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# Opposite Corners.

Extracts from this document...

Introduction

Opposite Corners

Introduction

Ellie Birch

## Opposite corners are really simple. Basically, you choose any number between 1 – 100. The only numbers that you won’t be able to choose are the numbers along the bottom row (91 – 100) and the ones running down the right hand side (the multiples of 10). Look at the green lines on the diagram and you’ll see that only rectangles can be made, and as they are only half squares,

2x2 Squares

1x12= 12 difference = 10

11x2= 22 difference = 10

57x68= 3876 difference = 10

67x58= 3886 difference = 10

81x92= 7452 difference = 10

91x82= 7462 difference = 10

Number in left hand corner of the box = n

L)[n(n+11)] + [n+1(n+9)]

En          n+1

N

n+10      n+11

The formula

Middle

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    DifferenceThe 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

Conclusion

### 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)

1. – 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.

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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