Problem 1

A line in the plane is called sunny if it is not parallel to the

x

-axis, the

y

-axis, or the line

x+y=0

. Let

n\ge3

be a given integer. Determine all nonnegative integers

k

such that there exist

n

distinct lines in the plane satisfying both of the following: for all positive integers

a

and

b

with

a+b\le n+1

, the point

(a,b)

lies on at least one of the lines; and exactly

k

of the

n

lines are sunny.

Step 6 of 6: Climb back up: the answer is k = 0, 1, 3 for every n

Previous step Next step

k\in\{0,1,3\}

Detailed analysis

By the reduction of the second step, adding long lines to the three base constructions for $n=3$ produces, for every $n\ge3$ , valid families of $n$ lines with exactly $k=0$ , $k=1$ , or $k=3$ sunny lines, and the necessity argument shows no other value of $k$ is possible. Therefore the complete answer is $k\in\{0,1,3\}$ .