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  • Level: GCSE
  • Subject: Maths
  • Word count: 2865

Opposite Corners. In this coursework, to find a formula from a set of numbers with different square sizes in opposite corners is the aim. The discovery of the formula will help in finding solutions to the tasks ahead as well as patterns involving Opposite

Extracts from this document...

Introduction

Introduction:

The mathematical investigations that are about to be undertaken are all under one puzzle called Opposite Corners. In this coursework, to find a formula from a set of numbers with different square sizes in opposite corners is the aim. The discovery of the formula will help in finding solutions to the tasks ahead as well as patterns involving Opposite Corners.

There are a few basic procedures to follow to achieve a basic understanding of the whole puzzle.

A box consisting of numbers from 1 to 100, a 10 by 10 grid (arranged in a regular pattern) will aid in initiating an understanding for this piece.

Procedure:

  1. Place borders of four lines in order to enclose numbers arranged in a given grid. The enclosed numbers should form a perfect square.
  1.   Multiply the numbers that are found diagonally opposite and placed in the four corners of the box.
  1. From the products obtained after multiplying, find the difference between them.

An example is demonstrated on the next page.

Below is a 10 by 10 grid. Here the numbers are arranged in 10 columns.

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...read more.

Middle

  • A 6 by 6 grid

image03.png

  • A 7 by 7 grid

For example, in the 3×3 square below obtained from a 6 by 6 grid, the difference is:  

image04.png

8×22=176

10×20=200

200 – 176=24

In another example of a 5×5 square (below) obtained from a 7 by 7 grid, the difference is:

image05.png

9×41=369

13×37=481

481 – 369=112

There is an obvious difference in the differences of products of the multiplied values, in the opposite corners from the above grids as compared to the 10 by 10 grid. Now the ruley (n−1)² is going to be tested on the above findings.

In a 3×3 square (from a 6 by 6 grid):

y (n−1) ² = difference

6 (3 – 1)² = difference

6 × 2² = difference

Therefore difference = 24

In a 5×5 square (from a 7 by 7 grid):

y (n−1) ² = difference

7 (5 – 1)² = difference

7 × 4² = difference

Therefore difference = 112

Unit 5 - Extra Tasks:

In this unit the difference for the following squares are going to be determined. For example:

(i)  3×3 from the 20 by 20 grid.

(ii) 8×8 from the 15 by 15 grid.

The difference obtained from square size n×n is 3240, from a 10 by 10 grid; the value of n is to be found.

...read more.

Conclusion

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In an 8×8 square (below) obtained from a 15 by 15 grid, the difference is:    

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From the previous page,

124 × 236 = 29264

131 × 229 = 29999

29999 - 29264 = difference

Therefore difference = 735

Solution Check:

y (n−1) ² = difference

15 (8 – 1)² = difference

15 × 7² = difference

Therefore difference = 735

  1. Solution:

3240 = 10 (n – 1) ²

3240÷10 = (n – 1) ²

image06.png

image07.png

19 = n

SolutionCheck:

y (n−1) ² = difference

10 (19 – 1)² = difference

10 × 18² = difference

Therefore difference = 3240

Below is a 13 by 13 grid.

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In a 10×10 square (below) obtained from a 13 by 13 grid, the difference is:

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1× 127 = 127

10 × 118 = 1180

1180 – 127 = difference

Therefore difference = 1053

SolutionCheck:

y (n−1) ² = difference

13 (10 – 1)² = difference

13 × 9² = difference

Therefore difference = 1053

Conclusion

                Therefore the main aim of this coursework has been dealt with. The formula y (n−1) ² = differenceis the link between size of square, the grid size and the difference.  

                                                                                                                                       

...read more.

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Here's what a teacher thought of this essay

4 star(s)

This is a well thought out and demonstrated algebraic investigation. To further develop this a general form for other rectangles within the grid (not just squares) should be investigated.

Marked by teacher Cornelia Bruce 18/04/2013

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