I read this puzzle recently on a book that suggested ways to improve lateral thinking. Simply stated the challenge is: draw four dots in a square on a piece of paper. Is it possible to connect the four points with three straight lines without ever lifting the pencil from the paper and ending up in the same place as you started?
Clearly I support thinking outside the box – but here is the challenge: can you perform the same task with only 1 straight line? I think the answer is: yes.
The key idea to solve this puzzle is to think about what a straight-line is. A straight line is the shortest way between two points. We all know what straight lines look like on flat surfaces (Eucledian spaces), but do they look like on other surfaces?On a sphere the shortest path between two points is along a geodesic. The imagine what this means, imagine you are at the North Pole and you would like to travel the shortest distance to the South Pole. The shortest distance will be to travel due South all the way until you reach the South Pole, the meridian that you end up traveling on is a geodesic.
With this in mind the question becomes: is it possible to lie the four corners of a square on a sphere so that all the points lie on the same geodisic? If we pick a sphere whose radius is such that the diameter of the sphere is exactly half as large as the diagonal of the square that the four dots lie on, then when the paper is laid onto a sphere, the four points will lie on the equator of the sphere. This is shown in the figure to the left: on top you can see what the problem looks like: on the top are the four points drawn on the flat paper. On the bottom is the situation when the paper is laid onto a sphere. Notice that the diagonals of the square (red and blue dashed lines) are straight lines in both the spherical and flat coordinate spaces. On the other hand, the geodisic along which all four points lie (the equator of the sphere as we look down), looks like a straight line in the spherical coordinates, but looks like a circle in the flat geometry.
So in conclusion, it is possible to connect four points describing a square by a single, straight line without lifting the pen from the paper, but you must be prepared to think outside the ball outside the box.