• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month
Page
  1. 1
    1
  2. 2
    2
  3. 3
    3
  4. 4
    4
  5. 5
    5
  6. 6
    6
  7. 7
    7
  8. 8
    8
  • Level: GCSE
  • Subject: Maths
  • Word count: 2284

Number Stairs - Up to 9x9 Grid

Extracts from this document...

Introduction

                                                                                                               Number Stairs

91

92

93

94

95

96

97

98

99

100

81

82

83

84

85

86

87

88

89

90

71

72

73

74

75

76

77

78

79

80

61

62

63

64

65

66

67

68

69

70

51

52

53

54

55

56

57

58

59

60

41

42

43

44

45

46

47

48

49

50

31

32

33

34

35

36

37

38

39

40

21

22

23

24

25

26

27

28

29

30

11

12

13

14

15

16

17

18

19

20

1

2

3

4

5

6

7

8

9

10

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

47image10.png

35

36

37

25

26

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

13

14

15

16image15.png

9

10

11

12

5

6

7

8

1

2

3

4

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

91

92

93

94

95

96

97

98

99

100

81

82

83

84

85

86

87

88

89

90

71

72

73

74

75

76

77

78

79

80

61

62

63

64

65

66

67

68

69

70

51

52

53

54

55

56

57

58

59

60

41

42

43

44image16.png

45image17.pngimage18.png

46image01.png

47

48

49

50

31

32

33

34

35

36

37

38

39

40

21

22

23

24

25image02.pngimage03.png

26

27

28

29

30

11

12

13

14

15

16

17

18

19

20

1

2

3

4

5

6

7

8

9

10

...read more.

Middle

10image07.png

11

12

5

6image08.png

7

8

1image03.pngimage09.png

2

3

4

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

91

92

93

94

95

96

97

98

99

100

81

82

83

84

85

86

87

88

89

90

71

72

73

74

75

76

77

78

79

80

61

62

63

64

65

66

67

68

69

70

51

52

53

54

55

56

57

58

59

60

41

42

43

44

45

46

47

48

49

50

31

32

33

34

35

36

37

38

39

40

21

22

23

24

25

26

27

28

29

30

11

12

13

14

15

16

17

18

19

20

1

2

3

4

5

6

7

8

9

10

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.

91

92

93

94

95

96

97

98

99

100

81

82

83

84

85

86

87

88

89

90

71

72

73

74

75

76

77

78

79

80

61

62

63

64

65

66

67

68

69

70

51

52

53

54

55

56

57

58

59

60

41

42

43

44

45

46

47

48

49

50

31

32

33

34

35

36

37

38

39

40

21

22

23

24

25

26

27

28

29

30

11

12

13

14

15

16

17

18

19

20

1

2

3

4

5

6

7

8

9

10

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).

19

13

14

7

8

9

1

2

3

4

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

28

29

19

20

21

10

11

12

13

1

2

3

4

5

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

22

23

15

16

17

8

9

10

11

1

2

3

4

5

  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.

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related GCSE Number Stairs, Grids and Sequences essays

  1. Number stairs

    + (x + n) + (x + n + 1) + (x + 1) + (x + 2) Now I will simply that to get: T=6x + 4n + 4 THE GENERAL FORMULA FOR ANY 3-STEP STAIR GRID SIZE IS: T=6x + 4n + 4 PART 2 Investigate further the

  2. Investigation of diagonal difference.

    72 73 74 81 82 83 84 91 92 93 94 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 I have now

  1. Maths - number grid

    I feel I have got this defined difference of 12 because I am using a 12x12 grid. I will use algebra to prove that my found result is correct: (r+1)(r+12) - r (r+13) r(r+12) +1 (r+12) - r -13r r +12r +r +12 - r -13r = 12 From the

  2. Maths coursework. For my extension piece I decided to investigate stairs that ascend along ...

    = 356 ? This must mean that my formula Un = 6n + 4g + 4 - where 'n' is the stair number, 'g' is the grid size, and 'Un' is the term which is the stair total - works for all 3-stair numbers on any size grid.

  1. Number Stairs

    sum of all figures inside the stairs gives you the stair total. With the stair number 1 we get a stair total of 1+2+3+9+10+17= 42 =T. The stair total is calculated accordingly to the stair number for any grid size.

  2. number grid

    The Difference for any 2 X 2 grid is always 10 inside a 10 X 10 grid. 3 X 3 Grid I have now found the difference for any 2 X 2 grid inside a 10 X 10 grid, now I will try to find out the difference for any 3 X 3 grid inside a 10 X 10 grid.

  1. Mathematics - Number Stairs

    + (n+2) + (n+11) + (n+12) + (n+22) = 6n + 48 8 9 10 11 12 1 2 3 T = 6n + 36 T = 6n + 40 T = 6n + 44 T = 6n + 48 4 5 3 Step-Staircase / Grid Width 12 25 13

  2. Mathematical Coursework: 3-step stairs

    now have found the pattern, this would allow me to predict the following 3-step stairs if I was still using the method. In order to check if my prediction is right I would have to again add up the numbers in the 3-step stair shape.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work