Friday, September 21, 2018 0

Do people still play pencil and paper games these days? I grew up with them in my family – all sorts from number games, word games, drawing games. A number of the mathematical games involve spots or dots and, believe me, they can be fiendishly difficult to win against a good opponent.

The simplest of all is boxes. A square grid of dots is created (the bigger you make it, the longer the game). The first player joins two  adjacent dots with either a vertical or horizontal line, the second player does the same, and so play continues. However, if a player can complete a box with his line he claims it by putting his initial inside it and then having another go. Sometimes this leads to a player creating a whole chain of boxes in one go. This diagram shows the principle using a simple nine spot grid:

In move 8, player B was able to create a box, but in move 9 player A created 3, thus winning the game.

However, as players get more experienced, they find that the most obvious strategies are not necessarily the best. Look at the following example:

After position 1, the obvious move for A would be position 2, but then B would get 5 boxes and win. If A creates just two boxes, B can also claim 2 but then his extra line would give all the rest to A.

Another game with grids of dots I call “the spotty game” but, apart from the person who showed it to me years ago, I’ve never come across it anywhere else. A grid of 9 dots across by 8 dots down is interspersed with a second grid, in a different colour, of 8 across by 9 down (see diagram). The rules are simple: each player, using a different colour, joins two adjacent dots on his own grid, either vertically or horizontally. You cannot cross a line drawn by the other player. The aim is to make a continuous line, following any route, from one side of the grid to the other (or the top to the bottom). The diagram shows the action near the start, with black trying to create a line from left to right, and red from top to bottom.

In the second diagram we can see that red has made it through, although black nearly got there. Strategy is difficult: players want to attack and create their own route, but at the same time must defend by blocking their opponent’s route. Just one slip can make all the difference. It’s a great game, with fortune swinging both ways during the game, the winner sometimes not apparent until the very end.

How about a game called sprouts which has even developed a branch of maths (sort of) called “sproutology”! The setup is simply a few dots scattered around – minimum 2. Each player goes in turn and draws one line (it doesn’t have to be straight) which either joins two dots, or starts and ends at the same dot. A new dot is drawn in the middle of the line. The only rules are that lines cannot cross, and no dot can have more than three lines attached. This means that if a dot is in the middle of a line, you cannot draw a new line looping back to the same dot – this would count as having 4 lines attached. A player wins when his opponent cannot draw a new line (or there is a version where the person drawing the last line loses). The diagram below shows a short game which started with two dots. Each new line is given a red dot. In the final image, the only “free” dots (i.e. which  don’t have three lines) are marked in green and can’t be used: they can’t be joined together, nor can you add a “to and from line” to either of them.

Go on, get the paper and pencil out, and enjoy yourself!

#### Image credits

Dots and boxes, via Wikimedia Commons (CC BY-SA 3.0)

Dots and boxes chains, via Wikimedia Commons (CC BY-SA 3.0)

Sprouts, via Wikimedia Commons (CC BY-SA 3.0)