Why buy designer games, when some of the best competitive pastimes need only a pen and paper? William Hartston reviews some old and new favourites
Are you bored with bridge and backgammon? Cheesed off with chess? Unmoved by Monopoly? Then why not get back to basics with some old-fashioned pencil-and-paper games. (Far better, of course, played with elegant fountain pen and parchment, which may be purchased with the money saved by not buying expensively packaged designer games.)

First comes noughts-and-crosses or tic-tac-toe. Yes, yes, we all know how to do it: if your opponent puts her cross in the middle, you have to reply in a corner, or if she starts in the corner, you have to go in the middle - and with only the slightest modicum of care, it always ends in a draw. But can you work out the right strategy for the reverse version, where completing a row of three loses the game? Again, there is a simple strategy that leads to a draw, but it's not at all obvious. See if you can work it out; there's an explanation at the end of this piece if you need it.

Here is another game, excellent for whiling away the time on train journeys. Usually called "Crosswords" or "Word Squares", it is also known, among true aficionados, as "Sexy Taxi", for reasons which will soon be apparent.

Played by any number of people (but two or three is best), the game begins with everyone drawing a square on a sheet of paper subdivided into smaller squares. The starting grid may be of any agreed size, from four-by-four upwards, depending on how long you want the game to last and how many long words the players know.

Play proceeds with each person in turn saying a letter, which must then be entered by each player on to an empty square on his grid. Each player knows only where he himself has placed the letter, not where the other players have put theirs. The object is to score points by forming words, read horizontally or vertically, with longer words scoring more points.

On a four-by-four grid, a good scoring system is 6 points for a 4-letter word, 4 points for a 3-letter word, and 2 points for a two-letter word. Each letter may be scored in only one word on each line. (So "BOAT" scores 6 as a 4-letter word, with nothing extra for "boa", "oat" or "at".)

It's a good idea to ensure that all players end the game having contributed the same number of letters, so with two or three playing on a five-by- five grid, it is sensible to start with an agreed letter already in the middle square. Once the game has started, sneaky players will think of words containing unusual letters, such as Q, X or Z, with which to embarrass their opponents towards the end of the game. Which is why the words SEXY and TAXI often crop up.

Another old favourite is Dots-and-Boxes, in which two players alternately fill in lines on a rectangular lattice of dots with the aim of completing squares. What gives this game an added tactical element is the rule that completion of a box is followed by another move by the same player. Well- played games thus tend to end with one player forced to make a move that lets his opponent complete a long chain of boxes. You can find a detailed account of the mathematics of Dots-and-Boxes in Winning Ways For Your Mathematical Plays by ER Berlekamp, JH Conway and RK Guy (Academic Press, 1982). There is also a good account of a $500 Dots-and-Boxes tournament played among mathematicians, written up by Julian West as "Championship- Level Play of Dots-and-Boxes" in Games of No Chance (Cambridge University Press, 1996), but you need to have read the earlier work to understand much of the terminology.

Mathematically simpler, but no less taxing in real life, is the game of "Snakes" or "Chains" or whatever else you want to call it. An example can be seen to the right of the noughts-and-crosses board above. You start with two interlocking rectangular grids, shown here as crosses and circles, but it's simpler playing in two different colours. The grids shown here are five-by-four, but any size is possible as long as one dimension is one unit longer than the other. If the crosses are (n+1) across and n down, then the circles must be (n+1) down and n across.

One player takes the circles, the other the crosses, and they move alternately by joining two neighbouring dots. The object is to form a connected path from one edge to the other in the direction of the longer dimension. So in the above, crosses are heading from top to bottom, circles from left to right. Circles have just won by completing a path from bottom left to top right.

Since this game can never end in a draw (exercise for reader: prove it!), and moving first cannot be a disadvantage, the player who starts ought to win, but even on a relatively small six-by-five grid, the game is surprisingly complicated - and can be very confusing if you cannot find two pencils of different colours.

Finally, there is the game of Sprouts: draw a handful of dots (six or seven is a good number to start with) on a sheet of paper, then, moving alternately, the two players draw a line joining two dots, or a loop joining one dot to itself. The move is then completed by adding another dot anywhere along the new line. The only rules are that no line may cross another (or itself); no line may be drawn through a dot; and each dot may have only three lines leading from it. The last player to make a move wins.

Finally, we cannot leave without mentioning Battleships. That's the game we all used to play at school, with agreed numbers of battleships, cruisers, destroyers and submarines depicted by lines of squares of different lengths on a grid, and the players naming squares in turn on which bombs were dropped in an attempt to locate and destroy the enemy flotilla. We mention it only because a reader has asked if anyone knows its origins. Our researches reveal two theories: 1) It was originally called "Jutland" and dates back to the First World War; 2) It was played by radio operators in the Second World War as a means of ensuring that communications were functioning properly. Can any reader throw light on this?

And before we forget, the correct strategy for losing-noughts-and-crosses is as follows: the first player must start in the middle, then reflect the opponent's moves through the centre. After any other opening move, you can be forced to complete a line of three and lose.