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

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    90

    71

    72

    73

    74

    75

    76

    77

    78

    79

    80

    61

    62

    63

    64

    65

    66

    67

    68

    69

    70

    51

    52

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    60

    41

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    43

    44

    45

    46

    47

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    49

    50

    31

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    33

    34

    35

    36

    37

    38

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    40

    21

    22

    23

    24

    25

    26

    27

    28

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    30

    11

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    20

    1

    2

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    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

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    30

    11

    12

    13

    14

    15

    16

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    20

    1

    2

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    5

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    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.

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