Number of Squares in an NxN Square
Date: 7/29/96 at 7:57:25 From: Anonymous Subject: How Many Squares in an N by N square? How many squares are there in a 8 x 8 square? How many rectangles are there? Find an equation that will find out the number of squares simply by knowing the size of the square. I can't find out the equation. Can you help? This equation has to apply to both squares and rectangles.
Date: 7/30/96 at 15:52:27 From: Doctor Robert Subject: Re: How Many Squares in an N by N square? There are quite a few! This is an interesting problem. Let's tackle the one about the squares first. There is only ONE 8x8 square. There are FOUR 7x7 squares. There are NINE 6x6 squares, and so on. You can see this by taking a square (say 6 by 6) and seeing how many squares you can move it horizontally and how many vertically and multiplying these two numbers together). If you see the pattern above, the answer to the question as to how many squares there are is 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 = 204 The "equation" you want is N = (9-s)^2 where N is the number of squares of side s. That is, for a square of N sides, the total number of squares = 1^2 + 2^2 + 3^2 + ... + N^2.] For rectangles, it's just a bit harder to see. Consider a rectangle 1 by 2, that is, 1 unit high and 2 units long. How many units can fit in the top row? Seven! How many rows are there? Eight. So there must be 56 1 by 2 rectangles. Of course, there are also 56 2 by 1 rectangles. If you proceed this way to count the possible number of rectangles of different sizes you can generate the following table. Across the top are the possible lengths of the rectangles. Down the side are possible heights of the rectangles. In the table are the number of possible rectangles with these dimensions. 1 2 3 4 5 6 7 8 1 64 56 48 40 32 24 16 8 2 56 49 42 35 28 21 14 7 3 48 42 36 30 24 18 12 6 4 40 35 30 25 20 15 10 5 5 32 28 24 20 16 12 8 4 6 24 21 18 15 12 9 6 3 7 16 14 12 10 8 6 4 2 8 8 7 6 5 4 3 2 1 There are quite a few interesting things about this table. Notice that the elements along the major diagonal represent the squares. The sum of all the numbers is 1296, so there must be 1296-204 = 1092 rectangles that are not squares. The "equation" must be N = (9-m)(9-n) where N is the number of rectangles, m is the height of the rectangle and n is the width of the rectangle. I hope that this helps. -Doctor Robert, The Math Forum Check out our web site! http://mathforum.org/dr.math/
Date: 10/02/98 at 11:24:31 From: John Pinkerton Subject: Re: Comment on Number of Squares in NxN Square Dear Dr. Math: Here's a quick derivation for the number of rectangles in an NxN chessboard. Each rectangle is defined by selecting two of the N + 1 vertical lines and two of the N + 1 horizontal lines that are the edges of the squares. So, the number of rectangles is the square of N + 1 "choose" 2, which is the square of (N + 1)*N/2. For an 8x8 square, you have ((8 + 1)*8/2)^2, or 36 squared, which is the 1296 that Dr. Robert gets by adding up his interesting table. Keep up the good medicine, John Pinkerton P.S. on "choose": Recall that the number of ways to choose 2 objects out of K is K "choose" 2, or (K)*(K - 1)/2. There are K choices for the first object and K - 1 choices for the second, so there are (K)*(K - 1) ordered pairs. But, order doesn't matter. So, we divide by 2 since you don't want to double count for the two orders you might choose a given pair of objects. In the above chessboard problem, K is N + 1.
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