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# Number grid

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Introduction

Uzair

Diagonal differences

Introduction - In this casework I am going to examine the difference between the products of opposite corners. I am firstly going to start with a 2x2 box then go further into my investigation by working out formulas and using different box sizes and grids.

Part 1 (Square boxes -10 column grid)

I am going to find the diagonal difference of any 2x2 box in a 10 column grid. From this I will try to find out if there are any correlations and to find differences for other

2 x 2 Box-Numeric’s 1   2 11 12 65 66 75 76 By looking at the 2 x 2 boxes in a 10 column grid, I have found out that if you place a   2 x 2 box, any where in the grid, the difference will always be 10.

Predict

Therefore I predict that, if I place any 2 x 2 box in a 10 column grid the difference will be 10.

 27 28 37 38 This estimate proves my aim, that if you place any 2 x2 box in a 10 column grid, the difference will always be 10

Difference in Algebra

Now I am going to show what the difference will be for any 2 x 2 box in a 10 column grid algebraically. n n+1 n+10 n+11

Using algebra shows that my prediction was correct that, for any 2x2 box in a 10 column grid the difference will always be 10.

Middle

42=16

Therefore I can predict that the difference for a 5 x 5 grid will be 80. Now I am going to prove this prediction algebraically.

5 x 5 box - Proving algebraically

 n n+1 n+2 n+3 n+4 n+5 n+6 n+7 n+8 n+9 n+10 n+11 n+12 n+13 n+14 n+15 n+16 n+17 n+18 n+19 n+20 n+21 n+22 n+23 n+24 This shows that I have proved the difference for a 5 x 5 box in a 5 column grid correctly.

Now to find out a general formula for the difference for any square box in a 5 column grid will be.

 Box size ( b ) Formula Difference 3 x 3 5 (3-1)2 = 20 5 x 5 5 (5-1)2 = 80 b x b 5(b–1)2

Therefore, the overall formula for any square box in a 5 column grid is 5(n–1)2

I am now going to test the general formula by doing the differences for all square boxes in a 5 x 5 column grid.

2 x 2 = 5(2–1)2=5

3 x 3 = 5(3–1)2=20

4 x 4 = 5(4–1)2=45

5 x 5 = 5(5–1)2=80

7 x 7 Column Grid

 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 n n+1 n+2 n+3 n+4 n+5 n+ 6 n+7 n+8 n+9 n+10 n+11 n+12 n+13 n+14 n+15 n+16 n+17 n+18 n+19 n+20 n+21 n+22 n+23 n+24 n+25 n+26 n+27 n+28 n+29 n+30 n+31 n+32 n+33 n+34 n+35 n+36 n+37 n+38 n+39 n+40 n+41 n+42 n+43 n+44 n+45 n+46 n+47 n+48

I am going to find the diagonal difference of any 2 x2 box in a 7 column grid. From this I will try to find out if there are any correlations.

2 x 2 Box-Numeric’s

 1 2 8 9 34 35 41 42 By looking at the 2 x 2 boxes in a 7 column grid, I have found out that if you place a   2 x 2 box, any where in the grid, the difference is 7.

Predict

Therefore I predict that, if I place any 2 x 2 box in a 7 column grid, the difference will always be 7. 29 30 36 37

This estimate proves my aim, that if you place any 2 x2 box in a 7 column grid, the difference will always be 7

Difference in Algebra

Now I am going to show what the difference will be for any 2 x 2 box in a 7 column grid algebraically

2 x 2 Box n n+1 n+7 n+8

I am now going to increase the size of my boxes to 3 x 3, 4 x 4 etc and going to work out the difference in algebraic forms.

3 x 3 Box

 n n+1 n+2 n+7 n+8 n+9 n+14 n+15 n+16 4 x 4 Box

 n n+1 n+2 n+3 n+7 n+8 n+9 n+10 n+14 n+15 n+16 n+17 n+21 n+22 n+23 n+24 I am going to display my results in a table, where I would try finding any patterns within the results I have.

7 x7 Grid Square Box sizes Difference Pattern 2 x 2 7 = 1x 7 12=1 3 x 3 28 = 4 x 7 22=4 4 x 4 63 = 9 x 7 32=9 Predict 7 x 7 252= 36 x 7 62=36

Therefore I can predict that the difference for a 7 x 7 grid will be 252. Now I am going to prove this prediction algebraically.

7 x 7 box - Proving algebraically

 n n+1 n+2 n+3 n+4 n+5 n+ 6 n+7 n+8 n+9 n+10 n+11 n+12 n+13 n+14 n+15 n+16 n+17 n+18 n+19 n+20 n+21 n+22 n+23 n+24 n+25 n+26 n+27 n+28 n+29 n+30 n+31 n+32 n+33 n+34 n+35 n+36 n+37 n+38 n+39 n+40 n+41 n+42 n+43 n+44 n+45 n+46 n+47 n+48 This shows that I have proved the difference for a 7 x 7 box in a 7 column grid correctly.

Now to find out a general formula for the difference for any square box in a 7 column grid will be.

 Box size ( b ) Formula Difference 3 x 3 7(3-1)2 = 20 7 x 7 7(7-1)2 = 252 b x b 7 (b–1)2

Conclusion

Therefore, by using this method, I will use the formula 10(c-1)(r-1) which I gained from part 3, however instead of the grid size 10, I will use (g) to represent my random grid size.

I will now represent this in a table. Grid size ( g ) Formula 10 x 10 10(c-1)(r-1) 7 x 7 7(c-1)(r-1) 6 x 6 6 (c-1)(r-1) g x g g (c-1)(r-1)

Proving overall formula algebraically

I am now going to explain how I got this formula algebraically.

 n  n+(c-1) n+g(r-1) n+g(r-1)+(c-1)

1) n[n+g(r-1)+(c-1)]

=n2+gn(r-1)+n(c-1)

[n+(c-1)][n+g(r-1)]

=n2+gn(r-1)+n(c-1)+g(c-1)(r-1)

2) n2+gn(r-1)+n(c-1)+g(c-1)(r-1) - n2+gn(r-1)+n(c-1)

=g(c-1)(r-1)

Therefore this overall equation “g(c-1)(r-1)”explains the difference, for any size rectangular box in any size grid.

Conclusion - In this task I have successfully found out general formula for any square box in any size grid which was to be ‘g(b-1)2and the general formula any rectangle box in any size grid which was to be ‘g(c-1)(r-1)’. These formulas were proven algebraically above.

If I were to extend this task further, I would change the grid numbers i.e. rather than it going up in 1s, it would go in 2s or 3s.

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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