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  • Level: GCSE
  • Subject: Maths
  • Word count: 2284

Number Stairs - Up to 9x9 Grid

Extracts from this document...

Introduction

                                                                                                               Number Stairs

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This is a 3-step stair.

The total of the numbers inside the stair shape above is:

  • 1st Line: 25+26+27
  • 2nd Line: 35+36
  • 3rd Line: 45

T=Total        T=194

The stair total for this 3-step stairs is 194.

Part 1

image00.png

  • 1st Line: 25+26+27
  • 2nd Line: 35+36                        Going up by 1
  • 3rd Line: 45

image00.png

45

46

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35

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27

Hypothesis: The number from left to right are going up by 1 and the numbers going from bottom to top are going up by 10, therefore if I was given the bottom left hand corner on a 10 by 10 square grid, I would know the rest of the number stair digits.

E.g. Bottom left hand corner number.
image00.pngimage13.png

88

89

90

78

79

80

68

69

70

On a different number square grid, e.g. 4 by 4 number square grid, the theory would be the same, except that the number above the bottom left hand corner number is going to go up by 4.
image14.png

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16image15.png

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The total of the numbers inside the stair shape is:

  • 1st Line: 1+2+3
  • 2nd Line: 5+6
  • 3rd Line: 9

T=Total        T=26

The stair total for this 3-step stair is 26.

Part 2

I have investigated further and I have found out that the number going diagonal in a 10 by 10 number square grid…

E.g. On a 10 by 10 number square grid

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...read more.

Middle

10image07.png

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1image03.pngimage09.png

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Top right corner number stair, the total is always higher.

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The total of the numbers inside the top right corner number stair shape is:

  • 1st Line: 78+79+80
  • 2nd Line: 88+89
  • 3rd Line: 98

T=Total        T=452

Bottom left hand corner number stairs, the total is always lower.

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E.g.

The total of the numbers inside the bottom left hand corner number stair shape is:

  • 1st Line: 1+2+3
  • 2nd Line: 11+12
  • 3rd Line: 21

T=Total        T=50

                                                                                                     Finding the formula

Hypothesis: I have found out that to find the formula for any number square you must firstly make one of the numbers in that stair pattern as ‘x’.

...read more.

Conclusion

Formula: 10x + 10 + 10g

I tested out this formula on any grid size I preferred (6 x 6 grid).

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1 + 2 + 3 + 4 + 7 + 8 + 9 + 13 + 14 + 19 =80

10 x 1= 10 + 10=20                10 x 6 = 60

60 + 20 = 80  Correct.

9x9 grid – 5 step stair

37image11.png

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Total = 215

Algeraic: x + x + 1 + x + 2 + x + 3 + x + 4 + x + g + x + g + 1 + x + g + 2 + x + g+ 3 + x + 2g + x + 2g + 1 + x + 2g + 2 + x + 3g + x + 3g + 1 + x + 4g = Formula: 15x + 20 + 20g

I will now test this formula on a 7 x 7 grid, but still staying with a 5 step stair.

29image11.png

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  Total = 175

So, 15 x 1 = 15 + 20 = 35        20 x 7 = 140

140 + 35 = 175

Again this shows that my 5 step stair formula for any grid size is correct.

Finding the algebraic formula

2 step stair = 3x + 1 + g

3 step stair = 6x + 4 + 4g

4 step stair = 10x + 10 + 10g

5 step stair = 15x + 20 + 20g

???             =21x + 35 + 35g

I have noticed a certain pattern which occurs constantly through the formulas. In the first column it goes up in triangle numbers:

3x, 6x, 10x , 15x

I believe that the next number will be 21, because 15 add the next triangle number in the pattern which is 6 is 21. Also for the last part of the formula I had to find the difference from the numbers at the end of the formulas so that I could notice a pattern.

image12.png

      … g                        

 3

      … 4g             3

 6

      … 10g             4

 10

      … 20g              5

15

21x + … + …?(35g)

Algebraic formulas for and grid size

2 step stair = 3x + 1 + g

3 step stair = 6x + 4 + 4g

4 step stair = 10x + 10 + 10g

5 step stair = 15x + 20 + 20g

6 step stair = 21x + 35 + 35g

...read more.

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