A lattice point in the coordinate plane with origin O is called invisible if the segment OA contains a lattice point other than O and A. Let L be a positive integer. Show that there exists a square with side length L and sides parallel to the coordinate axes, such that all points in the square are invisible.

*I had several problems here: *

*1) - The problem says: L must be positive as an integer length as the side of a square. I wondered about the "positive". T**here is no such thing as a square with negative side. *

*2) - The problem asks us to show that there is a square containing only "invisible" lattice points, but it is never stated if the lattice points on the side of a square are to be considered to be "in" the square or out. I will assume "within", or otherwise there is an easy solution of the 2x2 lattice (around the invisible point (2,2), where the square perimeter is: (1,1), (2,1), (3,1) **(3,2), (3,3), (2,3) (1,3), (1,2) *

*3) Why would we look for anything larger than a square of side 2 (4 points) to satisfy the problem? If there is a larger square with all invisible points, there will be a 2x2 square of invisible points within it. *

*4) It is not mentioned if all four quadrants are to be considered. However, s**ince the four lines: x=1, x=-1, y=1, y=-1 are made entirely of visible points, they can not be overlaid by a square holding all invisible points. Therefore, only squares positioned entirely within one quadrant can be candidates. So, by symmetry, we need only search quadrant I. *

*5) I understand that a point A=(x,y) is proven invisible if OA crosses another point, and that this happens when x and y have a mutual factor. E.g., **if A= (2,6), OA then also crosses (1,3) and A is invisible. *

*However, I don't see how the Chinese Remainder Theorem comes to the rescue, since it is more concerned with divisors (x) that the do _not_ share prime factors with y but rather leave a remainder in a division.*

*6) I have seen somewhere before this "comb" of lines (of multiples) used to search for primes (where new primes would lie off of any lines of multiples of known primes.) *

Anyway, I performed a computer search of Quadrant I out to x=100,000, y=100,000 and found no 2x2 square of invisible grid points.

The program is *here, *and the first 80 rows and columns are shown *here*, where where "V" means "visible". The rows and columns are labeled.

In summary, I did not find any of the squares asked for, but I have no proof that none exist.

Perhaps i have misunderstood the problem...

*Edited on ***February 5, 2021, 11:47 pm**