuel, is there a limit to the width of a desert the truck could cross?

Counterfeit Coins

You are given 12 identically looking coins and a two-pan balance scale. One of the coins is counterfeit and either lighter or heavier than the others.

How many weightings do you need to determine which one of the coins is counterfeit?

Bonus: In general, how many weightings are necessary to find a counterfeit coin among n coins? Think about error correction codes…

Problematic Peaks

Two Buddhist monks are located at the same elevation at opposite ends of a mountain range. They are trying to reach a monastery at the highest peak.

If the mountain range’s elevation never drops below their starting altitude, can the two monks reach the monastery while always staying at the same elevation?

Note: For certain periods, one of the monks may have to retrace their steps.