Opposite Corners

Prove formula works

Investigate rectangles(with same method).

Change grid size.

Squares:

I started by calculating the differences for 2x2-4x4 boxes and then predicting the difference for a 5x5.

2x2-         1x12=12        2x13=26        3x14=42        4x15=60        5x16=80

        2x11=22        3x12=36        4x13=52        5x14=70        6x15=90

(diff.)10                10                10                10                10

3x3-         1x23=23        2x24=48        3x25=75

        3x21=63        4x22=88        5x23=115

        (diff.)40                40                40

4x4-        1x34=34        2x35=70        3x36=108

        4x31=124        5x32=160        6x33=198

        (diff.)90                90                90

To make a prediction for a 5x5 I saw that the existing differences were firstly multiples of 10. Then if you divide them by 10 they are square numbers(1,4,9). The next logical number in the sequence would be 160 as it is a multiple of 10 and a square number when divided by 10.

To test my prediction I calculated two box locations on the 10x10 grid.

5x5-        1x45=45        2x46=92

        5x41=205        6x42=252

        (diff.)160                160

To prove using algebra why the difference is always the same for squares I used this diagram to represent the grid.

n        n+1

n+10        n+11

n(n+11)= n²+11n

(n+10)(n+1)=n²+10n+n

n²+11n+10

n²+11n

10

To find a formula for any square on a 10x10 grid I created this table from the parts above:

Looking at the last column I noticed that the numbers were 1 less than the dimension size in the 1st column.

Working backwards I took 1 dimension(d) then subtracted 1, squared it then multiplied by 10 to give this formula:

(d-1)²x10

I then proved my formula worked by testing it with a 3x3 box.

(3-1)²x10

2²x10

4x10=40

Rectangles:

As for squares I firstly calculated the differences for 4 different sized rectangles.

2x3-        1x13=13        2x14=28        

        3x11=33        4x12=48        

20. 20

2x4-        1x14=14        2x15=30

        4x11=44        5x12=60

30. 30

3x4-        1x24=24        2x25=50

        4x21-84        5x22=110

60. 60

3x5-        1x25=25        2x26=52

        5x21-105        6x22=132

                80                80

The first pattern I noticed was that all the differences were multiples of 10.

To prove that the difference is always the same for a rectangle I used algebra in the same way as for squares.

n        n+1        n+2

n+10        n+11        n+12

n(n+12)=n²+12n

(n+10)(n+2)=n²+12n+20

n²+12n

n²+12n+20

        20

I created this table for rectangles:

I worked out from the table and other evidence the formula:

(w-1)(l-1)x10

w=width        l=length

Grid Size:

I started to try and find a formula for any grid size for a square and rectangle.

I saw that in both formulae there was a x10 at the end. I thought that this may be the grid size.

I tested my prediction with a 3x3 box and a 2x3 rectangle on a 20x20 grid.

3x3-        (3-1)²x20                                        2x3-        (2-1)(3-1)x20

        2²X20                                                1x2x20

        4x20=80                                                2x20=40

Actual Differences:

3x3-        1x43=43                                        2x3-        1x42=42

        3x41=123                                                2x41=82

                80                                                        40