9. Points Near Points
a. What is the maximum number of points that can be chosen in a plane such
that when any three points are chosen, at least one pair of points has a
separation of one unit?
The particular distances in each part are irrelevant; there is simply “the right distance” and “other
distances”. Four points can form a rhombus made of two equilateral triangles. Two such rhombuses
can share one “far” point and pivot until the other “far” points are one unit apart, for a total of 7 points.
b. What is the maximum number of points that can be chosen in a plane such that when
any four points are chosen, at least one pair of points has a separation of one unit?
Joining three of those rhombuses end-to-end, the end points and joining points make
four points without a one-unit separation. We can bend the shape around the joining
points until the end points are one unit apart. This makes 10 points.
c. What is the maximum number of points that can be chosen in a plane such that when
any five points are chosen, at least one pair of points has a separation of one unit?
If you make two of the 7-point shapes from part a, you cannot choose five points without
choosing at least three from one of these 7-point shapes, which we already established must
include at least one pair one apart. These two shapes contain a total of 14 points. Placing these
two shapes exactly one unit apart in some direction appears to not allow the addition of another
point.
d. What is the maximum number of points that can be chosen in a plane such that when
any three points are chosen, at least two pairs of points have a separation of two units?
The two pairs may share a point or not.
Four points arranged in a rhombus will have this property.
e. What is the maximum number of points that can be chosen in a plane such that when
any four points are chosen, at least two pairs of points have a separation of two units? The
two pairs may share a point or not.
The arrangement of 7 points in part a seems to be the maximum.
f. What is the maximum number of points that can be chosen in a plane such that when
any five points are chosen, at least two pairs of points have a separation of two units? The
two pairs may share a point or not.
The arrangement of 10 points in part b seems to be the maximum. We thought we had
successfully added an 11th point, but we found a set of four that didn’t have two pairs.
g. What is the maximum number of points that can be chosen in space such that when any
three points are chosen, at least one pair of points has a separation of one unit?
If we build the figure from part a in 3D, the components are trigonal bipyramids instead of
rhombuses. Each rhombus gained a point from this, for a total of 9 points.
h. What is the maximum number of points that can be chosen in space such that when any
four points are chosen, at least one pair of points has a separation of one unit?
A similar analysis gives 10 + 3 = 13.
i. What is the maximum number of points that can be chosen in space such that when any
five points are chosen, at least one pair of points has a separation of one unit?
By the same logic as part c, 9 + 9 = 18.
j. What is the maximum number of points that can be chosen in space such that when any
three points are chosen, at least two pairs of points have a separation of two units? The
two pairs may share a point or not.
A square bipyramid satisfies these conditions, and contains 6 points.
k. What is the maximum number of points that can be chosen in space such that when any
four points are chosen, at least two pairs of points have a separation of two units? The two
pairs may share a point or not.
Two trigonal bipyramids sharing one “far” point works, for an answer of 9. Although there are
many degrees of freedom left, we haven’t been able to find an arrangement that allows another
point.
l. What is the maximum number of points that can be chosen in space such that when any
five points are chosen, at least two pairs of points have a separation of two units? The two
pairs may share a point or not.
Three trigonal bipyramids gives an answer of 13, with many degrees of freedom left that appear
unproductive.