The answer for the 4*4 squares are the same, just as the 2*2 and 3*3 squares had the same difference. I think that any 4*4 square would have a difference of 90. To prove this I will use algebra.
z-number in the top left corner
z z+1 z+2 z+3 z(z+33)=z²+33z
z+10z+11 z+12z+13 (z+3)(z+30)=z²+33z+90
z+20z+21 z+22z+23 (z²+33z+90)-(z²+33z)=90
z+30z+31 z+32z+33 Difference = 90
This proves that with any 4*4 square, the opposite corners multiplied then the answers subtracted = 90
I am now going to further my investigations by one step. I am going to use a 5*5 square and do the same as I did for 2*2, 3*3 and 4*4 squares.
1 2 3 4 5 1 5
11 12 13 14 15 * 45 * 41
21 22 23 24 25 45 205
31 32 33 34 35 205-45=160
41 42 43 44 45 Difference = 90
Opposite corners
6 7 8 9 10 6 10
16 17 18 19 20 * 50 * 46
26 27 28 29 30 300 460
36 37 38 39 40 460-300=160
46 47 48 49 50 Difference = 90
The answer for the 5*5 squares are the same, just as the 2*2, 3*3 and 4*4 squares had the same difference. I think that any 5*5 square would have a difference of 160. To prove this I will use algebra.
z z+1 z+2 z+3 z+4 z(z+44)=z²+44z
z+10 z+11 z+12z+13 z+14 (z+4)(z+40)= z²+44z+160
z+20 z+21z+22 z+23 z+24 (z²+44z+160)-(z²+44z)=160
z+30 z+31z+32 z+33 z+34 Difference = 160
z+40 z+41z+42 z+43 z+44
This proves that with any 5*5 square the opposite corners multiplied then subtracted from each other = 160
I will now put the results from my investigations in a table, and comment on them.
Square Size Difference of the corners multiplied then subtracted
2*2 10
3*3 40
4*4 90
5*5 160
I have noticed that my results are:
-
Multiples of 2,5 and 10.
- They are square numbers.
- They are one less than the size of the square.
From my results I can predict that the difference for a 6*6 square will be,
6*6 (6-1)²*10=250
Opposite corners
1 2 3 4 5 6 1 6
11 12 13 14 15 16 * 56 * 51
21 22 23 24 25 26 56 306
31 32 33 34 35 36 306-56=250
41 42 43 44 45 46 Difference
51 52 53 54 55 56
This proves my prediction is correct. Through my investigations I can say that the general rule for finding the difference is,
10(m-1)²
m-size of the square
m 10
m
Now I am going to prove my general
formula.
10
z-number in the top left corner. m-length of the square
z z+(m+1)
z(z+M)=z²+12z
(m+1)=M m (z+M)(z+10M)=y²+11Mz+10m²
(z²+11Mz+10m²)-(z²+11My)=10m²
z+10(m+1) z+10(m+1) 10(m-1)²
This proves that my general formula for finding the difference of any square in a 10*10 grid is correct.
Now I am going to further my investigation by investigating with a rectangle instead of a square. I will start with algebra to find the formula for the difference straight away.
z-number in the top left corner
z z+1 z+2 z(z+12)=z²+12z
z+10 z+11 z+12 (z+2)(z+10)=z²+12z+20
(z²+12z+20)-(z²+12z)=20
Difference
Opposite corner
This proves that with any 2*3 rectangular box the difference of the corners multiplied then subtracted always = 20.
Now I am going to further my investigation by investigating a 3*4 rectangular box. Again I will start with algebra.
z-number in the top left corner.
z z+1 z+2 z+3 z(z+23)=z²+23z
z+10 z+11 z+12 z+13 (z+3)(z+20)=z²+23z+60
z+20 z+21 z+22 z+23 (z²+23z+60)-(z²+23z)=60
Difference
This proves that with any 3*4 rectangular box the difference is always 60. I predict for a 4*5 rectangular box the difference of the corners multiplied then subtracted =120.
Now I am going to further my investigation by moving onto a 4*5 rectangular box. I will start with algebra.
z-number in the top left corner
z z+1 z+2 z+3 z+4 z(z+34)=z²+34z
z+10 z+11 z+12 z+13 z+14 (z+4)(z+30)=z²+34z+120
z+20 z+21 z+22 z+23 z+24 (z²+34z+120)(z²+34z)=120
z+30 z+31 z+32 z+33 z+34 Difference
This proves that my prediction is correct. This also proves that the difference of a 4*5 rectangular box = 120
I predict for a 5*6 rectangular box the difference will equal 200. Now I will investigate a 5*6 box to prove my prediction.
z-number in the top left corner.
z z+1 z+2 z+3 z+4 z+5 z(z+45)=z²+45z
z+10 z+11 z+12 z+13 z+14 z+15 (z+5)(z+40)=z²+45z+200
z+20 z+21 z+22 z+23 z+24 z+25 (z²+45z+200)-(z²+45z)=200
z+30 z+31 z+32 z+33 z+34 z+35 Difference
z+40 z+41 z+42 z+43 z+44 z+45
Opposite corners
This proves that my prediction is correct. This also proves that with any 5*6 rectangular box the difference equals 200.
I will now put the results from in a table, and comment on them.
Rectangle size Difference of the corners multiplied then subtracted
2*3 20
3*4 60
4*5 120
5*6 200
I have noticed that my results are:
-
Multiples of 2, 5, 10, 20
- The difference of the rectangle is the size of the previous rectangle.
From my results I predict that the difference for a 6*7 will be,
6*7 10((6-1)*(7-1))=300
1 2 3 4 5 6 7 1 7
11 12 13 14 15 16 17 * 57 * 51
21 22 23 24 25 26 27 57 357
31 32 33 34 35 36 37 357-57=300
41 42 43 44 45 46 47 Difference
51 52 53 54 55 56 57
This proves my prediction correct.
Through my investigations I found out that the general rule for finding the difference for a rectangular box is,
10((m-1)(n-1))
m m- length of the square
n- width of the square
n
Opposite corners
Now I am going to prove my formula.
z-number in the top left corner m-length of the rectangle
n-width of the rectangle
m
z z+(m-1) z(z+10N+M)=z²+10Nz+Mz
(z+M)(z+10N)=z²+10Nz+Mz+10MN
(n-1)=N (m-1)=M n
(z²+10Nz+Mz+10MN)-(z²+10Nz+Mz)
z+10(n-1) z+10(n-1)+(m-1) =10MN
10((m-1)(n-1))
This proves that my general formula for finding the difference of any rectangle is correct in a 10*10 grid is correct.
The general formula for finding the difference for a square and a rectangle consisted of a *10.
Square =(m-1)²*10 *10
Rectangle =(m-1)(n-1)*10
I suspect that it has a *10 because the square and the rectangle came from a 10*10 square grid.
To prove my theory I will use a different sized grid. This grid would be 9*.
Opposite corners
Instead of finding the formula for difference first I will find the general formula straight away.
m-length of the square
9
m
z z+(m-1) z(z+10M)=z²+10Mz
(z +M)(z+9M)=z²+9Mz+Mz+9M²
(m-1)=M m 9 (z²+10Mz+9M²)-(z²+10Mz)=9M²
9(m-1)²
z+9(m-1) z+10(m-1)
This proves that the general formula for finding the difference of any square in a 9*9 grid is 9(m-1)².
Now I am going to further my investigations by finding the general formula for finding the difference of any rectangle in a 9*9 grid.
m-length of the rectangle. n-width of the rectangle. z-number in the top
left corner
m
z z+(m-1)
z(z+9N+M)=z²+9Nz+Mz
(z+M)(z+9N)=z²+9Mz+Mz+9MN
(n-1)=N n
(m-1)=M (z²+10Mz+9MN)-(z²+10Mz)=9MN
9((m-1)(n-1)
z+9(n-1) z+9(n-1)+(m-1)
This proves that the general formula for finding the difference of any rectangle in a 9*9 is 9((m-1)(n-1)).
Now I am going to further my investigation further by finding the general formula for any square, rectangle or square grid.
Opposite corners
m-length of square or rectangle.
n-width of square or rectangle.
y-size of the grid.
z-number in the top left corner.
m
z z+(m-1) z(z+yN+M)=z²+yNz+Mz
(z+M)(z+yN)=z²+yNz+Mz+yNM
(z²+yNz+Mz+yNM)-(z²+yNz+Mz)=yNM
(n-1)=N y((n-1)(m-1)
(m-1)=M n
z+y(n-1) z+y(n-1)+(m-1)
This proves that the general formula for finding the difference for any square, rectangle and square grid.