Algebra Investigation - Grid Square and Cube Relationships

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        Algebraic Investigation 1: Square Boxes on a 10x10 Grid        

In this first investigation, the difference in products of the alternate corners of a square, equal-sided box on a 10x10 gridsquare will be investigated. It is believed that the products and their differences should demonstrate a constant pattern no matter what dimensions are used; as long as they remain equal. In order to prove this, both a numeric and algebraic method will be used in order to calculate this difference. The numeric method will help establish a baseline set of numbers for testing, and to help in the establishment of a set of algebraic formulae for use on an n x n gridsquare.

In the example gridsquare below, the following method is used in order to calculate the difference between the products of opposite corners.

Stage A:        Top left number x Bottom right number =        (a) multiplied by (d)

                                                

Stage B:        Bottom left number x Top right number =         (c) multiplied by (b)

Stage B – Stage A:        (c)(b) - (a)(d)          =  The difference

The overall, 10 x 10 grid that is used for the first investigation will be a standard, cardinal gridsquare, which progresses in increments of 1. The formulae calculated will mainly be applicable to this grid, as other formats of gridsquares will require others formulae to provide valid results.

This first investigation will focus only on gridsquares with equal widths and heights, which will at this stage be represented by the universal, constant term ‘w’.

The top left number in the grid (letter (a) in the above example) will be represented by the term ‘n’, which will be referred to in this manner in all proceeding investigations also.

This is only the first section of the investigation. In calculating the formula, enough information will be gained to progress and investigate other factors, variables and measurements that affect the difference between products.

In order to see a trend (in the form of an n x n grid) from the sample numeric grids (2 x 2, 3 x 3 etc.) it is necessary to use summary tables. Using these summary tables, it will be possible to establish and calculate the algebraic steps needed to gain an overall formula.


2 x 2 Grid

Prediction

I believe that this difference of 10 should be expected to remain constant for all 2x2 number boxes. I will now investigate to check if all examples of 2x2 grid boxes demonstrate this trend in difference. I will conduct this research using another 2 of these boxes from the overall cardinal10x10 number grid.

My predication also seems to be true in the cases of the previous 2 number boxes. Although it is quite certain that this trend would be observed in all number boxes of this instance, it is necessary to find an algebraic formula to prove that the difference remains invariable.

Any 2x2 square box on the 10x10 grid can be expressed in this way:

Stage A:        Top left number x Bottom right number =        n(n+11) =        n2+11n

Stage B:        Bottom left number x Top right number =         (n+10)(n+1)=        n2+1n+10n+10

                                                                              =        n2+11n+10

Stage B – Stage A:        (n2+11n+10)-(n2+11n) = 10

When finding the general formula for any number (n), both answers begin with the equation n2+11n, which signifies that they can be manipulated easily. Because the second answer has +10 at the end, it demonstrates that no matter what number is chosen to begin with (n), a difference of 10 will always be present.


3 x 3 Grid

Prediction

I believe that this difference of 40 should be expected to remain constant for all 3x3 number boxes. I will now investigate to check if all examples of 3x3 grid boxes demonstrate this trend in difference. I will conduct this research using another 2 of these boxes from the overall cardinal10x10 number grid.

My predication also seems to be true in the cases of the previous 2 number boxes. Although it is quite certain that this trend would be observed in all number boxes of this instance, it is necessary to find an algebraic formula to prove that the difference remains invariable.

Any 3x3 square box on the 10x10 grid can be expressed in this way:

Stage A:        Top left number x Bottom right number =        n(n+22) =        n2+22n

Stage B:        Bottom left number x Top right number =         (n+20)(n+2)=        n2+2n+20n+40

                                                                              =        n2+22n+40

Stage B – Stage A:        (n2+22n+40)-(n2+22n) = 40

When finding the general formula for any number (n), both answers begin with the equation n2+22n, which signifies that they can be manipulated easily. Because the second answer has +40 at the end, it demonstrates that no matter what number is chosen to begin with (n), a difference of 40 will always be present.


4 x 4 Grid

Prediction

I believe that this difference of 90 should be expected to remain constant for all 4x4 number boxes. I will now investigate to check if all examples of 4x4 grid boxes demonstrate this trend in difference. I will conduct this research using another 2 of these boxes from the overall cardinal10x10 number grid.

My predication also seems to be true in the cases of the previous 2 number boxes. Although it is quite certain that this trend would be observed in all number boxes of this instance, it is necessary to find an algebraic formula to prove that the difference remains invariable.

Any 4x4 square box on the 10x10 grid can be expressed in this way:

Stage A:        Top left number x Bottom right number =        n(n+33) =        n2+33n

Stage B:        Bottom left number x Top right number =         (n+30)(n+3)=        n2+3n+30n+90

                                                                              =        n2+33n+90

Stage B – Stage A:        (n2+33n+90)-(n2+33n) = 90

When finding the general formula for any number (n), both answers begin with the equation n2+33n, which signifies that they can be manipulated easily. Because the second answer has +90 at the end, it demonstrates that no matter what number is chosen to begin with (n), a difference of 90 will always be present.


5 x 5 Grid

Prediction

I believe that this difference of 160 should be expected to remain constant for all 5x5 number boxes. I will now investigate to check if all examples of 5x5 grid boxes demonstrate this trend in difference. I will conduct this research using another 2 of these boxes from the overall cardinal10x10 number grid.

My predication also seems to be true in the cases of the previous 2 number boxes. Although it is quite certain that this trend would be observed in all number boxes of this instance, it is necessary to find an algebraic formula to prove that the difference remains invariable.

Any 5x5 square box on the 10x10 grid can be expressed in this way:

Stage A:        Top left number x Bottom right number =        n(n+44) =        n2+44n

Stage B:        Bottom left number x Top right number =         (n+40)(n+4)=        n2+4n+40n+160

                                                                              =        n2+44n+160

Stage B – Stage A:        (n2+44n+160)-(n2+44n) = 160

When finding the general formula for any number (n), both answers begin with the equation n2+44n, which signifies that they can be manipulated easily. Because the second answer has +160 at the end, it demonstrates that no matter what number is chosen to begin with (n), a difference of 160 will always be present.

Summary

Summary Table

Algebraic Summary of Square Boxes

(The numbers in bold typeface are the ones concerned with the calculations of a general formula)

It is possible to see that the numbers that are added to n (mainly in the corners of the grids) follow certain, and constant sets of rules, which demonstrates confirmation of a pattern.

As evident from the algebraic summary boxes above, the bottom right number is directly linked to the top right and bottom left numbers. It is the sum of these that equal the bottom right, or:

Formula 1:                Bottom Right (BR)  =  Top Right (TR)  +  Bottom Left (BL)

 

As also shown by the summary boxes and examples above, the formula for the top right number remains constant, and is linked with the width, w, of the box in the following way:

Formula 2:                Top Right (TR)  =  Width (w)  -  1

It is also evident from the examples calculated that the bottom left number is also linked with the height, w, (the width and height are always equal, due to the dimensions of the box producing a  square) of the box using a formula that remains constant:

Formula 3:                Bottom Left (BL)  =  (Width (w)  -  1)  x  10


The formulas stated above can be used to calculate these terms in any given table:

Calculating a General Formula Using the Term ‘n’

The formulas stated above can also be used to calculate the general terms that would appear in a box with any selected top left number (n). To obtain the top right number, it is necessary to implement formula number 2 into the generic term n, which would provide the expression n+w-1. In order to find the bottom left number, it is required to implement formula 3 into terms of n, which would create the expression n+10(w-1), which simplifies by multiplying out the brackets to n+10w-10. The bottom right number is always gained by finding the sum of the top right and bottom left, which (in algebraic terms) produces n+11w-11.

In order to find the difference in previous boxes, the difference between the product of the top left and bottom right, and the top right and bottom left was calculated. This step also needs to be taken to find the algebraic expression for the difference for a box with any width and height, w.

Stage A:        Top left number x Bottom right number         =        n(n+11w-11)

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                                                                =        n²+11nw-11n

Stage B:        Bottom left number x Top right number

                

=        (n+10w-10)(n+w-1)

                =        n²+nw-n+10nw+10w²-10w-10n-10w+10

                =        n²+11nw-11n-20w+10w²+10

Stage B – Stage A:        =        [ n²+11nw-11n-20w+10w²+10] - (n²+11nw-11n)

                        =        10w²-20w+10

The result of these calculations means that for any size square box, the difference should be easily calculated using the formula. For example, if one of the previously tested 4x4 boxes is examined:

This means that if the width (w) of 4 is inserted into the formula, the difference of 90 should be returned.

Difference (d)  =  10 x 4² - 20 x 4 + 10

Difference (d)  = ...

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