Question

How many rectangles (including squares) are on an 8x8 chessboard? Bonus imaginary points for the general form for any size chessboard.

Answer 1

(Thinking Mathematically, page 39)

Answer 2

is it 263?
i'll skip on the imaginary points

Answer 3

It is not.

Answer 4

261?

Answer 5

Nopes.

Answer 6

oh well you stumped me on this one :)

Answer 7

Talk me through your reasoning :)

Answer 8

1296

Answer 9

i went through all the 1Xn rectangles, then all the 2Xn, up to 8Xn
where n is from 1 to 8

Answer 10

???????/

Answer 11

How did you figure out how many 1xn rectangles there were? Ashkay, you are quite correct!

Answer 12

64 1X1
32 1X2
20 1X3
16 1X4
....

Answer 13

oh wait i didn't allow for overlapping ..haha

Answer 14

Aha, there are 56 1x2 rectangles due to overlapping :P

Answer 15

There are still bonus points going for the general form for any size chessboard. I'll post the answer in about 30 minutes if people want to try to get it!

Answer 16

actually it was a guess... i was getting something around 1200 but i rounded it to 1296 as it is generally the answer:)

Answer 17

Hah, well played sir.

Answer 18

i got 204 squares

Answer 19

still i want the solution.... i am trying again though for perfect answer

Answer 20

got it. thanks to my dad

Answer 21

it is [(n)(n+1)/2]^2....

Answer 22

What's n?

Answer 23

n*n chess board

Answer 24

right?

Answer 25

It's a different formula to the one I have, but it seems to be correct!

Answer 26

whats yours?

Answer 27

Aha, I see how they link together now. Mine is \[\sum_{n=1}^{8}\sum_{m=1}^{8}(9-m)(9-n)\]

Answer 28

its the same!!!!

Answer 29

Or the more general form for any size chessboard (dimensions x and y) is \[\sum_{n=1}^{y}\sum_{m=1}^{x}((x+1)-m)((y+1)-n)\]

Answer 30

how did you get it?

Answer 31

m and n are the dimensions of the rectangles.

Answer 32

Take a rectangle of width 1, and height 1, and place it on the chessboard. You'll be able to fit 8 of the rectangles across the board, and 8 rectangles down the board. The total of those rectangles that will fit on the board is the product of those: 64. Move onto another size rectangle mxn. You can fit \((9-m)\) rectangles across the board, and \((9-n)\) rectangles down the board. The total number of these rectangles that can fit on the board is the product: \((9-m)(9-n)\). To take into account all values of m, and n, you have to do a summation where m, and n can be 1-8.

Answer 33

Summations starting at 1 go as \(\frac{(n)(n+1)}{2}\), which is what your dad presumably figured out.

Answer 34

The square comes into it because the board is square I thin. If the board has dimensions \(x\times y\), then it would be more like \[ \frac{(x)(x+1)}{2} \times \frac{(y)(y+1)}{2} \]