+5 A four by four grid of unit squares contains squares of various sizes (1 by 1 through 4 by 4), each of which are formed entirely from squares in the grid. In each of the 16 unit squares, write the number of squares that contain it. For instance, the middle numbers in the top row are 6s because they are each contained in one square, two ,two , and the one .
What is the sum of all sixteen numbers written in this grid?
+2 1 + 6 + 6 + 6 9 + 12 + 9 + 6 6 + 9 + 9 + 4 The sum of all sixteen numbers written in the grid is 92.
In this problem, we are given a 4x4 grid of unit squares. We need to find the sum of the numbers written in each square, which represents the number of squares that contain it.
To solve this problem, let's start with the top-left square. It is a 1x1 square, so it only contains itself. Therefore, we write 1 in this square.
Moving to the adjacent squares in the top row, the numbers are the same. The middle numbers in the top row are 6 squares. Let's see why.
The top-center square is a 1x1 square, so it contains itself, just like the top-left square. Additionally, it is also contained in two 2x2 squares, two 3x3 squares, and one 4x4 square. So, we write 6 in this square.
Similarly, the top-right square is contained in the same number of squares as the top-center square. So, we write 6 in this square as well.
Now, let's move to the squares in the middle row. The numbers in the middle row are 9, 12, and 9.
The middle-left square is contained in three 2x2 squares, three 3x3 squares, and one 4x4 square. So, we write 9 in this square.
The middle-center square is contained in four 2x2 squares, four 3x3 squares, and one 4x4 square. Therefore, we write 12 in this square.
The middle-right square is contained in the same number of squares as the middle-left square. Hence, we write 9 in this square.
Finally, let's move to the squares in the bottom row. The numbers in the bottom row are 6, 6, and 4.
The bottom-left square is contained in two 2x2 squares, two 3x3 squares, and one 4x4 square. So, we write 6 in this square.
The bottom-center square is contained in the same number of squares as the bottom-left square. Thus, we write 6 in this square.
The bottom-right square is contained in one 2x2 square, one 3x3 square, and one 4x4 square. Hence, we write 4 in this square.
To find the sum of all sixteen numbers, we add up the numbers in each square:
1 + 6 + 6 + 6 + 9 + 12 + 9 + 6 + 6 + 9 + 9 + 4 = 92 
