Finding 'x' and other curiosities

Students should be exposed to various perspectives on a topic, and especially ones that draw on Bloom’s Taxonomy. Here is a list of great questions (whether you are involved in education or not) that you, and your students can benefit from:

A. What does X mean? (Definition)
B. What are the various features of X? (Description)
C. What are the component parts of X? (Simple Analysis)
D. How is X solved or done? (Process Analysis)
E. How should X be solved or done? (Directional Analysis)
F. What is the essential function of X? (Functional Analysis)
G. What are the causes of X? (Causal Analysis)
H. What are the consequences of X? (Causal Analysis)
I. What are the types of X? (Classification)
J. How is X like, or unlike, Y? (Comparison)
K. What is the present status of X? (Comparison)
L. What is the significance of X? (Interpretation)
M. What are the facts about X? (Reportage)
N. How does X occur? (Reframing)
O. What kind of thing is X? (Attribution/Profile)
P. What is personal experience with X? (Reflection)
Q. What is my memory of X? (Reminiscence)
R. What is the importance of X? (Evaluation)
S. What are the essential major points or features of X? (Summary)
T. What case can be made for or against X? (Persuasion)

(source: http://owl.english.purdue.edu/owl/owlprint/673/)

Four Colour Theorem

Imagine any simple map, where there are many countries bordering each other, every which way. It takes at most four colours to colour the map such that no two adjacent countries have the same colour.

(source: http://upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Four_Colour_Map_Example.svg/300px-Four_Colour_Map_Example.svg.png)

Ham Sandwich Theorem

Take a three ingredient sandwich (i.e. bread, meat, and cheese), constructed in any manner. You can always make exactly one straight cut and divide all of the ingredients exactly in half.

(source: http://www.flickr.com/photos/mhaithaca/2469421935/sizes/l/in/photostream/)

Hairy Ball Theorem

You can’t brush a hairy ball and not create at least one whorl.

(source: http://upload.wikimedia.org/wikipedia/commons/e/ec/Hairy_ball.png)

Pick’s Theorem

Take a regular square grid of points and create a simple polygon by connecting any number of points. Count the number of points that lie exactly on the lines ( $p$ ) and the number of points that are contained by the shape ( $i$ ). Amazingly, no matter what shape you have, the area is given by

$A = \frac{p}{2}+i-1$

(sourceL http://upload.wikimedia.org/wikipedia/commons/thumb/8/84/Pick_theorem.svg/593px-Pick_theorem.svg.png)

British Flag Theorem

Very similar to the Pythagorean Theorem (which is also an amazingly simple and powerful theorem). Given any point inside a rectangle, and lines connecting the corners to this point, the sum of the squares of one set of opposing lines is equal to the sum of the squares of the other opposing lines. In less words:

$AP^2 + PC^2 = BP^2 + DP^2$

(source: http://www.artofproblemsolving.com/Forum/latexrender/pictures/f/1/4/f148d023e951fe3d398337ca88ad82cb7cd372de.png)

Futurama Theorem

Suppose you would like to swap brains with a friend (why wouldn’t you want to?). After enjoying the body of someone else, you want to put your brain back into your own body, but can’t because the immune system is now resistant to your brain. But, you could swap your brain with a different friend, and then put your brain back into your own body. If you played this with a group of people, how many people would you need in addition to swap and return the brains in this manner?

(source: http://pool.theinfosphere.org/images/4/4e/Prisoner_of_Benda_Theorem_on_Chalkboard.png)

Euler’s Formula (for Euler’s characteristic)

Because of his prolific nature, Euler’s formula refers to several theorems. This particular one relates the number of vertices, edges and faces of polyhedron. Here it is:

$V - E + F = 2$

For any convex polyhedra, $V$ is the number of vertices, $E$ is the number of edges, and $F$ is the number of faces. Very subtle, but wonderful result.

(source: http://upload.wikimedia.org/wikipedia/en/5/57/Uniform_polyhedron-43-s012.png)

Infinite Monkey Theorem

Yeah, there’s a theorem for that. The infinite monkey theorem states that a monkey typing on a typewriter will almost surely type all of the classic literature masterpieces (such as the works of William Shakespeare). The probability is astoundingly low. About 1 in $4.4 \times 10^{360783}$ .

(source: http://upload.wikimedia.org/wikipedia/commons/f/f1/Monkey-typing.jpg)

Pigeonhole principle

Another seemingly obvious, but subtly interesting theorem. It simply states, that if you have $n$ objects and $n-1$ containers to place these objects into in any manner, then there will be at least one container that will have two (2) or more objects in. As a simple example, if there are more than 365 people in a room, then it is guaranteed that there are at least two people with the exact same birthday (in fact, the number of people that will most likely have the same birthday can be much, much, lower. See the Birthday Problem).

(source: http://upload.wikimedia.org/wikipedia/commons/thumb/5/5c/TooManyPigeons.jpg/740px-TooManyPigeons.jpg)

Pizza theorem

Pick a point on the interior of a pizza, and choose how many piece of pizza you would like: 2, 4, 8, 16, etc. (this is $n$ ). Now, this means that you will make $n/2 - 1$ cuts to divide the pizza, and each cut will be $360/n$ degrees at the centre. Now, if you choose every other piece on the pizza, it will equal the amount of pizza leftover (no matter where $p$ is!)

(source: http://upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Pizza_theorem_1.svg/450px-Pizza_theorem_1.svg.png)

Road coloring problem

“In the real world, this phenomenon would be as if you called a friend to ask for directions to his house, and he gave you a set of directions that worked no matter where you started from.” (wikipedia.org)

Technically, “every finite strongly-connected aperiodic directed graph of uniform out-degree has a synchronizing coloring.” (wikipedia.org)

(sourceL http://upload.wikimedia.org/wikipedia/commons/thumb/6/69/Road_coloring_conjecture.svg/601px-Road_coloring_conjecture.svg.png)

Theorem on friends and strangers

From wikipedia.org:

“Suppose a party has six people. Consider any two of them. They might be meeting for the first time—in which case we will call them mutual strangers; or they might have met before—in which case we will call them mutual acquaintances. The theorem says:

In any party of six people either at least three of them are (pairwise) mutual strangers or at least three of them are (pairwise) mutual acquaintances.“

(source: http://upload.wikimedia.org/wikipedia/commons/e/ef/Friends_strangers_graph.gif)

Finding 'x' and other curiosities

Taxonomy of Questions

Math theorems that sound like kids should know about them