Four Corners

Previous post: Games and Mathematics

The first game I would like to analyse is Four Corners. I played this game with my friends at school on our playground when we were between 8 and 13 years old.

Rules

There are 4 runners and 1 catcher. The runners stand at the 4 corners of a square and attempt to exchange places with other runners. The catcher positions himself/herself at the centre of the square and tries to catch a runner while he/she is exchanging places by:

• either tapping them when they are running or
• reaching the safe circles in the corners before they do.

If the catcher manages to do any one of the above, then he/she becomes one of the runners in the next round and the person that he/she caught becomes the catcher.

Let's dive into some of the math behind the game...

Why a square?

The square is one of the most symmetric of all geometrical shapes. It was intuitive to understand and easy to create on the mud playground at school. Using a square ensured a game that was, by design, inherently fair to all players. Any other quadrilateral would place some runners or the catcher at a distinct disadvantage.

For instance, a rectangle (see the image below) would make an exchange between {A, D} and {B, C} nearly impossible (assuming the runners and catcher have similar running speeds) while exchanges between {A, B} and {C, D} would be rather difficult for the catcher to prevent.

Thus, the basic layout of the game rests on the fundamental properties of a square - a shape that children intuitively grasp even before they begin to formally learn and prove its properties in higher grades.

Why not a regular pentagon or hexagon?

Four Corners involved 5 players; could it be extended to a 6 player game (Five Corners) or a 7 player game (Six Corners) using a regular pentagon or hexagon respectively?

Technically, it could. However, estimating angles and dimensions for such polygons is significantly harder for children (and for most adults too!). A square has all 4 angles equal to 90⁰ (right angles) - this is easily estimated and recognised by both children and adults as they encounter them everyday. In fact, take a minute or two and look around the room/place you're reading this in. You are sure to find numerous examples of right angles! 

On the other hand, each angle of a regular pentagon is 108⁰ - it is quite difficult to estimate this, right? Even if children are able to arrive at a good approximation of a regular pentagon, deciding where the catcher must stand isn't easy because the centre of the pentagon isn't as intuitive to figure out as the centre of a square.

This logic can be extended to regular hexagons, octagons, decagons and so on.

Besides these geometrical difficulties, there are a couple of practical constraints:

• the catcher will find it difficult to keep track of the runners as their numbers increase
• the area required to play the game would become quite large if the number of players increased and space was at a premium in my school playground!

For instance, the area of a square of side 10 m is 100 m² while the area of a hexagon with an equal side length of 10 m is ~260 m².

How do runners exchange places?

You might recall your math teacher at school telling you something about the shortest distance between two points being the length of the straight line joining those two points. Well, we 8-year-olds knew this fact well before formally learning it in geometry!

We (the runners) intuitively realised that running along the imaginary straight line joining the corners would be the quickest way to exchange places as the path traversed would be the shortest. Any other route would increase the chances of the catcher either tapping us or reaching the safe circle before us.

Why a safe circle?

By this age, we had gained an intuitive understanding of a point through our chapters on geometry. We learned that, to denote a point in our diagrams, we needed to mark a visible dot with our pencils.

Building on this representation, it made sense to regard the four corners of the imaginary square on the school playground as four points. However, marking a dot (like we did in our books in class) on the school playground was impractical in the context of the game. Think about it - would a safe dot have any meaning? How would the runners stand inside a safe dot?

To our young minds, the logical enlargement of the point (or dot) was a circle; this is geometrically inaccurate as points are dimensionless elements with no length, width or height. However, it made sense to us at the time because, to make points visible in our books, we used to darken them such that the points ended up resembling tiny coloured circles. We simply extrapolated this crude visual understanding of a point in building a key feature of the game.

~ o ~ x ~ o ~

So, that concludes the elements of math that I could find in Four Corners. Can you think of any more mathematical aspects of this game?