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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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