Here’s the question again:
Apart from the corners of a square, can you find other ways of arranging four points in the plane, such that there are only two distinct distances between them?
Go ahead and try scribbling some patterns. Please do not click “Continue” until you’ve come up with at least two new arrangements. As you think about this, you may stumble upon one of the following Frequently Asked Questions:
- “Can I put two points on top of each other, as in the same place in the plane?” – No, you can’t. The four points must reside in different places on the plane.
- “Must I draw only quadrilaterals of various shapes?” – No, we never said that. You need to place four points anywhere on the plane, and then measure all possible distances between any pair of those points, but the points aren’t necessarily the corners of a quadrilateral.
- “What do you mean by ‘other ways of arranging’? I can take your square and make it bigger!” – That’s technically true, but our intention here is for you to find truly different arrangements of points. What does “truly different” means? Well, if you take the square we started out with, you can certainly make it bigger or smaller, move it up or down, and rotate it without changing it being a square, and thus without changing the fact that there are only two distinct distances. So for our purposes, a new arrangement is one that is not merely a rotation, translation (move) or re-sizing of an existing arrangement.
- “I think a pyramid works!” – well, it does. In fact the correct pyramid (which one?) can get you four points offering a single unique distance between them, but we’re now being strictly two-dimensional. Please stick to the plane!
- “There are infinitely many ways to do that – any rectangle works!” – No, it does not. Make sure you see why the corners of a rectangle admit three distinct distances between them. There are no infinite families of solutions to this problem, we guarantee that.
No more excuses… now scribble away, and please spend at least 15 minutes coming up with some new shapes!
Tags: Math Circles, SJMC
My 8-year old daughter came up with an additional solution. She made the points colinear. Your challenge does not seem to preclude that.
@Tumblemark: you can’t have four collinear points having just two distinct distances. If the points are ABCD in order, then AB < AC < AD.