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  • Level: GCSE
  • Subject: Maths
  • Word count: 3711

Number Stairs

Extracts from this document...

Introduction

image32.pngimage31.pngimage30.pngimage29.png

image00.png


I have been given a number grid that counts in ascending order from one to a hundred, beginning at the bottom left hand corner to end at the top right corner with the number one hundred. With this grid I have been given the task of investigating the relationship/s between the grid size, the stair size, the stair total and the ‘n’ number.

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

35

16

17

18

19

20

1

2

3

4

5

6

7

8

9

10

The step shape given above is a 3-step stair, simply because it consists of 3 steps.

The stair total is labelled as being the sum of all of the numbers in the stair shape:

        24 + 25 + 26 + 34 + 35 + 44 = 212

                The Stair total for this 3-step shape is 212

The ‘n’ number is defined as being the smallest number of the stair shape, in the grid above it is specified as ‘ 24’.

I will systematically work my way through this problem to find appropriate algebraic solutions to simplify the workings of the stair totals. In doing so, I will use different size grids and use different size stairs. Again, investigating relationships and discovering formulae for each problem I encounter.

I am going to start with a 10 by 10 grid with a 3-step stair.

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

35

16

17

18

19

20

1

2

3

4

5

6

7

8

9

10

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

35

16

17

18

19

20

1

2

3

4

5

6

7

8

9

10

n

T

1

56

2

62

3

68

4

74

5

80

6

86

...read more.

Middle

Using the same method as the previous, I will develop a formula.

Again, you can see that the ‘T’ column is increasing by 6 each time the ‘n’ number is increased by 1.

The ‘T’ column increases by 6 each time, so we multiply the ‘n’ number by the increase:

  • 6n, and then add the remaining amount to end up with the ‘T’ number.
  • 6n + 36 = T

To check this formula I will calculate another sum to which I know the solution to:

n

T

1

42

2

48

3

54

4

60

5

66

37

258


image33.png

Using another method, I will ensure that my formula is correct and if possible simplify the expression given.

n+3G

n+3G+1

n+3G+2image12.png

n+2G

n+2G+1

n+2G+2

n+G

n+G+1

n+G+2

n

n+1

n+2

n+n+1+n+2+ n+G+n+G+1+n+2G=T

Simplifying the above equation using means of collecting up will give me:

6n+4G+4=T

                For this grid, it would be:        6n+4*8+4=T

I have developed formulas for each grid (8 by 8, 9 by 9 and 10 by 10), I have come to the simple conclusion that the general formula 6n+4G+4=T applies to all grids as long as the stair number remains the same at 3 (3-step stair).

Developing further on the generic subject matter, I will find solutions on moving the ‘n’ number either along the x

...read more.

Conclusion

n+5G+2

n+5G+3

n+5G+4

n+5G+5

n+5G+6

n+5G+7

n+5G+8

n+5G+9

n+4G

n+4G+1

n+4G+2

n+4G+3

n+4G+4

n+4G+5

n+4G+6

n+4G+7

n+4G+8

n+4G+9

n+3G

n+3G+1

N+3G+2

n+3G+3

n+3G+4

n+3G+5

n+3G+6

n+3G+7

n+3G+8

n+3G+9

n+2G

n+2G+1

N+2G+2

N+2G+3

n+2G+4

n+2G+5

n+2G+6

n+2G+7

n+2G+8

n+2G+9

n+G

n+G+1

N+G+2

N+G+3

n+G+4

n+G+5

n+G+6

n+G+7

n+G+8

n+G+9

n

n+1

n+2

N+3

n+4

n+5

n+6

n+7

n+8

n+9

Above I will develop a 6-step stair shape formula that will work with any grid:

N+n+1+n+2+ n+3+n+4++n+5+ n+G+ n+G+1+ n+G+2+ n+G+3+n+G+4+n+2G+ n+2G+1+n+2G+2+n+2G+3+ n+3G+n+3G+1+n+3G+2+n+4G+n+4G+1+n+5G=T

Using means of collecting up I can simplify this formula to:  21n+35G+35 = T

I will check that this formula is correct using the 10 by 10 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

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

35

16

17

18

19

20

1

2

3

4

5

6

7

8

9

10

n

T

1

406

2

427

3

448

4

467

5

486

6

505

                        Here I am using the formula to see whether it works…

  1. 21*1+35*10+35=406
  1. 21*2+35*10+35=427
  1. 21*3+35*10+35=448

From the sums I have calculated above you can see that the formula works and is accurate as it follows my table completely.

I assume and believe that the formula will work with all problems that use a 6-step stair.

...read more.

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