GCSE Advanced Mathematics

Coursework

Number Grids

Ahmed Nafi


Squares

Hypothesis

The difference of products of the diagonally opposite numbers will be constant throughout the grid and the difference of 2 by 2 square boxes in any grid will be equal to the grid length.

Example



Analysis

10 by 10 Grids


8 by 8 Grids

11 by 11 Grids

12 by 12 Grids

13 by 13 Grids

14 by 14 Grids


Conclusion

From the analysis, we can observe that the difference of products of the diagonally opposite numbers is always constant, which proved the first part of the hypothesis as true. The difference of 2 by 2 squares for any grid is always equal to the grid-length and thus proves the second part of the hypothesis as true.

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From the series of these results, a formula can be developed, as shown below.

 

        ……. (i)

The above formula is of quadratic format,. It may be pointed out here that if the difference between D & D1 (Figure 1 & 2) is constant, then it would be linear. And if the difference between D1 & D2 (Figure 1 & 2) is constant, then it would be a quadratic. Here the second condition holds true and thus the formula is quadratic.

Figure 1        Figure 2


Difference

  • I substituted the values for the quadratic,
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