a standard checkerboard with 8 blocks to a side contains 204 squares of various sizes. in how many such squares are there an equal number of red blocks and black blocks?
im in a quick hurry please help
1by1 --- 64 of them
2by2 --- 49
3by3 --- 36
4by4 --- 25
5by5 --- 16
6by ----- 9
7by7 ---- 4
8by8 ---- 1
total = 204
There can only be an equal number of reds and blacks in the even-numbered squares, so
64 + 36 + 16 + 4 + 1 = 121
121 blocks have the same number of reds and blacks.
( e.g. a 3 by 3 square can have either
a) 4 reds and 5 blacks, or
b) 5 reds and 4 blacks. )
To find the number of squares that contain an equal number of red and black blocks on a standard checkerboard with 8 blocks on each side, we can use the concept of combinations.
Let's break down the problem step by step:
1. Determine the number of squares that have an odd number of blocks on each side:
- For a square to have an odd number of blocks on each side, the size of the square's side must be an odd number between 1 and 8.
- We can calculate the number of odd-sized squares by counting the number of odd integers between 1 and 8:
1, 3, 5, 7 = 4 squares with an odd number of blocks on each side.
2. Calculate the number of squares that have an even number of blocks on each side:
- To calculate the number of squares that have an even number of blocks on each side, we need to consider the sizes of squares that can be formed by combining the blocks.
- In a 1x1 square, there are 64 possibilities (8 rows × 8 columns).
- In a 2x2 square, there are 49 possibilities because we lose one row and one column on each side.
- Continuing this pattern, we can calculate the numbers for each square size until we reach the maximum size of 8x8.
- Adding up the possibilities for each square size, we get:
64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204 squares with an even number of blocks on each side.
3. Determine the number of squares that have an equal number of red and black blocks:
- Since we have an equal number of red and black blocks on the checkerboard, the squares with an odd number of blocks on each side will not have an equal number of red and black blocks.
- We need to count how many of the squares with an even number of blocks on each side have an equal number of red and black blocks.
- Counting the possibilities that satisfy this condition, we find that there are 4 squares with an equal number of red and black blocks.
Therefore, there are 4 squares on the checkerboard that contain an equal number of red and black blocks.