Games and Math go really well together, although a lot of people don't make that connection. Math really can be fun if some of its principles are taught through games. We see this in quite a number of free and commercial video games that teach math concepts, typically through solving arithmetic problems.

In this article I present 8 games of a different variety. They use paper and pencil, are free, are easy to learn, and teach logical thinking and strategy that involves looking ahead. They're great fun and they can usually be played by young and old alike. And, I've bet there are some games on my list that many of you have never heard of.

After reading this list you may realize that many of the games that we typically buy we can easily make ourselves with paper, pencil, index cards, and maybe some dice. For the games that require graph paper read on to learn how to easily generate your own on your computer with a free web-based program and then print them out.

Here's my list. Please let us all know of others paper and pencil games you know of.

1. Battleship. This is a great two-player game that gives practice in grids and their related coordinate system. Each player secretly draws ships on a grid. Different ships have different dimensions. Then each player has to guess all of the coordinates of each of the opponent's ships. There's a popular commercial version of this game but you can make your own games with graph paper and pencil.

You can save yourself some money on graph paper by printing your own using the great free graph paper web-site.

Wikipedia has instructions for playing battleship.

2. Sprouts. This is an incredibly fun two person game, with very simple rules and devel oping a strategy for the game serves as a great exploration for the mathematically inclined. The game involves connecting dots with curves while not violating some basic conditions.

This game was popularized by Martin Gardner. I wrote a review of the CD that contains all of Gardner's articles from 30 years of his column in Scientific American. The CD is a remarkable source of many many Math games and diversion, including a number on this list and I highly recommend it for the mathematically curious. It'll take me years to explore only a fraction of its content.

You can learn to play sprouts from this Wikipedia article.

3. Bulls and Cows. This is the precursor to the game of Mastermind, whose object is to guess the opponent's secret code. Each player makes up their own numeric code and give hints to the opponent about the number of correct digits, either in the correct position (called a "bull") or in the wrong position ("cow"). Once again, Wikipedia has instructions. You, could, of course, play the full-blown version of Matermind with paper and pencil using these Wikipedia instructions. Bulls and Cows and Mastermind are very interesting games from the perspective of developing a strategy, especially one that can be programmed into a computer.

4. Nim. Nim is another two-player game popularized by Martin Gardner. There are a number of variations to this very basic game with a strategy that's straightforward to implement, once it's learned. The game involves players taking turns removing objects from one of a number of heaps of similar objects. Each player must take one or more objects from only one of the heaps. The player who is forced to take the last object loses. While this is, strictly speaking, not a paper and pencil game, it can be played on paper if one represents objects and heaps as lines surrounded by circles. Lines can be crossed out as they are "removed" from a heap. If one has toothpicks or matches on hand the game can certainly be more easily played with those as objects.

More information about nim can be found in .... Wikipedia.

5. Hex. Here's another game popularized by Martin Gardner that's not strictly a paper and pencil game but can easily be turned into one. The game is played on a board made up of hexagons. The goal is to connect your playing pieces from one side of the board to the other while keeping your opponent from doing the same. Read the instructions for play at Wikipedia then make and print your own hex graph paper for the board using the free hexagonal graph paper generator, use symbols for each of the two players, like O and X, instead of colored pieces and you're set.

6. Connect Four. Here's a game that can be easily turned into a paper and pencil game with a small amount of imagination and paying attention to "gravity" during play. The goal is for two players, taking turns, to connect four disks of their color in a row. Using graph paper one can make a board. Then, during play, keep the rule in mind that you must place a piece "on top" of other pieces, just as if you had dropped them into one of the slots in the board game. Yes, Wikipedia gives instructions.

7. Racetrack. Racetrack is a game I have very fond memories of playing in high scho ol. It was written about by Martin Gardner although I'm not sure that it's particularly well-known. It's a fantastic game for learning very basic physics principles of vectors, velocity and acceleration. But, don't let the physics scare you. You won't notice any physics or Math here unless you're looking for it.

The game is played by two people, using a sheet of graph paper on which one of the players draws a "racetrack." The players "move" around the track, drawing line segments to mark their movement, following simple rules, and trying not to "crash". This really is an amazing game!

Read all about it, rules and variations, in Wikipedia.

8. Dots and Boxes. This is a very simple game, suitable for the younger ones. Using graph paper each of two players taks turns drawing line segments connecting two adjacent corners. The goal is to make boxes. The player who is able to make the most boxes wins. The game is easy to play and a little bit of strategy goes a really long way.

See Wikipedia's article for instructions, strategy and variations.

One other game, which I won't count in this list (a free bonus!) is one I wrote about before, a fun arithmetic game that sparks exploration. It's another paper and pencil game that helps kids to enjoy exploring simple patterns with numbers.

There you have it. 8 really fun games (plus the bonus game), no expensive purchases, all easy to learn and easy to play. Plus, there's a tremendous depth to these games. Play them, enjoy them, and figure out their strategies.

Share this article with your friends and share your favorite pencil and paper games with all of us.

If you enjoyed this article you might also enjoy some of the other ones listed in my articles page. There are all kinds of articles about fun Math games, Math projects, mental Math tricks, Math education, and more. Plus, I've made a few videos. Check them out at my videos page.

[ Update: 1/20/08: I've discovered a couple of books and a web page that folks who enjoyed this article might like. The first book is A Gamut of Games. It has some pencil and paper games and is mentioned at the bottom of this post. The second book is Games with Pencil and Paper. The interesting web page is one that gives instructions for a novel game called Connect Capture. ]