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My text book had a question to find the no. of rectangles in a chessboard. My attempt was that out of $64$ points if we choose any $4$ points, a rectangle is formed and hence no. of rectangles in $C(64,4)$ [$64$ choose $4$]. But in the book its given, to form a rectangle $2$ vertical and horizontal lines need to be selected, and there are $9$ vertical and horizontal lines, so number of rectangles are $C(9,2)\times C(9,2)$.

$2$ questions: Why is my attempt wrong, and aren't there only $8$ lines in a chessboard? I know its a little basic for Stackexchange level of questions but any answer would be appreciated greatly.

BrainNuke
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    If you chose 4 points, who guarantees that the lines connecting them form right angles? – YJT Sep 01 '20 at 09:45
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    Moreover, choosing four points, you will count the rectangles more than once. – TheSilverDoe Sep 01 '20 at 09:45
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    try to draw a chessboard and I think you will automatically get why there are 9 lines instead of 8. – Devansh Kamra Sep 01 '20 at 09:46
  • @Peter The lines are the edges of the squares of the chessboard. – N. F. Taussig Sep 01 '20 at 09:48
  • @N.F.Taussig I noticed that and deleted my comment. – Peter Sep 01 '20 at 09:50
  • Why do you not pick one vertex and see how many rectangles can you get? Once you do that, you will see the pattern and how to avoid duplicate counts. If you can show your attempt, it will help. – Math Lover Sep 01 '20 at 09:55
  • Thanks guys, i just realised it. I'm very sorry to ask such a question, because I just understood how stupid of a question it was – BrainNuke Sep 01 '20 at 10:03
  • Does this count include skew rectangles such as ${(1,0), (0,1), (1,2), (2,1) }$ ? Or is it only including rectangles with sides that are parallel to the sides of the board ? – gandalf61 Sep 01 '20 at 10:20

3 Answers3

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If you choose 4 squares from among the $64$ on a chessboard, odds are great that they are not the four corners of a rectangle.

As to why there are 9 lines, we are counting the thin border lines between the squares, not the actual rows and columns that the chess pieces are placed on. Those are the lines that form the possible boundaries of a rectangle on a chessboard, and there are nine horizontal and nine vertical such boundary lines.

Arthur
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  • Thank you very much! I just realised how stupid my question is, so very sorry for wasting your time – BrainNuke Sep 01 '20 at 10:05
  • @BrainNuke No worries. And I chose to take the time to write an answer. That's not you wasting my time, that's me wasting my own time. And I'm perfectly fine with that. – Arthur Sep 01 '20 at 10:08
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Hint : To define uniquely a rectangle, you just have to choose two points that don't belong to the same lign or to the same column.

TheSilverDoe
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This might be considered a silly error as all the 4 points you would have chosen from your method may be on the same line i.e., left side of the chessboard as I've shown in first figure. But counting on the second method, we would only take two of the line horizontally and vertically just to form a rectangle. Now, I assume that you just needed the mistake in your method and have understood the textbook method. I hope it's clear now

Arpit Goel
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