Simple problems that can be carried around in your head

What are some problems that can easily be carried around in your head and require no need of ink and paper? Problems like irrationality of $\sqrt2$ or infinitude of primes or Gauss's first initiation with number theory.

I had and lost Mathematical Snippets. So I was hoping for something similar reference. Simple problems that I can take to shower, or during rush hour traffic, or for a walk, or waiting in a line in grocery or during a math lecture to entertain me. /nodisrespecttoprof. :]

Thank you so much in advance!

EDIT: I am sorry, I actually had Math-A-Day in mind. But being a fan of Theoni Pappas I also bought the other one I mentioned.

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Leonhard Euler, Lev Pontryagin and Louis Antoine might argue that virtually any mathematical problem can be carried around in your head without need of ink or paper. – Arthur Fischer Dec 3 '12 at 6:30
I retagged it with reference request. Just so there is a focus, an example would be the January 8th entry here. It is as beautiful, simple and pure as it can be. – Sniper Clown Dec 3 '12 at 6:36

3 Answers

Look at Lewis Carroll's "Pillow Problems and a Tangled Tale" (available on Amazon - published by Dover). These are problems he thought of to distract his mind from other things and were intended to be worked on while lying in bed. The problems concern themselves mainly with geometry, algebra, and probability.

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You may enjoy Q.E.D: Beauty in Mathematical Proof by Polster. Here it is on Amazon: http://www.amazon.com/Q-D-Beauty-Mathematical-Wooden/dp/0802714315

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Geometry is especially adapted to reflexion without paper.
For example if you want to see a non-flat morphisms of rings, nothing is simpler than visualizing the normalization of a non-normal algebraic variety (like, say, a cusp), which is never flat.
Another easy example is the immersion of a simple point into a double point.

A more elementary example: suppose you want an example of a sequence of continuous functions such that the pointwise limit of the integrals is not the integral of the limit : visualize a bump, the graph of a function $f\geq 0 \:$ with compact support , such that $\int _\mathbb R f (x)dx=1$.
If you take a sequence $(f_n)$ of such bumps with support of $f_n$ equal to $[-1,+1]$, it is visually clear that $$lim \int _\mathbb R f_n (x)dx=lim 1=1\neq \int _\mathbb R (lim f_n) (x)dx=\int _\mathbb R 0 \:dx=0$$

And if I may be excused a personal example illustrating math without pencil in geometry: the last two sections of this answer were conceived during a walk yesterday afternoon.

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