## Five Points on a Sphere

Posted on: January 8th, 2013 by
Given any five points on a sphere, show that there exists a closed hemisphere containing four of them.

## Trisection of an Angle via Origami

Posted on: January 8th, 2013 by
Angle trisection is another of the classical problems that cannot be solved using a compass and unmarked ruler but can be solved using origami. This construction is due to Hisashi Abe. via Mathematics of paper folding - Wikipedia, the free encyclopedia. To prove that angle trisection cannot be solved using an unmarked ruler and a compass, one must invoke Galois theory. What's interesting however is that Continue reading the story "Trisection of an Angle via Origami"

## Steiner points

Posted on: January 8th, 2013 by
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: http://www.unige.ch/~gander/Preprints/BDM56-GanderE.pdf

## Band around the Earth - An Easy Puzzle

Posted on: January 6th, 2013 by
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"

## Button Trap Room

Posted on: January 4th, 2013 by
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"

## The blue-eyed islanders puzzle

Posted on: January 2nd, 2013 by
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"

## Brownian Motion

Posted on: January 1st, 2013 by

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.

via Puzzles.

## Locks and Switches

Posted on: January 1st, 2013 by
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"

## Sphagetti Breakfast

Posted on: January 1st, 2013 by

A bowl of spaghetti contains n strands. Thor picks two ends at random and joins them together. He does this until no ends remain.

What is the

a) expected number of spaghetti loops in the bowl?

b) expected average length of the loops? (in strands)

c) expected number of k-hoops? ( a k-hoop is a loop made from k strands)

## Lion in a Circular Cage Puzzle

Posted on: January 1st, 2013 by
1

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.)

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