Upper Left No.

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.
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