how many rectangles are there on a chess board?
i've tried twice and got 168 is this right? my friend said i'm way short.
any help would be much appreciated.
thank you
Check this for a start (squares):
http://mathforum.org/library/drmath/view/56760.html
The end of the article shows what to do for rectangles.
Thanks MathMate,
boy looks like i was way off.
thanks for the help
cc
To find the number of rectangles on a chessboard, you need to consider all possible shapes and sizes of rectangles that can be formed using the grid of squares on the chessboard.
There are two main types of rectangles on a chessboard: horizontal and vertical rectangles.
1. Let's start with horizontal rectangles:
- Each row on the chessboard has 8 squares, and you can select any two rows out of the 8 available (use combinations). So, there are \(\binom{8}{2}\) ways to choose two rows.
- Each column also has 8 squares, and you can select any two columns out of the 8 available. Again, there are \(\binom{8}{2}\) ways to choose two columns.
- Now, multiply the number of choices for rows and columns: \(\binom{8}{2} \times \binom{8}{2} = 28 \times 28 = 784\) horizontal rectangles.
2. Next, let's consider vertical rectangles:
- Similar to horizontal rectangles, you can select any two columns and any two rows. Multiply the choices: \(\binom{8}{2} \times \binom{8}{2} = 784\) vertical rectangles.
3. Finally, we need to consider squares as rectangles as well:
- There are 8 rows and 8 columns, so you have 8 choices for the row and 8 choices for the column. Multiply the choices: \(8 \times 8 = 64\) squares as rectangles.
To find the total number of rectangles:
Total = horizontal rectangles + vertical rectangles + squares as rectangles
Total = 784 + 784 + 64 = 1632 rectangles on a chessboard.
So, it seems that your calculation of 168 rectangles was way off. The correct answer is 1632 rectangles on a chessboard.