The 12 cm square was divided into 4 cm square cells and 37 points were marked.

The 12 cm square was divided into 4 cm square cells and 37 points were marked. Prove that in any one cell there will be at least 5 points.

To divide a 12 cm square into 4 cm squares, divide each side into 12/4 = 3 parts. As a result, we get 3 ∙ 3 = 9 cells.

We will use the contradiction method for the proof. Suppose there is no cell with at least 5 points. This means that each cell has a maximum of 4 points. Since we have 9 cells, then a maximum of 9 ∙ 4 = 36 points are placed in total. But according to the condition of the problem, 37 points were posed. This means that our assumption is incorrect, and at least one cell will contain at least 5 points, which is what we had to prove.