Today’s question is probably the shortest I have ever set:
Find all the ways to arrange four points so that only two distances occur between any two points.
In other words, how many ways are there to draw four dots on a piece of paper such that whichever two dots you choose, the distance between these two points is one of two values?
I’ll give you one solution for free, the most obvious one, in which the points are arranged in a square:
The four blue lines are one length, and the two green ones are another length. (With four dots, there are six ways to chose a pair of dots, so there are six ‘distances’ between dots.)
I learned of this puzzle from maths juggler Colin Wright, who learned it from puzzle maven Peter Winkler. What’s beguiling about the problem is not just the satisfying brevity of the statement, but the deceptiveness of the answer.
“Nearly everyone misses at least one [solution], and for each possible solution, it’s been missed by at least one person,” said Winkler.
I think it’s a lovely problem because it embraces many levels of difficulty. If you struggle with geometry, you’ll be overjoyed to find one or two other solutions. (There are more.) The business end of the problem, however, is proving that you have found every solution.
I’ll be back with the solutions at 5pm UK time today. Meanwhile, NO SPOILERS. Although do discuss how you might attack the problem. Is there a ‘simple’ strategy that gets them all or is it just down to intuition and trial-and-error?
Clarifications: The four points must be distinct. That is, no point is allowed to be superimposed on another point. For each solution, we exclude all reflections, rotations and different sizes of that solution. Also, if the two distances that occur between any two of the four points in one solution are a and b, then the two distances that appear in another solution are not necessarily a and b.
I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.
My new book So You Think You’ve Got Problems? is out on November 7. It’s a puzzle book, but also a book of true stories and mathematical ideas. The 200 puzzles I selected all, in different ways, contain an element of surprise. If you enjoy my column, I hope you’ll enjoy this book.
