(N² + 12N + 20) – (N² + 12N)
= 20
What I have noticed:
I have noticed that when a square that is 3x3, is taken from a 5x5 grid, the difference is always 20.
Prediction
I predict that a 4x4 square from a 5 wide grid, it will have a final difference of 45.
Proof:
Comparison:
N N+3
x
N+15 N+18
N (N+18) = N² + 18N
(N+3) (N+15) = N² + 15N + 2N + 45
= N² + 18N + 45
Difference:
(N² + 18N + 45) – (N² + 18N)
= 45
What I have noticed:
I have noticed that when a square that is 4x4, is taken from a 5x5 grid, the difference is always 45.
Prediction
I predict that a 5x5 square from a 5 wide grid, it will have a final difference of 80.
Proof:
Comparison:
N N+4
x
N+20 N+24
N (N+24) = N² + 24N
(N+4) (N+20) = N² + 20N + 4N + 80
= N² + 24N + 80
Difference:
(N² + 24N + 80) – (N² + 24N)
= 80
What I have noticed:
I have noticed that when a square that is 5x5, is taken from a 5x5 grid, the difference is always 80.
I have found out the difference for a five wid grid is g(x-1)².
I will now see if the same equation works for a six wide grid.
I will now carry out the same investigation but on a 6 wide grid
Prediction
I predict that a 2x square from a 6 wide grid, it will have a final difference of 6.
Comparison:
N N+1
x
N+6 N+7
N (N+7) = N² + 7N
(N+1) (N+6) = N² + 6N + N + 6
= N² + 7N + 6
Difference:
(N² + 7N + 6) – (N² + 7N)
= 6
What I have noticed:
I have noticed that when a square that is 2x2, is taken from a 6x6 grid, the difference is always 6.
Prediction
I predict that a 3x3 square from a 6 wide grid, it will have a final difference of 24.
Comparison:
N N+2
x
N+12 N+14
N (N+14) = N² + 14N
(N+2) (N+12) = N² + 12N + 2N + 24
= N² + 14N + 24
Difference:
(N² + 14N + 24) – (N² + 14N)
= 24
What I have noticed:
I have noticed that when a square that is 3x3, is taken from a 6x6 grid, the difference is always 24.
Prediction
I predict that a 4x4 square from a 6 wide grid, it will have a final difference of 54.
Comparison:
N N+3
x
N+18 N+21
N (N+21) = N² + 21N
(N+3) (N+18) = N² + 18N + 3N + 54
= N² + 21N + 54
Difference:
(N² + 21N + 54) – (N² + 21N)
= 54
What I have noticed:
I have noticed that when a square that is 4x4, is taken from a 6x6 grid, the difference is always 54.
Prediction
I predict that a 5x5 square from a 6 wide grid, it will have a final difference of 96.
Comparison:
N N+4
x
N+24 N+28
N (N+28) = N² + 28N
(N+4) (N+24) = N² + 24N + 4N + 96
= N² + 28N + 96
Difference:
(N² + 28N + 96) – (N² + 28N)
= 96
What I have noticed:
I have noticed that when a square that is 5x5, is taken from a 6x6 grid, the difference is always 96.
I have found out the difference for a six wid grid is g(x-1)².
I will now see if the same eqyuation works for a seven wide grid.
I will now carry out the same investigation but on a 7 wide grid
Prediction
I predict that a 2x2 square from a 7 wide grid, it will have a final difference of 7.
Comparison:
N N+1
x
N+7 N+8
N (N+8) = N² + 8N
(N+1) (N+7) = N² + 7N + N + 7
= N² + 8N + 7
Difference:
(N² + 8N + 7) – (N² + 8N)
= 7
What I have noticed:
I have noticed that when a square that is 2x2, is taken from a 7x7 grid, the difference is always 7.
Prediction
I predict that a 3x3 square from a 7 wide grid, it will have a final difference of 28.
Comparison:
N N+2
x
N+14 N+16
N (N+16) = N² + 16N
(N+2) (N+14) = N² + 14N + 2N + 28
= N² + 8N + 7
Difference:
(N² + 14N + 28) – (N² + 14N)
= 28
What I have noticed:
I have noticed that when a square that is 3x3, is taken from a 7x7 grid, the difference is always 28.
Prediction
I predict that a 4x4 square from a 7 wide grid, it will have a final difference of 63.
Comparison:
N N+3
x
N+21 N+24
N (N+24) = N² + 24N
(N+3) (N+21) = N² + 21N + 3N + 63
= N² + 24N + 63
Difference:
(N² + 24N + 63) – (N² + 24N)
= 63
What I have noticed:
I have noticed that when a square that is 4x4, is taken from a 7x7 grid, the difference is always 63.
Prediction
I predict that a 5x5 square from a 7 wide grid, it will have a final difference of 112.
Comparison:
N N+4
x
N+28 N+32
N (N+32) = N² + 32N
(N+4) (N+28) = N² + 28N + 4N + 112
= N² + 32N + 112
Difference:
(N² + 32N + 112) – (N² + 32N)
= 112
What I have noticed:
I have noticed that when a square that is 5x5, is taken from a 7x7 grid, the difference is always 112.
I have found out the difference for a seven wid grid is g(x-1)².
I now have found a formula that will work for and size number grid over five wide and square of 2x2. this formula is g(x-1)
Table of results:
5 wide grids
6 wide grids
7 wide grids
Conclusion
During my investigation, I have cancelled out, expanded and simplified brackets, and compared my results to ‘N’. I did this to find a final equation to be able to calculate the difference of any square of any size drawn on a number grid. I found this to be g (x-1) ². This formula works for any grid size over 5 and square size over 2x2.
If I had more time I would want to find out if the same formula works with a rectangle on a number grid and experiment with other quadrilaterals and see if the same equation works.
I would also see if I could find any other equation that would work.
Overall I think my investigation went well and I have found the final equation.
