Suppose there are 4 cities, each city located at the vertex of a square having a side of unit length. What is the smallest road network that can connect all 4 cities?
The solution is explained very well in this video:
The following article also gives more insight on Steiner points:
The circumference of the Earth is approximately 40,000 kilometers, and someone has just made a metal band that circles the Earth, touching the ground at all locations.
You come along at night, as a practical joke, and add just 10 meters to its length (one hundredth of one kilometer !)
It is now one four-millionth longer, and sits magically just above the ground at all locations
How far … Continue reading the story "Band around the Earth - An Easy Puzzle"
You are trapped in a small room with four walls. Each wall has a button that is either in an ON or OFF setting, although you can never tell what the setting is. When you push a button, you switch its setting. If you can get all the buttons to have the same setting, you are set free. Each time step, you can use your … Continue reading the story "Button Trap Room"
There is an island upon which a tribe resides. The tribe consists of 1000 people, with various eye colours. Yet, their religion forbids them to know their own eye color, or even to discuss the topic; thus, each resident can and does see the eye colors of all other residents, but has no way of discovering his or her own there are no reflective surfaces. … Continue reading the story "The blue-eyed islanders puzzle"
Suppose the starting point of a particle undergoing Brownian motion in 2 dimensions is chosen uniformly at random on an imaginary circle C1. Suppose there is a solid circle C2 completely inside C1, not necessarily concentric. Show that the particle hits the boundary of C2 with the uniform distribution.
There is a lock which is an N by N grid of switches. Each switch can be in one of two states (on/off). The lock is unlocked if all the switches are on. The lock is built in such a way that, if you toggle some switch, all the switches in its row and its column toggle too
Give an algorithm which, given N and a … Continue reading the story "Locks and Switches"
A lion and a lion tamer are enclosed within a circular cage. If they move at the same speed but are both restricted by the cage, can the lion catch the lion tamer? (Represent the cage by a circle, and the lion and lion tamer as two point masses within it.)