# Pencil and paper

Can you see the light? That is, can you see the patterns in light bulbs that are left on, after repeatedly following a simple rule to switch them on and off?

A twist on the game of Dots and Boxes – but instead of capturing as many squares as possible, you and your opponent are going after gold doubloons!

A simple game of strategy, where each player attempts to claim spaces to form an unbroken chain between opposite sides of the board. (See attached image for an example.)

List all the divisors of *N*, including 1 and *N*, and take turns crossing out a divisor (and all of its divisors that are not yet crossed out). Win by forcing your opponent to cross out the final number on the list. Can you come up with a winning strategy?

Imagine building a sponge by removing portions of a cube, then removing similar portions of the cubic pieces that remain, repeating this process *n* times. Can you build such a sponge out of smaller versions of the sponge, or other building blocks?

Let your inner artist out, to paint a canvas using colored rectangles. But watch out: there are a few catches.

Try to find patterns that allow you to predict colors that will appear in rows of dots. Each row is produced from the one above it, following simple rules.

In Evenland, the citizens never invented the number one; instead, they started with 2, and built up sums and products from that starting point. What can we say about prime numbers in Evenland?

How fast can the cookie monster eat all of the cookies, if he must follow some simple rules?

Can you arrange a set of numbers in pairs, so that the sum of each pair is a perfect square? What if the sums must be primes? What if the sums must all be distinct? Discover the surprising variations in this simple challenge.