Pencil and paper games for two players
Classic games · · 5 min read
A pencil, a sheet of paper and a friend: that is all these seven games need. They are played in classrooms, on trains and on the back of receipts all over the world, and most of them hide more strategy than they show.
For each game you get the rules in a few lines, where it comes from when that is known, and the one idea that turns a beginner into the player who usually wins.
The seven games at a glance
From the quickest to the longest:
- Tic-tac-toe: three in a row on a 3×3 grid. A minute a game.
- Nim and the 21 game: take stones, matches or numbers; whoever takes the last one wins.
- Dots and boxes: join dots and claim the squares you close.
- Sprouts: connect spots with curves that may never cross.
- Hangman: guess the word one letter at a time.
- Five in a row: tic-tac-toe on squared paper, but you need five.
- Battleship: find and sink a hidden fleet on a 10×10 grid.
Tic-tac-toe
Draw a 3×3 grid. One player writes X, the other O, taking turns; three of your marks in a row, column or diagonal wins.
With perfect play on both sides every game is a draw, so the game is won by spotting the other player’s mistakes: take the centre or a corner first, block every two-in-a-row at once, and look for the fork, a move that makes two threats so only one can be blocked. The full method is in how to win at tic-tac-toe.
Nim and the 21 game
Put down 21 marks, or several piles of them. Players take turns crossing out marks; in the 21 game you cross out one, two or three, and in classic Nim any number from a single pile. Whoever takes the last mark wins.
It looks like luck and is pure arithmetic: in the 21 game, leave your opponent a multiple of 4 after every move and you cannot lose. Why that works, and the binary trick for several piles, is in how to always win the 21 game.
Dots and boxes
Draw a grid of dots, five by five is a good start. Players take turns joining two neighbouring dots with a short line, across or down. Whoever draws the fourth side of a box writes their initial in it and must draw another line. When every box is closed, the player with more boxes wins. The French mathematician Édouard Lucas described it in 1889 under the name “la pipopipette”.
The beginner’s rule: never draw the third side of a box, because that hands it to your opponent. Sooner or later every line left gives something away, and the board is made of chains, rows of boxes that fall one after another once the first is opened.
That is where the real game starts. When your opponent opens a long chain, take every box except the last two and close them off with one line, leaving that pair for them. They take two boxes and must then open the next chain for you. Giving up two to win many is the heart of the game; the mathematician Elwyn Berlekamp wrote a whole book about it.
Sprouts
Invented by the mathematicians John Conway and Michael Paterson at Cambridge in 1967. Start with two or three spots. A move is a curve joining two spots, or a spot to itself, with a new spot drawn somewhere on it. A curve may not cross another curve, and no spot may have more than three lines touching it. The player who makes the last move wins.
Every spot has room for three lines and each move uses two of those places but adds a spot with one, so a game that starts with n spots lasts at most 3n − 1 moves. Counting the moves still possible, and closing off parts of the paper so the remaining spots cannot reach each other, is how strong players plan the ending.
Hangman
One player thinks of a word and draws a dash for each letter. The other guesses letters: a right guess is written into every dash where it belongs, and a wrong one adds a line to a little drawing. Guess the word before the drawing is finished to win.
Start with the vowels, then the most common consonants of your language; a word’s length and the letters already in place narrow the choice quickly. When it is your turn to choose, short words with rare letters are the hardest to guess.
Five in a row
On squared paper, players take turns drawing X and O in empty squares; the first to get five in a row, across, down or diagonally, wins. In Japan it is called gomoku. The board has no edge to speak of, so games can roam across the whole page.
Attack in pairs: a row of three with both ends open has to be blocked, and a move that makes two such threats at once wins. The first player has a real advantage — in 1994 the computer scientist Victor Allis proved that on a 15×15 board the first player can always force a win — so take turns starting.
Battleship
Each player draws two 10×10 grids, lettered A to J and numbered 1 to 10, and secretly places a fleet on one of them: in the usual version one ship of 5 squares, one of 4, two of 3 and one of 2. Players take turns calling a square, such as B7, and hear “hit”, “miss” or “sunk”. Sink the whole fleet first to win. It was a pencil and paper game for decades before the plastic board version of 1967.
Search on a chessboard pattern. The smallest ship covers two neighbouring squares, so it always lies on one light and one dark square: firing only at the dark ones is enough to find every ship, with half the shots. After a hit, try the four squares around it, then follow the line.
The same games on a screen
Two of them are on Duelto, free and without an account: tic-tac-toe and Nim, each against a bot at three levels or with a friend on the same screen. When you want an opponent from somewhere else, tic-tac-toe online and Nim online are rated, so every win counts.
The others still belong on paper. Try dots and boxes with the chain rule next time you have a pencil handy; it is the one that rewards a little thinking most.