1 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
1 32 33 34 35 36 37 38 39 40
1 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
12 x 56 = 672
52 x 16 = 832
832 - 672 = 160
I shall now use letters to prove this correct
X
X+4
X+40
X+44
X(X+44)=X²+44X
(X+40)(X+4)=X²+44x+160
(X²+44x+160) - (X²+44X)=160
This proves my equation correct!
Now I shall try to answer the question "does this differ with a
different size grid?"
I shall try to fill the table shown underneath to get a clear view on
all the data and try to see a formula from this data.
Grid Width Difference
4x4 10
3x3 9
2x2 8
Square size 2x2 grid width 10
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
12 x 23 = 276
22 x 13 = 286
286 - 276 = 10
I shall now use letters to prove this correct
X
X+1
X+10
X+11
X(X+11) = X² + 11X
(X+1)(X+10) = X²+11x+10
(X²+11x+10) - (X²+11X) = 10
Square Size 2x2, Grid Width 9
1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18
19 20 21 22 23 24 21 26 27
11 x 21 = 231
20 x 12 = 240
240 - 231 = 9
I shall now use letters to prove this correct
X
X+1
X+9
X+10
X(X+10) = X² + 10X
(X+9)(X+1)=X²+10X+9
(X²+10X+9) - (X²+10X) = 9
Square size 2x2, Grid Width 8)
1 2 3 4 5 6 7 8
9 10 11 12 13 14 15 16
17 18 19 20 21 22 23 24
10 x 19 = 190
18 x 11 = 198
198 - 190 = 8
I shall now use letters to prove this correct
X
X+1
X+8
X+9
X(X+9) = X² + 9X
(X+8)(X+1) = X²+9X+8
(X²+9X+8) - (X²+9X) = 8
Square Size 3x3, Grid Width 10)
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
12 x 34 = 408
32 x 14 = 448
448 - 408 = 40
I shall now use letters to prove this correct
X
X+2
X+20
X+22
X(X+22)=X²+22X
(X+20)(X+2)=X²+22X+40
(X²+22X+40) - (X²+22X) = 40
Square size 3x3, Grid Width 9)
1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18
19 20 21 22 23 24 25 26 27
28 29 30 31 32 33 34 35 36
11 x 31 = 341
29 x 13 = 377
377 - 341 = 36
I shall now use letters to prove this correct
X
X+2
X+18
X+20
X(X+20)=X²+20X
(X+18)(X+2)=X²+20X+36
(X²+20x+36) - (X²+20)= 36
Square Size 3x3, Grid size 8)
1 2 3 4 5 6 7 8
9 10 11 12 13 14 15 16
17 18 19 20 21 22 23 24
25 26 27 28 29 30 31 32
10 x 28 = 280
26 x 12 = 312
312 - 280 = 32
I shall now use letters to prove this correct
X
X+2
X+16
X+18
X(X+18)=X²+18X
(X+16)(X+2)=X²+18X+36
(X²+18X+36) - (X²+18X)=36
Square Size 4x4, Grid Size 10)
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
12 x 45 = 540
42 x 15 = 630
630 - 540 = 90
I shall now use letters to prove this correct
X
X+3
X+30
X+33
X(X+33)=X²+33X
(X+30)(X+3)=X²+33X+90
(X²+33X+90) - (X²+33X)=90
Square Size 4x4, Grid Size 9)
1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18
19 20 21 22 23 24 25 26 27
28 29 30 31 32 33 34 35 36
37 38 39 40 41 42 43 44 45
11 x 41 = 451
38 x 14 = 532
532 - 451 = 81
I shall now use letters to prove this correct
X
X+3
X+27
X+30
X(X+30)=X²+30X
(X+27)(X+3)=X²+30X+81
(X²+30X+81) - (X²+30X) = 81
Square Size 4x4, Grid Size 8)
1 2 3 4 5 6 7 8
9 10 11 12 13 14 15 16
17 18 19 20 21 22 23 24
25 26 27 28 29 30 31 32
33 34 35 36 37 38 39 40
10 x 37 = 370
34 x 13 = 442
442 - 370 = 72
I shall now use letters to prove this correct
X
X+3
X+24
X+27
X(X+27)=X²+27X
(X+24)(X+3)=X²+27X+72
(X²+27X+72) - (x²+27X)=72
I now have enough data to fill the table and look for an equation.
2x2 Grid Width Difference
10 10
9 9
8 8
3x3 Grid Width Difference
10 40
9 36
8 32
4x4 Grid Width Difference
10 90
9 81
8 72
I can see from these results that the width or height of the square
minus 1 and then squared and multiplied by the grid width gives me the
difference.
(S-1)² x W = D
To prove this equation correct I will do a different square and grid
size and see if the equations answers and the practical answer are
alike.
Square Size 5x5, Grid Width 7)
(S-1)² x W = D
(5-1)² x 7 = D
4² x 7 = D
16 x 7 = D
112 = D
1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31 32 33 34 35
36 37 38 39 40 41 42
9 x 41 = 369
37 x 13 = 481
481 - 369 = 112
I shall now use letters to prove this correct
X
X+4
X+28
X+32
X(X+32)=X²+32X
(X+28)(X+4)=X²+32X+112
(X²+32X+112) - (X²+32X) = 112
This proves my equation correct, as the answers are the same.
Now I shall try to find an equation that works with rectangles, to do
this I will substitute all 4 numbers used with only letters and see
what each of these letters relate to.
e.g. Instead of:
X
X+4
X+28
X+32
It will be:
X
X+Y
X+Z
X+(Y+Z)
The bottom right corner is as it is because I have noticed that Y + Z
is always equal to the extra addition to X in that corner.
Looking at my previous diagrams it is noticeable the Y is equal to the
width of the square minus 1 and that Z is equal to the width of the
grid multiplied by the squares heights after taking 1 away from it.
X = top left number for square
Y = width of square - 1
Z = Width of Grid x (height of square - 1)
D = Difference
Now that the height and width of the square are represented by
different letters I can use rectangles in the grid. I have also
noticed that the bottom left corner multiplied by the top right is
always the bigger of the two numbers. Because of this an equation can
easily be formed.
[(X+Z)(X+Y)] - [X(X+{Y+Z})] = D
Test: This test should prove my equation correct. I will use a
rectangle of height 4 and width 3 and a grid with a width of 6 and I
shall start at number 8 in the grid.
So: X = 8
Y = 2
Z = 18
[(X+Z)(X+Y)] - [X(X+{Y+Z})] = D
[(8+18)(8+2)] - [8(8+{2+18})] = D
[(8+18)(8+2)] - [8(8+20)] = D
[64+16+144+36] - [64+160] = D
260 - 224 = D
36 = D
1 2 3 4 5 6
7 8 9 10 11 12
13 14 15 16 17 18
19 20 21 22 23 24
25 26 27 28 29 30
8 x 28 = 244
26 x 10 = 260
260 - 244 = 36
X
X+2
X+18
X+20
X(X+20)=X²+20X
(X+18)(X+2)=X²+20X+36
(X²+20X+36) - (X²+20X) = 36
This proves the formula correct! However, the formula is very long
therefore it takes a long time to do it, I shall multiply out the
brackets and cancel it down to see if it reduces the length.
[(X+Z)(X+Y)] - [X(X+{Y+Z})] = D
[X²+XY+ZX+ZY] - [X²+X(Y+Z)] = D
[IMAGE]
[IMAGE][X²+XY+ZX+ZY] - [X²+XY+XZ] = D
[IMAGE]
ZY=D
As you can see cancelling down the equation made it much simpler as
all I need to do now is multiply 2 numbers together. I shall now check
to see if I have made any mistakes while cancelling down the equation
by doing the same test as before using the same size grid (6) and the
same size box (4 high, 3 wide).
The answer to this is 36 as I have already worked this out previously.
X=2
Y=2
Z=18
ZY=D
2x18=D
36=D
The equation is correct this makes it very quick and easy to work out
the answer for any size box on any grid widths without needing to draw
anything.
Multiplication Tables:
If I change the multiplication table of the grid, 1 being the default,
does it change the difference and can I find a formula to work with
all my factors?
To start with I shall keep the grid width the same (10) but change it
when I have a formula. I shall aim to fill the table below.
Multiplication Table Box Size
1 2x2
2 3x3
3 4x4
Box Size 2x2, Multiplication Table 1)
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
12 x 23 = 276
22 x 13 = 286
286 - 276=10
I shall now use letters to prove this correct
X
X+1
X+10
X+11
X(X+11) = X² + 11X
(X+1)(X+10) = X²+11X+10
(X²+11X+10) - (X²+11X) = 10
Box Size 2x2, Multiplication Table 2)
2 4 6 8 10 12 14 16 18 20
22 24 26 28 30 32 34 36 38 40
42 44 46 48 50 52 54 56 58 60
24 x 46 = 1104
44 x 26 = 1144
1144 - 1104 = 40
I shall now use letters to prove this correct
X
X+2
X+20
X+22
X(X+22) = X² + 22X
(X+2)(X+20) = X²+22X+40
(X²+22X+40) - (X²+22X) = 40
Box Size 2x2, Multiplication Table 3)
3 6 9 12 15 18 21 24 27 30
33 36 39 42 45 48 51 54 57 60
63 66 69 72 75 78 81 84 87 90
36 x 69 = 2484
66 x 39 = 2574
2574 - 2484 = 90
I shall now use letters to prove this correct
X
X+3
X+30
X+33
X(X+33)=X²+33X
(X+30)(X+3)=X²+33X+90
(X²+33x+90) - (X²+33X)= 90
Box Size 3x3, Multiplication Table 1)
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
12 x 34 = 408
32 x 14 = 448
448 - 408 = 40
I shall now use letters to prove this correct
X
X+2
X+20
X+22
X(X+22)=X²+22X
(X+20)(X+2)=X²+22X+40
(X²+22X+40) - (X²+22X) = 40
Box Size 3x3, Multiplication Table 2)
2 4 6 8 10 12 14 16 18 20
22 24 26 28 30 32 34 36 38 40
42 44 46 48 50 52 54 56 58 60
62 64 66 68 70 72 74 76 78 80
24 x 68 = 1632
64 x 28 = 1792
1792 - 1632 = 160
I shall now use letters to prove this correct
X
X+4
X+40
X+44
X(X+44)=X²+44X
(X+40)(X+4)=X²+44X+160
(X²+44X+160) - (X²+44X) = 160
Box Size 3x3, Multiplication Table 3)
3 6 9 12 15 18 21 24 27 30
33 36 39 42 45 48 51 54 57 60
63 66 69 72 75 78 81 84 87 90
93 96 99 102 105 108 111 114 117 120
36 x 102 = 3672
96 x 42 = 4032
4032 - 3672 = 360
I shall now use letters to prove this correct
X
X+6
X+60
X+66
X(X+66)=X²+66X
(X+60)(X+6)=X²+66X+360
(X²+66X+360) - (X²+66X)= 360
Box Size 4x4, Multiplication Table 1)
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
12 x 45 = 540
42 x 15 = 630
630 - 540 = 90
I shall now use letters to prove this correct
X
X+3
X+30
X+33
X(X+33)=X²+33X
(X+30)(X+3)=X²+33x+90
(X²+33X+90) - (X²+33X) = 90
Box Size 4x4, Multiplication Table 2)
2 4 6 8 10 12 14 16 18 20
22 24 26 28 30 32 34 36 38 40
42 44 46 48 50 52 54 56 58 60
62 64 66 68 70 72 74 76 78 80
82 84 86 88 90 92 94 96 98 100
24 x 90 = 2160
84 x 30 = 2520
2320 - 2160 = 360
I shall now use letters to prove this correct
X
X+6
X+60
X+66
X(X+66)=X²+66X
(X+60)(X+6)=X²+66X+360
(X²+66X+360) - (X²+66X) = 360
Box Size 4x4, Multiplication Table 3)
3 6 9 12 15 18 21 24 27 30
33 36 39 42 45 48 51 54 57 60
63 66 69 72 75 78 81 84 87 90
93 96 99 102 105 108 111 114 117 120
123 126 129 132 135 138 141 144 147 150
36 x 135 = 4860
126 x 45 = 5670
5670 - 4860 = 810
I shall now use letters to prove this correct
X
X+9
X+90
X+99
X(X+99)=X²+99X
(X+90)(X+9)=X²+99X+810
(X²+99X+810) - (X²+99X) = 810
I have now gathered enough data to put in my table and look for a
formula.
Multiplication Table Box Size
1, 2, 3 2x2
10, 40, 90 3x3
40, 160, 360 4x4
From looking at this I can see a pattern quite easily. To start with
the most obvious, they are all multiples of 10 which may be to do with
the grid size and so implies that within the equation you will
probably have to multiply the grid size at some point. Secondly, if
they are all divided by ten they are all square numbers. This makes it
likely that the last two "commands" in the equations are to square a
number and multiply it by the grid size. Thirdly they are all
multiples of the multiplication tables, suggesting you need to
multiply it by the multiplication table at some point.
M = Multiplication Table
X = top left number for square
Y = width of square - 1
Z = Width of Grid x (height of square - 1)
D = Difference
X
X + MY
X + MZ
X + (MY+ MZ)
If this is true then: [(X + MZ)(X + MY)] - [X(X + {MY + MZ})] = D
I shall now cancel down the equation.
[(X + MZ)(X + MY)] - [X(X + {MY + MZ})] = D
[X²+XMY + MZX +ZYM²] - [X² + X(MY +MZ)] = D
[IMAGE] [IMAGE]
[IMAGE][X²+XMY + MZX +ZYM²] - [X² + XMY + XMZ] = D
ZYM² = D
Test: I shall test to see if my formula is right by using both the
formula and the practical way of figuring out the difference. The box
size, grid width and Multiplication table will be as follows: Box
Height: 3
Box Width: 4
Grid Width: 8
Multiplication Table: 4
X: 40
ZYM² = D
16 x 3 x 4² = D
16 x 3 x 16 = D
768 = D
4 8 12 16 20 24 28 32
36 40 44 48 52 56 60 64
68 72 76 80 84 88 92 96
100 104 108 112 116 120 124 128
40 x 116 = 4640
104 x 52 = 5408
5408 - 4640 = 768
I shall now use letters to prove this correct.
X
X+12
X+64
X+76
X(X+76)=X²+76X
(X+64)(X+12)=X²+76X+768
(X²+76X+768) - (X²+76X) = 768
This proves my formula correct!