Question

A 6 by 6 grid is composed of 34 black squares and 2 white squares. all squares are randomly placed. in how many ways can the white squares be placed so that they have only one point in common?

Answer 1

There are 6 distinct ways to place two white squares on a 6 by 6 grid so that they have only one point in common. This includes positions inside the grid and avoids counting overlaps at corners.

Step-by-step explanation:

To place two white squares so that they have only one point in common on a 6 by 6 grid, we need to consider the positions where two squares can meet at a corner. At each corner of a square, there are four different squares that meet. Since there are 6 squares along one side of the grid, there will be 5 corners along that side. Multiplying the number of corners along one side by itself gives us the total number of intersecting points in the entire grid, except the border. This is because the squares on the border of the grid do not have four squares meeting at their corners.

So there are 5 corners horizontally × 5 corners vertically = 25 intersecting points within the grid where four squares meet. However, each intersecting point is shared by four squares, so to avoid counting the same position of the white squares multiple times, we divide by 4, yielding 25 ÷ 4 = 6.25, but since we can't have a fraction of a position, we consider only whole numbers, so there are 6 distinct ways to place the two white squares on such that they have only one point in common.