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Opposite Corners.

Extracts from this document...

Introduction

Opposite Corners

Introduction

Ellie Birch

image41.pngimage01.pngimage00.pngimage01.png

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, image02.pngimage00.pngimage03.pngimage39.pngimage19.pngimage29.pngimage02.pngimage11.pngimage04.pngimage11.pngimage04.pngimage03.pngimage08.png

image05.png

2x2 Squares

image41.pngimage01.pngimage01.pngimage00.png

1x12= 12 difference = 10

11x2= 22 difference = 10image00.png

57x68= 3876 difference = 10

67x58= 3886 difference = 10image00.pngimage01.pngimage01.png

81x92= 7452 difference = 10image00.png

91x82= 7462 difference = 10image00.pngimage06.pngimage06.png

image00.png

Number in left hand corner of the box = n

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

En          n+1  image07.pngimage09.png

N

n+10      n+11

The formula

...read more.

Middle

1x13= 13

11x3= 33image10.pngimage10.png

18x10= 180

8x20= 160image10.pngimage12.pngimage12.png

image10.png

66x58= 3828

56x68= 3808

The difference is always 20.

Nn          n+1      n+2      n+3

image14.pngimage13.png

Ln+10   n+11    n+12    n+13

Hn(n+13)                Difference is 20image15.png

(n+3) (n+10)

3x3 Squares

image10.pngimage16.pngimage16.pngimage41.png

1x23= 23

21x3= 63

image10.png

43x65= 2795image10.pngimage17.pngimage17.png

63x45= 2835

image16.pngimage10.pngimage16.png

68x90= 6120image10.png

88x70= 6160

image10.png

The difference is always 40.

n          n+1          n+2

image18.pngimage20.png

n+10   n+11        n+12

n+20    n+21       n+22

Hn(n+22)                Difference is 40image15.png

(n+2) (n+20)

2(w)xL    DifferenceThe numbers step up in 10image21.png

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=44image22.pngimage23.png

                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=55image25.pngimage24.png

11    12    13   14    15

The prediction was correct, the difference is 40.

So that means that if you want

...read more.

Conclusion

Square number         (w) _        Formulaimage37.png

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)]

image38.png

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)]

image38.png

5 – 1

5(4x10) – (4x10)

        4image40.png

40x5 = 200 – 40 = 160

160 divided by 4 = 40

The formula proves to be correct.

...read more.

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