Stage Four
4 Points in a
Plane
Mark four points on a flat surface so that there are only two different distances between them.
One arrangement is shown.
Find another arrangement.
How many arrangements are there?
Reference: The Amazing Mathematical Amusement Arcade
Brian Bolt, Cambridge University Press. p21
Solution: Points in a Plane
For this problem Roger, my problem solving mentor, thought that there were 2 possible arrangements.
The first one was where the points were at the corners of a square:
Looking at this you can see that the distances between any 2 points is either the length of the sides or the length of the diagonals - so there is only 2 possible distances between any 2 points.
Roger's other idea was that the 4 points could be equally spaced along a line like this:
But this doesn't work. Why? Because there can be three different possible distances between dots. Can you see them?
We though we had the only solutions. But then I found another when I was trying to see what would happen if you squeezed the square to form a rhombus.
While squeezing the figure it came to a point where the shorter diagonal is the same length as the sides of the rhombus. Then all the distances between the points are the same, except for those at either end of the long diagonal. So for this figure there are definitely only 2 different distances between the points:
This is a special rhombus for it is made up of 2 equilateral triangles that are fitted together.
We may have missed other solutions. If you can find any, let me know.
