## 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.

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 a ruler and compass can only solve quadratic equations whereas angle trisection can be proven to involve solving cubic equations. More about origami can be found here: http://plus.maths.org/content/power-origami

Thanks to Rakesh Misra for bringing this to my attention.

## 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 has it risen ... could a flea, a rabbit or even a man squeeze underneath it?

## 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 hands to either press two buttons simultaneously, or just one button. As soon as this occurs, the room spins around violently, disorienting you so that you can no longer tell which side is which. How can you escape?

## 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. If a tribesperson does discover his or her own eye color, then their religion compels them to commit ritual suicide at noon the following day in the village square for all to witness. All the tribespeople are highly logical and devout, and they all know that each other is also highly logical and devout and they all know that they all know that each other is highly logical and devout, and so forth.[Added, Feb 15: for the purposes of this logic puzzle, "highly logical" means that any conclusion that can logically deduced from the information and observations available to an islander, will automatically be known to that islander.]Of the 1000 islanders, it turns out that 100 of them have blue eyes and 900 of them have brown eyes, although the islanders are not initially aware of these statistics each of them can of course only see 999 of the 1000 tribespeople.One day, a blue-eyed foreigner visits to the island and wins the complete trust of the tribe.One evening, he addresses the entire tribe to thank them for their hospitality.However, not knowing the customs, the foreigner makes the mistake of mentioning eye color in his address, remarking “how unusual it is to see another blue-eyed person like myself in this region of the world”.What effect, if anything, does this faux pas have on the tribe?

## 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 configuration of the N^2 switches, will tell you whether the lock can be unlocked by a sequence of switch toggles

Note 1: Can be done in O(N^2) time and O(1) space.

Note 2: You just need to tell if a sequence which unlocks the lock exists (and not the actual sequence)

Update (Dec 20, 2010):

## 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.)

## Coin Puzzle: Predict the Other's Coin

Posted on: January 1st, 2013 by

Assume the following 3-player game consisting of several rounds. Players A and B build a team, they have one fair coin each, and may initially talk to each other. Before starting the first round, however, no more communication between them is allowed until the end of the game. (Imagine they are separated in different places without any communication infrastructure.)

A round of the game consists of the following steps:

1. The team gives one dollar to player C.
2. Both A and B toss their coins independently.
3. Both A and B try to predict the other's coin by telling the guess to C. (No communication: A does not know the outcome of B's coin toss, and vice versa, nor the guess).
4. If C verifies that both A and B guess the other's coin correctly, then C has to give 3 dollars back to the team.

Should C play this game?

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