If this Formula is correct it should allow us to work out what the difference of a 5x5 square is:
D = (5-1)² x 5
16 x 5
80
If the formula I have worked out is correct then the difference for my 5x5 square should be 80. I will now do the only 5x5 square I can do to show it is correct.
5x5
21 x 5 = 105
1 x 25 = 25
Difference = 80
The Difference I predicted for my 5x5 square was correct so I have shown that my formula is correct and works for the 5x5 square.
Squares - 10x10 grid
2x2
21 x 12 = 252
11 x 22 = 242
Difference = 10
25 x 16 = 400
15 x 26 = 390
Difference = 10
87 x 78 = 6786
77 x 88 = 6776
Difference = 10
Formula
A = (X + 10) (X + 1)
= X² + X + 10X + 10
= X² + 11X + 10
B = X (X + 11)
= X² + 11X
Difference = A – B
= (X² + 11X + 10) – (X² + 11)
= 10
3x3
42 x 24 = 1008
22 x 44 = 968
Difference = 40
82 x 64 = 5248
62 x 84 = 5208
Difference = 40
37 x 19 = 703
17 x 39 = 663
Difference = 40
Formula
A = (X + 20) (X + 2)
= X² + 2X + 20X + 40
= X² + 22X + 40
B = X (X + 22)
= X² + 22X
Difference = A – B
= (X² + 22X + 40) – (X² + 22X)
= 40
4x4
63 x 36 = 2268
33 x 66 = 2178
Difference = 90
31 x 4 =124
1 x 34 = 34
Difference = 90
86 x 59 = 5074
56 x 89 = 4984
Difference = 90
Formula
A = (X + 30) (X + 3)
= X² = 3X + 30X +90
= X² + 33X + 90
B = X (X + 33)
= X² + 33X
Difference = A – B
= (X² + 33X + 90) – (X² + 33X)
= 90
5x5
85 x 49 = 4165
45 x 89 = 4005
Difference = 160
43 x 7 = 301
3 x 47 = 141
Difference = 160
82 x 46 = 3772
42 x 86 = 3612
Difference = 160
Formula
A = (X + 40) (X + 4)
= X² + 4X + 40X + 160
= X² + 44X + 160
B = X (X + 44)
= X² + 44X
Difference = A – B
= (X² + 44X + 160) – (X² + 44X)
=160
10x10 grid - square formula
These are the differences from my different squares on the 5x5 grid:
If we look at the difference’s we notice that they all end in zero’s so we can summarise that the formula involves a multiplication of 10. We also know that the grid is 10x10 so if G = Grid Size (Which is 10 NOT 100). Therefore if D = Difference and G = Grid Size we can make the following formula:
D = (Square size - 1)² x G
This table will show the formula in action to show how it produces the formula:
If this Formula is correct it should allow us to work out what the difference of a 6x6 square is:
D = (6-1)² x 10
25 x 10
250
Therefore, if this formula is correct the difference of a 6x6 square would be 250. I will now do 2 6x6 square to show the formula is correct.
6x6
62 x 17 = 1054
12 x 67 = 804
Difference = 250
95 x 50 = 4750
45 x 100 = 4500
Difference = 250
Both the Differences for my 6x6 matched my prediction so we can summarise that the formula is correct.
Rectangle – 5x5 Grid
3x2
6 x 13 = 78
11 x 8 = 88
Difference = 10
18 x 25 = 450
23 x 20 = 460
Difference = 10
8 x 15 = 120
13 x 10 = 130
Difference = 10
Formula
A = (X +5) (X + 2)
= X² + 2X + 5X + 10
= X² + 7X + 10
B = X (X + 7)
= X² + 7X
Difference = A – B
= (X² + 7X + 10) – (X² + 7X)
= 10
4x3
11 x 24 = 264
21 x 14 = 294
Difference = 30
2 x 15 = 30
12 x 5 = 60
Difference = 30
12 x 25 = 300
22 x 15 = 330
Difference = 30
Formula
A = (X + 10) (X + 3)
= X² + 3X + 10X + 30
= X² + 13X + 30
B = X (X + 13)
= X² + 13X
Difference = A – B
= (X² + 13X + 30) – (X² + 13X)
= 30
5x5 Rectangle Formula
These are my differences from my different rectangles on the 5x5 grid:
From looking at the differences we can see that each is a multiple of 5 so once again 5 must be involved in the formula. 5 is the size of the grid so if G = Grid Size and D = the difference we are seeking we can make this formula:
D = G x (length of rectangle – 1) (height of rectangle – 1)
The following table will show that this formula works:
To prove this formula is correct if will use the formula to work out the difference for a 5 x 4 rectangle then do it to show it is correct.
D = (5 – 1) (4 – 1) x 5
= 4 x 3 x 5
= 60
If the formula is correct the difference for the rectangle I am about to do should be 60.
6 x 25 = 150
21 x 10 = 210
Difference = 60
The difference I predicted was correct which means my formula is correct.
Rectangles – 10x10 Grid
3x2
23 x 35 = 805
33 x 25 = 825
Difference = 20
83 x 95 = 7885
93 x 85 = 7905
Difference = 20
7 x 19 = 133
17 x 9 = 153
Formula
A = (X + 10) (X + 2)
= X² + 2X + 10X + 20
= X² + 12X + 20
B = X (X + 12)
= X² + 12X
Difference = A – B
= (X² + 12X + 20) – (X² + 12X)
= 20
4X3
23 x 46 = 1058
43 x 26 = 1118
Difference = 60
1 x 24 = 24
21 x 4 = 84
Difference 60
7 x 30 = 210
27 x 10 = 270
Difference 60
Formula
A = (X + 20) (X + 3)
= X² + 3X + 20X + 60
= X² + 23X + 60
B = X (X + 23)
= X² + 23X
Difference = A – B
= (X² + 23X + 60) – (X² + 23X)
= 60
5x4
61 x 95 = 5795
91 x 65 = 5915
Difference = 120
66 x 100 = 6600
96 x 70 = 6720
Difference = 120
6 x 40 = 240
36 x 10 = 360
Difference = 120
Formula
A = (X + 30) (X + 4)
= X² + 4X + 30X + 120
= X² + 34X + 120
B = X (X + 34)
= X² + 34X
Difference = A – B
= (X² + 34X + 120) – (X² + 34X)
= 120
10x10 Grid – Rectangle Formula
These are the differences from my rectangles on my 10x10 Grid:
Looking at the formula we again see that all the numbers are multiples of 10. All multiples of 10 are multiples of 5 so we can try the same formula used for rectangles of a 5x5 grid and adapt it were appropriate. The formula I used for the 5x5 Grid is:
D = G x (length of rectangle – 1) (height of rectangle – 1)
D = Difference and G = Grid Size so out new formula is:
D = 10 x (length of rectangle – 1) (height of rectangle – 1)
This table will show how my formula matches my results:
If this formula is correct then I can predict a 6x5 rectangle with 100% accuracy:
D = 10 x (6 -1) (5 – 1)
= 10 x 5 x 4
= 200
If this formula is correct than the difference of a 6x5 rectangle on a 10x10 grid should be 200.
13 x 58 = 754
53 x 18 = 954
Difference = 200
My prediction for what the difference was correct so we can summarize that the formula is correct.
10x10 Grid – Rhombus
2x2
32 x 24 = 768
23 x 33 = 759
Difference = 9
87 x 79 = 6873
78 x 88= 6864
Difference = 9
11 x 3 = 33
2 x 12 = 24
Difference = 9
Formula
A = (X + 9) (X + 1)
= X² + X + 9X + 9
= X² +10X + 9
B = X (X + 10)
= X² + 10X
Difference = A – B
= (X² + 10X + 9) – (X² + 10X)
= 9
3x3
24 x 8 = 192
6 x 26 = 156
Difference = 36
21 x 5 = 105
3 x 23 = 69
Difference = 36
82 x 66 = 5412
64 x 84 = 5376
Difference = 36
Formula
A = (X + 18) (X + 2)
= X² + 2X + 18X + 36
= X² +20X + 36
B = X (X + 20)
= X² + 20X
Difference = A – B
= (X² + 20X + 36) – (X² + 20X)
= 36
4x4
31 x 7 = 217
4 x 34 = 136
Difference = 81
34 x 10 = 340
37 x 7 = 259
Difference = 81
41 x 17 = 697
14 x 44 = 616
Difference = 81
Formula
A = (X + 27) (X + 3)
= X² + 3X + 27X + 81
= X² +30X + 81
B = X (X + 30)
= X² + 30X
Difference = A – B
= (X² + 30X + 81) – (X² + 30X)
= 81
10x10 Grid – Rhombus Formula
These are the differences from my results:
Looking at the differences we notice that they are all multiples of 9 so we can assume 9 is part of the formula. In this formula we can use D for difference and N will be the size of the rhombus. E.g. if the rhombus is 4x4 then N will be 4.
The formula I have found out for the rhombus is:
D = 9 (n – 1)²
This table shows the formula working:
To prove that the formula is correct I will now predict the difference for a 5x5 rhombus on a 10x10 grid now and then do the shape and show that they match up.
D = 9 (5 – 1)²
= 9 x 16
= 144
If my formula is correct then when I do the 5x5 rhombus the difference will be 144.
41 x 9 = 369
5 x 45 = 225
Difference = 144
The difference I predicted was correct so I can successfully summarize that the formula I have found is correct.
