# 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

## 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.

For example,

In a 2×2 square (from the 10 by 10 grid above),

7 × 18 = 126

8 × 17 = 136

The difference between the products above is 10

Investigations to see if any rules or patterns connecting the size of square chosen and the difference can be found.

When a rule has been discovered, it will be used to predict the difference for larger squares.

A test of the rule will be done by looking at squares like 8 × 8 or 9 × 9

Another test will be run to see how well the rule can be generalised.

[For example what is the difference for a square of size n×n?]

The rule will be tested in other situations

Hint:

• In a 3 × 3 square ……

• What happens if the numbers are arranged in six columns or seven columns?

Firstly, before the tasks are considered, other squares will be tried using the procedure in the example of the 2×2 square on Page 2.

The following squares have been obtained from the 10 by 10 grid on Page 2.

In the 4 × 4 square:

34 × 67 = 2278

37 × 64 = 2368

The difference between the products is 90.

In a 5×5 square:

25 × 69 = 1725

29 × 65 = 1885

The difference between the products is 160.

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#### Here's what a teacher thought of this essay

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.