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#23 Four points, two distances

Set by Colin Wright (with thanks to Peter Winkler)

When you have four distinct points arranged on a flat surface (such as coins on a table) there are six possible pairs and hence six possible distances you could measure between pairs of points. It is possible to arrange the points (or coins) so that all six distances are different.

It isn’t possible to make all the distances the same – but can you arrange them in a way that only uses two distances?

Once you have found one arrangement that uses only two distances, can you find a different one?

How many arrangements that use two distances can you find in total?

Solution next week

#22 Even rows

Solution

There are lots of solutions, and each condition is a subset of the previous. The exact arrangement of the numbers doesn’t matter too much – only where the odd and even numbers are. For example, this is one that meets all three criteria.

16

3

5

14

1

4

12

11

2

7

9

8

15

6

10

13