1 / 5

Counting Squares and Rectangles on a Grid: A Comprehensive Analysis

In this study by Gloria Gonzalez and Ieni Vargas, we explore the configurations of 1x1 to 8x8 squares on a grid. The left and top edges of squares can occupy up to 8 positions, leading to a total of 64 configurations for 1x1 squares. We derive similar counts for larger squares, culminating in a total of 204 squares on an 8x8 grid. Additionally, we extend our findings to calculate the number of rectangles formable on a chessboard using combinatorial analysis, revealing a total of 1296 rectangles.

lavonn
Télécharger la présentation

Counting Squares and Rectangles on a Grid: A Comprehensive Analysis

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Squares By: Gloria Gonzalez And Ieni Vargas

  2. Consider the left hand vertical edge of a square of size 1 x 1. This edge can be in any one of 8 positions. Similarly, the top edge Can occupy any one of 8 positions for a 1 x 1 square. So the total Number of 1 x 1 squares = 8 x 8 = 64. For a 2 x 2 square the left hand edge can occupy 7 positions and the Top edge 7 positions, giving 7 x 7 = 49 squares of size 2 x 2. Continuing in this way we get squares of size 3 x 3, 4 x 4 and so on.

  3. We can summarize the results as follows: Sizes of squares Numbers of squares 1 x 1 8^2 = 64 2 x 2 7^2 = 49 2 x 2 7^2 = 49 4 x 4 5^2 = 25 5 x 5 4^2 = 16 6 x 6 3^2 = 9 7 x 7 2^2 = 4 8 x 8 1^2 = 1 Total = 204

  4. There is a formula for the sum of squares of the integers 1^2 + 2^2 + 3^2 + ... + n^2 N (n+1) (2n+1) Sum = ------------ 6 In our case, with n = 8, this formula gives 8 x 9 x 17/6 = 204. As an extension to this problem, you might want to calculate the Number of rectangles that can be drawn on a chessboard. There are 9 vertical lines and 9 horizontal lines. To form a rectangle You must choose 2 of the 9 vertical lines, and 2 of the 9 horizontal Lines. Each of these can be done in 9C2 ways = 36 ways. So the number Of rectangles is given by 36^2 = 1296.

  5. Tanks for you time Gloria Gonzalez and Ieni Vargas

More Related