# This is a 3-step stair. The total of the numbers inside the step square shape is 25+26+27+35+36+45=194. The total for this step square is 194.

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Introduction

David Drew

MATHS COURSEWORK

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 step square shape is 25+26+27+35+36+45=194. The total for this step square is 194.

Part 1

For other 3-step step squares investigate the relationship between the stair total and the position of the stair shape on the grid.

In red is another example which there doesn’t appear to be a relationship with the step-square in blue.

As we look into the Step Square we say the bottom left hand corner is X and the others come after in the same formula

X+2 0 | ||

X+10 | X+11 | |

X | X+1 | X+2 |

When all the numbers and X’s are put together we get the following formula – 6X + 44 as there is 6 squares with X in them and we add the numbers that follow to get an otherall formula. This formula can be used to find out the total of that the Step Square using the bottom left hand corner.

We can prove this with an example.

Middle

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

If use the formula for all the numbers in red we get 10 x 5 + 110 = 160 and if we add up all the numbers: -

5+6+7+8+15+16+17+25+26+35=160

So as we can see the formula works.

Now I will do a 5x5 step square on a 10x10 grid

5x5

I will now draw a 10x10 grid and mark on in blue the 5x5 step square.

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 formula for all these numbers are as shown below.

X+40 | |||||

X+30 | X+31 | ||||

X+20 | X+22 | X+23 | |||

X+10 | X+12 | X+13 | X+14 | ||

X | X+1 | X+2 | X+3 | X+4 |

Here as shown all the X’s add up to 15 and all the numbers add up to 220

Conclusion

So the formula for this part is

½ N

To get the next bit we need to substitute N into ½ N and take that away from X ( the amount of X’s in a Step Square) to see how much we are off from X

N 3 4 5

1/2N 4 ½ 8 12 1/2

X 6 10 15

X- ½ N ½ 2 2 1/2

As we can see the first difference of X- ½ N is ½ this means that it Is ½ N so the altogether formula for the X’s is

½ n + ½ n

We can now put together the whole formula together

(½ n + ½ n) + (N --- N) x 11 b6 6 6

To see if this works we now check it .

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 |

Marked off is a 3x3 step square the total of this square is 1+2+3+11+12+21=50

Using the formula to check this it’s

(½ 3 + ½ 3) + (3 --- 3) x 11 b6 6 6

= 4.5 + 1.5 + 11(4.5 – 0.5)

= 50

So as we can see the formula works

So the formula for any size step square in a 10x10 grid is

(½ n + ½ n) + (N --- N) x 11 b6 6 6

Which has been proved and tested to show that it works so we know that this is right and will work for future use.

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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