Z² + 44Z + 160 - Z² + 44Z = 160
Table
The Square formula for a 10X10 grid
X times X square
X
X
2 X 2 square
Z = top left number = 12 (in this case)
Z+(x-1) = 13
X = 2 – it is the size of the square, therefore
Z+ (X-1) = 13 (top right number)
12+ (2-1) = 13
Z + 10(X-1) = 22 (bottom left number)
12 +10(2-1) = 22
Z+11(X-1) = 23 (bottom right number)
12+11(2-1) = 23
Z + 11 (X-1) = Z + 11X – 11 (bottom right number)
Z (Z +11X -11) = Z² + 11XZ – 11Z
Z + 10(X-1) = Z + 10X – 10 (bottom left number)
(Z + X-1) (Z + 10X – 10) = Z² + 10XZ – 10Z + XZ + 10X² - 10X – Z – 10X +10
= Z² + 11XZ – 11Z + 10X² - 20X + 10
(Z² + 11XZ – 11Z + 10X² - 20X + 10) – (Z² + 11XZ – 11Z)
= 10X² - 20X + 10
10X² - 20X + 10 = 10 (X-1)²
10X² - 20X + 10 ÷ 10 = X²- 2X + 1
X² - 2X + 1 = (X-1) (X+1)
10 (X-1)² = 10X² - 20X + 10
The formula for a square on a 10 X 10 grid is 10(X-1)²
To get the formula 10 (X-1)² I multiplied the corners of the square algebraically then I took away the terms, ending up with
10X² - 20X + 10 which factorises to give 10 (X-1)²
Ex 6
5 X 5 square = 160
10 (5-1)² = 160
4 X 4 square = 90
10 (4-1)² = 90
This formula works for squares on a 10 X 10 grid
Rectangles
Ex 7
2 X 3 rectangle on a 10 X 10 grid
1 X 13 = 13
3 X 11 = 33
33-13 = 20
Z (Z+12) = Z² +1 2Z
(Z+2) (Z+10) = Z² +10Z + 2Z +20
= Z² + 12Z + 20
Z² +12Z + 20 - Z² +1 2Z = 20
Ex 8
2 X 4 rectangle on 10 X 10 grid
25 X 28 = 950
28 X 35 = 980
980 – 950 = 30
Z (Z+13) = Z² + 13Z
(Z+3) (Z+10) = Z² +10Z +3Z +30
= Z² +13Z +30
Z² +13Z +30 - Z² + 13Z = 30
Ex 9
3 X 4 rectangle on a 10 X 10 grid
71 X 94 = 6674
91 X 74 = 6734
6734 – 6674 = 60
Z (Z+23) = Z² +23Z
(Z+3) (Z+20) = Z² + 20Z +3Z +60
= Z² + 23Z +60
Z² + 23Z +60 - Z² +23Z = 60
Ex 10
3 X 5 rectangle on a 10 X 10 grid
41 X 65 = 2665
45 X 61 = 2745
2745 – 2665 = 80
Z (Z+24) = Z² +24Z
(Z+4) (Z+20) = Z² +20Z +4Z +80
= Z² + 24Z +80
Z² + 24Z +80 - Z² +24Z = 80
Rectangle formula for a 10 X 10 grid
Y = 3
X = 2
Z = 1 (in this case, it is the top left number)
Z+(Y-1) = 3 (top right number)
Z+ (3-1) =3
Z-10 +10X = 11 (bottom left number)
Z-10 +10 x 2 = 11
Z-10+10X+Y-1 = 13 (bottom right number)
Z-10 +10 x 2 + 3 – 1 = 13
Z (Z-10+10X+Y-1) = Z (Z-11+10X+Y)
= Z² - 11Z + 10XZ +YZ
(Z-10+10X) (Z+Y-1) = Z² + YZ –Z -10Z – 10Y +10 +10XZ +10XY -10X
Z² + YZ –Z -10Z – 10Y +10 +10XZ +10XY -10X - Z² - 11Z + 10XZ +YZ
= -10Y +10 +10XY -10X
Formula = 10(Y-1) (X-1)
-10Y +10 +10XY – 10X ÷ 10 = -Y +1 +XY –X
-Y +1 +XY –X
(X-1) (Y-1)
10 (X-1) (Y-1) = -10Y +10 +10XY -10X
I worked out the formula 10(X-1) (Y-1) by multiplying
Z (Z-11+10X+Y) + (Z-10+10X) (Z+Y-1)
Then took them away from eachother.
Z² + YZ –Z -10Z – 10Y +10 +10XZ +10XY -10X - Z² - 11Z + 10XZ +YZ
To get -10Y +10 +10XY – 10X
Which factorises to give 10(Y-1) ( X-1)
Ex 11
3 X 5 rectangle on 10 X 10 grid = 80
10(Y-1) (X-1)
10(5-1) (3-1) = 80
2 X 4 rectangle on 10 X 10 grid = 30
10(Y-1) (X-1)
10 (4-1) (2-1) = 30
The formula 10(Y-1) (X-1) works for rectangles on a 10 X 10 grid
Rectangle formula for a rectangle on any size number grid
The formula for any size rectangle on a 10X10 grid is 10(Y-1) (X-1) therefore to find out the formula for a rectangle on any size number grid we simply have to increase or decrease the number but this would not be a formula.
9X9 number grid
The formula for this grid would be 9(Y-1) (X-1)
EX 12
Y =3
X =2
1 X 12 = 12
3 X 10 = 30
30 – 12 = 18
9(Y-1) (X-1)
9(3-1) (2-1) =18
General formula for a rectangle on any size grid explanation
W
W
W represents the size of number grid
X and Y represent the sides of the rectangle
The general formula for any rectangle on any number grid = W(Y-1) X-1)
Ex 13
1 X 12 = 12
3 X 10 = 30
30 – 12 = 18
Z = top left number (1 in this case)
Z+2 = 3 (top right number) 1+2 = 3
Z+9 = 10 (bottom left number) 1+9 = 10
Z+11= 12 (bottom right number) 1+11 = 12
Z (Z+11) = Z² + 11Z
(Z+9) (Z+2) = Z² + 2Z + 9Z + 18
= Z² + 11Z + 18
Z² + 11Z + 18 - Z² + 11Z = 18
9(Y-1) (X-1)
9(3-1) (2-1) =18
Ex 14
7X7
4X2
4X14 = 56
7X11 = 77
77-56 = 21
Z = 4 (in this case)
Z+3 = 7 (top right number) 4 + 3 = 7
Z+7 = 11 (bottom left number) 4 + 7 = 11
Z+10 = 14 (bottom right number) 4 + 10 = 14
Z (Z+10) = Z² + 10Z
(Z+3) (Z+7) = Z² + 7Z +3Z +21
= Z² + 10Z + 21
Z² + 10Z + 21 - Z² + 10Z = 21
Ex 15
5X5
4 X 3 rectangle
1 X 14 = 14
4 X 11 = 44
44-14 = 30
Z=1 (in this case) top left number
Z+3 = 4 (top right number) 1+3 = 4
Z+10 = 11 (bottom left number) 1+10 = 11
Z+13 = 14 (bottom right number) 1+13= 14
Z (Z+13) = Z² + 13Z
(Z+3) (Z+10) = Z² + 10Z + 3Z+ 30
= Z² + 13Z + 30
Z² +13Z + 30 - Z² + 13Z = 30
W(Y-1) (X-1)
5(4-1) (3-1) = 30
Table
Working out for the rectangle formula for any size grid
W
W
10 X 10 grid
3 X 2
W= 10 (this is the size of the grid – 10 X 10 in this case)
Y = 3 (the width of the rectangle – 3 in this case)
X = 2 (the height of the rectangle – 2 in this case)
Z = 1 (in this case)
Z+Y-1 = 3 (1+ 3 -1 = 3)
Z-W+10X = 11 (1-10+20 = 11)
Z-W+10X+Y-1 = 13 (1-10+20+3-1 = 13)
Z (Z-W+10X+Y-1) = Z² - ZW + 10XZ + ZY – Z
(Z-W+10X) (Z+Y-1) = Z² + ZY – Z – ZW – YW – W + 10XZ + 10 XY – 10 X
(Z² + ZY – Z – ZW – YW – W + 10XZ + 10 XY – 10 X) – (Z² - ZW + 10XZ + ZY – Z)
= -WY – W + 10XY – 10X
-WY – W + 10XY – 10X ÷ 10 = -WY – W + XY – X
Which factorises to give W(Y-1) (X-1)
The formula at work
4 X 3 rectangle on a 5 X 5 grid = 30
5X5
4 X 3 rectangle
1 X 14 = 14
4 X 11 = 44
44-14 = 30
Z=1 (in this case) top left number
Z+3 = 4 (top right number) 1+3 = 4
Z+10 = 11 (bottom left number) 1+10 = 11
Z+13 = 14 (bottom right number) 1+13= 14
Z (Z+13) = Z² + 13Z
(Z+3) (Z+10) = Z² + 10Z + 3Z+ 30
= Z² + 13Z + 30
Z² +13Z + 30 - Z² + 13Z = 30
W(Y-1) (X-1)
5(4-1) (3-1) = 30
7X7
4X2
4X14 = 56
7X11 = 77
77-56 = 21
Z = 4 (in this case)
Z+3 = 7 (top right number) 4 + 3 = 7
Z+7 = 11 (bottom left number) 4 + 7 = 11
Z+10 = 14 (bottom right number) 4 + 10 = 14
Z (Z+10) = Z² + 10Z
(Z+3) (Z+7) = Z² + 7Z +3Z +21
= Z² + 10Z + 21
Z² + 10Z + 21 - Z² + 10Z = 21
W(Y-1) (X-1)
7(4-1) (2-1) = 21
Further investigation
Further steps could be taken in the investigation process, unfortunately time is not on our side. For example the shape of the grid could have been altered to a rectangle and then a formula for a certain rectangle could be found, then a formula for any rectangle could be found.
