Opposite Corners.

2x3 squares

                           (3 = width)

1x13= 13

11x3= 33

18x10= 180

8x20= 160

66x58= 3828

56x68= 3808

The difference is always 20.

Nn          n+1      n+2      n+3

Ln+10   n+11    n+12    n+13

Hn(n+13)                Difference is 20

(n+3) (n+10)

3x3 Squares

1x23= 23

21x3= 63

43x65= 2795

63x45= 2835

68x90= 6120

88x70= 6160

The difference is always 40.

n          n+1          n+2

n+10   n+11        n+12

n+20    n+21       n+22

Hn(n+22)                Difference is 40

(n+2) (n+20)

2(w)xL    Difference              The numbers step up in 10

2         10

3                            20    >10

4                        30    >10

I predict that a 2x5 square will have a difference of 40.

4x4

1.    2    3    4                         1x14=44

                11x4=44

11  12  13  14

The difference is 40. Now I need to test out my prediction:

1       2      3      4    5            1x15= 15

        11x5=55

11    12    13   14    15

The prediction was correct, the difference is 40.

So that means that if you want to work out the answer for a 2x11 square, you’ll look at the above results and find a formula.

(Wx10) –10 = difference

So a 2x11 square would have a difference of 100, and that can be worked out by doing the following:

11x10 = 110

-10 = 100

3(w) xL     Difference

2                20

3                 40

4                       60

3x4

      1     2    3                                 1x33= 33        

                            31x3= 93

11.    12  13

21     22   23

31     32   33

The difference is 40.

I predict that a 3x5 square will have a difference of 80.

1        2        3                1x43=43

                                               41x3=123

11        12        13        

21        22        23        

31        32        33        

41        42        43        

The difference is 80 which means that my prediction was correct.

4x4 Squares

1x34= 34

31x4= 124        

27x60= 1620

30x57= 1710

52x85= 4420

55x82= 4510

The difference is always 90

n          n+1          n+2            n+3

n+10   n+11        n+12    n+13

n+20    n+21       n+22    n+23

n+30    n+31       n+32    n+33

Hn(n+33)                Difference is 90

(n+3) (n+30)

        

General Formula

The general formula is the formula that helps you work out the formula needed to calculate the difference for the square numbers.

 

Square number         (w) _        Formula

2                                (wx10) –10  = difference

3                                (wx20) – 20 = difference

4                                (wx30) – 30 = difference

5                                  (wx40) – 40 = difference

I predict that a 5x5 square will have a formula of:

(wx40) – 40

This is because I have made a formula to work out the general formula, and tested it out:

w[10(w – 1)] – [10(w – 1)]

w – 1

 Square number = 2 (w)

2. – 1 = 1

 x 10 = 10

x 2 = 20

– 1 = 1

x 10 = 10

20 – 20 = 10

divided by 2 – 1 = 10

All you do is write out this equation: (wx__) - __ and replace the two blank spaces with the number that you have found

That proves to be correct, so now I will see if the theory for the 5x5 square is correct.

I predicted that the formula would be:

 (wx40) – 40 = difference

5[10(5 – 1)] – [10(5 – 1)]

5 – 1

5(4x10) – (4x10)

        4

40x5 = 200 – 40 = 160

160 divided by 4 = 40

The formula proves to be correct.