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# Number grid - Essay Example

Summary
The first question that arises when we begin to draw squares around the numbers is: How many unique squares can be drawn within the grid It is dependent upon the length of the sides of the square. A table is given below:
Using the same grid, we will calculate the difference in the product of the corners…

## Extract of sampleNumber grid

UL*LR
UR*LL
Difference
1
23
63
40
26
1248
1268
40
33
1815
1855
40
48
3360
3400
40
55
4235
4275
40
62
5208
5248
40
78
7800
7840
40
TABLE 2
As the table indicates, no matter where we place the square in the grid the difference in the product of the corners for a 3 x 3 square is always 40.

Table 3 below are the results of a square that is 4 x 4 placed randomly on the grid.
Upper Left No.
UL*LR
UR*LL
Difference
1
34
124
90
17
850
940
90
24
1368
1458
90
36
2484
2574
90
41
3034
3124
90
53
4558
4648
90
67
6700
6790
90
TABLE 3
As the table indicates, no matter where we place the square in the grid the difference in the product of the corners for a 4 x 4 square is always 90.

Table 4 below are the results of a square that is 5 x 5 placed randomly on the grid.
Upper Left No.
UL*LR
UR*LL
Difference
1
45
205
160
16
960
1120
160
23
1541
1701
160
35
2765
2925
160
42
3612
3772
160
56
5600
5760
160
TABLE 4
As the table indicates, no matter where we place the square in the grid the difference in the product of the corners for a 5 x 5 square is always 160.
Table 5 below are the results of a square that is 6 x 6 placed randomly on the grid.
Upper Left No.
UL*LR
UR*LL
Difference
1
56
306
250
15
1050
1300
250
23
1794
2044
250
23
1794
2044
250
21
1596
1846
250
32
2784
3034
250
45
4500
4750
250
41
3936
4186
250
TABLE 5
As the table indicates, no matter where we place the square in the grid the difference in the product of the corners for a 6 x 6 square is always 250.

Table 6 below are the results of a square that is 7 x 7 placed randomly on the grid.
Upper...
This is true for a 2 x 2 square and all other sizes. However, the difference in the product of the corners is dependent upon the size of the square. As the size of the square gets larger, the difference in the product of the corners also increases.
But is there an algebraic relationship between the size of the square and the difference of the product of the corners Can we calculate the difference by knowing the size of the square Table 10 lists the results from the previous investigations.
As we have seen, no matter what size square is used, we can use algebra to calculate the number of possible squares and the difference in the product of their corners. This applies to all possible combinations placed on the grid. ...Download file to see next pagesRead More
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