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Squares/rectangles on a chessboard
By Rachelle Najman (T4009) on Sunday, April 15, 2001 - 02:08 am:

These two challenging questions are giving me a tough time, can anyone help?

1) It was once claimed that there are 204 squares on an ordinary chessboard. Can you justify this claim?

2) How many rectangles are there on a chessboard?

Thanks in advance for all of your help and explanations!

By Brad Rodgers (P1930) on Sunday, April 15, 2001 - 03:44 am:

1)-

1×1 squares - 64
2×2 squares - 49
3×3 - 36
4×4 - 25
.
.
.

(can you see why this is true?)
64+49+36+...=204

A similar process of logic holds for the rectangles, and if you organize your data right, you should see another pattern. I think you'll get answer of 1296. I'll go ahead and let you see if you can discover this pattern, and then I'll try and explain why it holds true. (Hint:if you want to figure out why it holds true by yourself, think of triangular numbers inside what is being multiplied-that probably doesn't help much, sorry)

Brad

By Rachelle Najman (T4009) on Tuesday, April 17, 2001 - 02:26 am:

Any more hints or explanations?

By Brad Rodgers (P1930) on Wednesday, April 18, 2001 - 08:46 pm:

Think about each size of rectangle in turn. You might find it helpful to use a chessboard and a piece of paper of the right size. Hopefully you can see how this table builds up:

Rectangle size Number of ways
1×1 8×8 64
1×2 8×7 56
1×3 8×6 48
1×4 8×5 40
... ... ...
2×1 7×8 56
2×2 7×7 49
2×3 7×6 42
... ... ...

Do you see how I am finding that many rectangle arrangements for these specific rectangle sizes? If so, I'll let you complete the rest of the table.

Brad
By The Editor:

Here's another way of thinking of it for the rectangles one.

If we want to create a rectangle on the chessboard, what we have to do is draw two vertical lines and two horizontal lines.
How many ways are there of doing this?

For the first horizontal line, we have 9 choices of where to draw it. For the second, we have eight choices left. That gives 9×8=72 combinations. But this counts each combination twice (once with the top line chosen first, once with the top line chosen second), so we divide by two, getting 36.

The same argument holds for the vertical lines. So overall we have 36×36=1296 possibilities.