6 dots & 5 slopes

Question

I read this question somewhere:

Is it possible to mark 6 dots on a surface, and connect them two by two to make lines, in which the total number of slopes is 5 ?

There is also the same question but with different numbers. They were easier to solve than this one:

$4$ dots & $4$ slopes

$5$ dots & $4$ slopes

I don't know if I said that correctly, but my point is that if you count all parallel lines as one line, you will have 5 lines in total.

I want to know if possible, how and if not, how can we prove that it's impossible.

Answer 1

It's not possible: Any $6$ non-collinear points determine at least $6$ different slopes.

More generally, for $n\geq 4$ non-collinear points the minimum number of slopes is

• $n-1$, if $n$ is odd
• $n$, if $n$ is even.

See P. Ungar: $2N$ Noncollinear points determine at least $2N$ directions