# Number Stairs - Up to 9x9 Grid

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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

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

45 | 46 | 47 |

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.

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.

13 | 14 | 15 | 16 |

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 | 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 |

Middle

10

11

12

5

6

7

8

1

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’.

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

37 | ||||

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.

29 | ||||

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.

… 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

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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