Functions Problem 8


Divide the sheet of agar into 361 squares with side length 1. We'll consider that each square includes the boundary of the square as well as its interior, so the squares overlap slightly.

We need to choose 380 points in the larger square as locations for mold samples. By the Pigeonhole principle, with 380 choices to make over 19*19=361 regions, at least two of the samples must be in the same square region.

Recall the Distance Formula for 2D points. If two points are part of the same square region of area 1, the maximum distance between them is \(\sqrt{2}\). Because we have said that there are at least two points that are part of the same square region, two of the mold samples must have been placed too close together. So the sheet of agar wasn't large enough.