Date: 03/24/2003 at 12:26:32 From: Veronica Subject: Rectangles A checkerboard has 8 horizontal boxes and 8 vertical boxes. How many rectangles are possible inside that board? I can't figure out what the right size is for the rectangles. If you split the board in two you get two rectangles.
Date: 03/25/2003 at 14:45:21 From: Doctor Ian Subject: Re: Rectangles Hi Veronica, That's not quite what the question is asking. Instead of a checker board, let's use a tic-tac-toe board, which is smaller and easier to think about: A---B---C---D | | | | E---F---G---H | | | | I---J---K---L | | | | M---N---O---P Now, each square is a rectangle, right? And there are 9 small (1x1) squares: ABEF BCGF CDHG EFJI FGJK GHKL IJNM JKON KLPO And there are also four larger (2x2) squares: ACKI BDLJ EGOM FHPN And the whole thing is a (3x3) square: ADPM Now, what about non-square rectangles? There are some 1x2 rectangles: ACGE BDHF EGKI FHJL IKOM JLPN and some 2 x 1 rectangles: ABJI BCKJ CDLK EFNM FGON GHPO There are also 1 x 3 rectangles, and 3 x 1 rectangles, and 2 x 3 rectangles, and 3 x 2 rectangles. I'll leave those for you to find. In other words, in a 3x3 square, you can find rectangles with these sizes: 1x1 1x2 1x3 2x1 2x2 2x3 3x1 3x2 3x3 In a 4x4 square, you could find some additional sizes: 1x1 1x2 1x3 1x4 2x1 2x2 2x3 2x4 3x1 3x2 3x3 3x4 4x1 4x2 4x3 4x4 But the question isn't asking how many _sizes_ of rectangles there are, but the _number_ of rectangles. For the 3x3 case, that's 1x1 1x2 1x3 (9) (6) (3) 2x1 2x2 2x3 Total = 9 + 6 + 3 + 6 + 4 + 2 + 3 + 2 + 1 (6) (4) (2) = 36 3x1 3x2 3x3 (3) (2) (1) For some explanations of how to go about figuring out the total number of rectangles for a square of any size, see Rectangles on a Chessboard http://mathforum.org/library/drmath/view/55431.html But if you feel like finding the answer on your own, a good way to do it is to start looking at how things change as you go from a 2x2 board, to a 3x3, to a 4x4, and so on. The idea is to spot a pattern that lets you express the number of rectangles of each size as a function of N, where NxN is the size of the board. For example, in the 3x3 case, we could replace the numbers with 1x1 1x2 1x3 (N^2) (2N) (N) 2x1 2x2 2x3 (2N) (N+1) (N11) 3x1 3x2 3x3 (N) (N-1) (1) Does that pattern work for the 2x2 and 4x4 cases as well? Or would you need a different pattern to describe all three cases? I'll leave that for you to think about. I hope this helps. Write back if you'd like to talk more. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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