Number Stairs
Introduction
In this investigation I plan to find out how grid size affects the relationship between the total of a step stair and the position of the step stair shape on various sized number grids, e.g.
91
92
93
94
95
96
97
98
99
00
81
82
83
84
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86
87
88
89
90
71
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74
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80
61
62
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70
51
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60
41
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50
31
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40
21
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28
29
30
1
2
3
4
5
6
7
8
9
20
2
3
4
5
6
7
8
9
0
The shaded squares show a 3 step stair on a 10 x 10 grid, but I shall be investigating various sized stairs relationships between their positions and their totals on various grid sizes.
The grid sizes I will be using in my investigation are as follows:
. 10 x 10
2. 9 x 9
3. 8 x 8
The total for the stair is:
+ 2 + 3 + 11 + 12 + 21= 50
The total for the stair is always worked out by adding up all the values inside of the stair shape.
The position of the stair is 1.
The stairs position is always worked out by the number in the bottom left hand corner.
The position= n
Investigation for 3 step stairs on a 10 by 10 grid
21
1
2
2
3
22
2
3
2
3
4
23
3
4
3
4
5
24
4
5
4
5
6
n
Total
50
2
56
3
62
4
68
As the stair position goes up by 1 the total goes up by 6 each time. So for the stair in position 5 I predict that its total will be 74.
25
5
6
5
6
7
n+2g
n+g
n+g+1
n
n+1
n+2
The nth term can be worked out by changing the numbers in the 3 step stair in to algebra by making the position number n, then relating all other numbers to the position number, as you up a level on the step stair you add the grid sizes number (in this case 10) to the number below it so is represented by g (grid size) e.g.
The sum of this stair= (n) + (n+1) + (n+2) + (n+g) + (n+g+1) + (n+2g) = 6n + 4g + 4
(6 x 1) + (4 x 10) + 4= 50, this represents the 1st term.
Using this formula I can now work out any term for a 3 step stair on a 10 x 10 grid, e.g. the 55th term:
(6 x 55) + (4 x 10) + 4= 374
The formula also works out any term for a 3 step stair on any sized grid e.g.
3 step stair on a 9 x 9 grid
9
0
1
2
3
(6 x 1) + (4 x 9) + 4= 46
Investigation for 4 step stairs on a 10 x 10 grid
31
21
22
1
2
3
2
3
4
32
22
23
2
3
4
2
3
4
5
33
23
24
3
4
5
3
4
5
6
34
24
25
4
5
6
4
5
6
7
n
Total
20
2
30
3
40
4
50
As the stair position goes up by 1 the total goes up by 10 each time. So for the stair in position 5 I predict that its total will be 160.
35
25
26
5
6
7
5
6
7
8
My prediction was correct.
The nth term can be worked out by changing the numbers in the 4 step stair in to algebra by making the position number n, then relating all other numbers to the position number, as you up a level on the step stair you add the grid sizes number (in this case 10) to the number below it so is represented by g (grid size) e.g.
n+3g
n+2g
n+2g+1
...
This is a preview of the whole essay
35
25
26
5
6
7
5
6
7
8
My prediction was correct.
The nth term can be worked out by changing the numbers in the 4 step stair in to algebra by making the position number n, then relating all other numbers to the position number, as you up a level on the step stair you add the grid sizes number (in this case 10) to the number below it so is represented by g (grid size) e.g.
n+3g
n+2g
n+2g+1
n+g
n+g+1
n+g+2
n
n+1
n+2
n+3
The sum of this stair= (n) + (n+1) + (n+2) + (n+3) + (n+g) + (n+g+1) + (n+g+2) + (n+2g) + (n+2g+1) + (n+3g) = 10n + 10g + 10
(10 x 1) + (10 x 10) + 10= 120, this represents the 1st term.
Using this formula I can now work out any term for a 4 step stair on a 10 x 10 grid, e.g. the 55th term:
(10 x 55) + (10 x 10) + 10= 670
Investigation for 5 step stairs on a 10 x 10 grid
41
31
32
21
22
23
1
2
3
4
2
3
4
5
42
32
33
22
23
24
2
3
4
5
2
3
4
5
6
43
33
34
23
24
25
3
4
5
6
3
4
5
6
7
44
34
35
24
25
26
4
5
6
7
4
5
6
7
8
n
Total
235
2
250
3
265
4
280
As the stair position goes up by 1 the total goes up by 15 each time. So for the stair in position 5 I predict that its total will be 295.
45
35
36
25
26
27
5
6
7
8
5
6
7
8
9
My prediction was correct.
The nth term can be worked out by changing the numbers in the 5 step stair in to algebra by making the position number n, then relating all other numbers to the position number, as you up a level on the step stair you add the grid sizes number (in this case 10) to the number below it so is represented by g (grid size) e.g.
n+4g
n+3g
n+3g+1
n+2g
n+2g+1
n+2g+2
n+g
n+g+1
n+g+2
n+g+3
n
n+1
n+2
n+3
n+4
The sum of this stair= (n) + (n+1) + (n+2) + (n+3) + (n+4) + (n+g) + (n+g+1) + (n+g+2) + (n+g+3) + (n+2g) + (n+2g+1) + (n+2g+2) + (n+3g) + (n+3g+1) (n+4g)= 15n + 20g + 20
(15 x 1) + (20 x 10) + 20= 235, this represents the 1st term.
Using this formula I can now work out any term for a 5 step stair on a 10 x 10 grid, e.g. the 55th term:
(15 x 55) + (20 x 10) + 20= 1045
Investigation for 6 step stairs on a 10 x 10 grid
51
41
42
31
32
33
21
22
23
24
1
2
3
4
5
2
3
4
5
6
52
42
43
32
33
34
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23
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25
2
3
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5
6
2
3
4
5
6
7
53
43
44
33
34
35
23
24
25
26
3
4
5
6
7
3
4
5
6
7
8
n
Total
406
2
427
3
448
As the stair position goes up by 1 the total goes up by 21 each time. So for the stair in position 4 I predict that its total will be 469.
54
44
45
34
35
36
24
25
26
21
4
5
6
7
8
4
5
6
7
8
9
My prediction was correct.
The nth term can be worked out by changing the numbers in the 6 step stair in to algebra by making the position number n, then relating all other numbers to the position number, as you up a level on the step stair you add the grid sizes number (in this case 10) to the number below it so is represented by g (grid size) e.g.
n+5g
n+4g
n+4g+1
n+3g
n+3g+1
n+3g+2
n+2g
n+2g+1
n+2g+2
n+2g+3
n+g
n+g+1
n+g+2
n+g+3
n+g+4
n
n+1
n+2
n+3
n+4
n+5
The sum of this stair= (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) + (n+5g)= 21n + 35g + 35
(21 x 1) + (35 x 10) + 35= 406, this represents the 1st term.
Using this formula I can now work out any term for a 6 step stair on a 10 x 10 grid, e.g. the 55th term:
(21 x 55) + (35 x 10) + 35= 1540
Investigation for 7 step stairs on a 10 x 10 grid
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43
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34
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6
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7
62
52
53
42
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44
32
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35
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7
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8
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53
54
43
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36
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8
3
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5
6
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8
9
n
Total
644
2
672
3
700
As the stair position goes up by 1 the total goes up by 28 each time. So for the stair in position 4 I predict that its total will be 728.
64
54
55
44
45
46
34
35
36
37
24
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28
4
5
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7
8
9
4
5
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8
9
0
My prediction was correct.
The nth term can be worked out by changing the numbers in the 7 step stair in to algebra by making the position number n, then relating all other numbers to the position number, as you up a level on the step stair you add the grid sizes number (in this case 10) to the number below it so is represented by g (grid size) e.g.
n+6g
n+5g
n+5g+1
n+4g
n+4g+1
n+4g+2
n+3g
n+3g+1
n+3g+2
n+3g+3
n+2g
n+2g+1
n+2g+2
n+2g+3
n+2g+4
n+g
n+g+1
n+g+2
n+g+3
n+g+4
n+g+5
n
n+1
n+2
n+3
n+4
n+5
n+6
The sum of this stair= (n) + (n+1) + (n+2) + (n+3) + (n+4) + (n+5) + (n+6) + (n+g) + (n+g+1) + (n+g+2) + (n+g+3) + (n+g+4) + (n+g+5) + (n+2g) + (n+2g+1) + (n+2g+2) + (n+2g+3) + (n+2g+4) + (n+3g) + (n+3g+1) + (n+3g+2) + (n+3g+3) + (n+4g) + (n+4g+1) + (n+4g+2) +(n+5g) + (n+5g+1)= 28n + 56g +56
(28 x 1) + (56 x 10) + 56= 644, this represents the 1st term.
Using this formula I can now work out any term for a 7 step stair on a 10 x 10 grid, e.g. the 55th term:
(28 x 55) + (56 x 10) + 56= 2156
Investigation for any sized stair on a 10 x 10 grid
Step Stair Size
Formula for Finding Total
3
6n + 4g + 4
4
0n + 10g + 10
5
5n + 20g + 20
6
21n + 35g + 35
7
28n + 56g + 56
The 'n' numbers in the formula are all triangle numbers. A triangular number is a number that can be arranged in the shape of an equilateral triangle.
The sequence of triangular numbers for n = 1, 2, 3... is:
, 3, 6, 10, 15, 21, 28, 36, 45, 55...
In order to find 'n' for the desired step stair you must add another of the same sized step stair to it so it forms a rectangle e.g.:
An example for a three step stair.
This then becomes a 4 x 3 rectangle.
Then sum of this rectangle is:
(n) + (n+1) + (n+2) + (n+g) + (n+g+1) + (n+2g) + (n) + (n+1) + (n+2) + (n+g) + (n+g+1) + (n+2g) = 12n +8g +8
The amount of 'n's' is 12, this is double the required number so you must divide it by 2 in order to make it 6.
OR
You can multiply the two sides together:
4 x 3= 12
This is always double the required number so you must divide it by 2:
2 ÷ 2= 6
The nth term for this is: n= (s) (s+1)
2
s= step stair size
For example using a three step stair:
6= (3) (4)
2
The 'g' numbers in the formula are all tetrahedral numbers. A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron.
The first few tetrahedral numbers are:
, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, ...
The usual formula for finding tetrahedral numbers is n (n+1) (n+2)
6
This formula does not work for finding 'g' with our formulee e.g.
If we use a step stair size of 3 and use this as n we get the third tetrahedral number, 10 but we only need the second.
In order to find the 'g' for the desired step stair you must change the formula to:
n=s
s= step stair size
g= (s-1) (s) (s+1)
6
For example using a three step stair:
4= (2) (3) (4)
6
The added numbers at the end of the formula are always the same as the 'g' numbers making them tetrahedral numbers as well so they must be worked out in the same way:
Added numbers= (s-1) (s) (s+1)
6
For exampe using a three step stair:
4= (2) (3) (4)
6
Therefore the formula for working out any sized step stair's total on a 10 x 10 grid is:
This formula also works for any sized step stair's total on any sized grid, for example:
3 step stair on an 8 x 8 grid:
(3) (4) = 6
2
6 x n= 6n
2 (3) (4) = 4
6
4 x g= 4g
2 (3) (4) = 4
6
I can now use this general formula to find out any sized stair on any sized grid, for example a 55 step stair on a 100 x 100 grid:
(55) (56) = 1540
2
540 x n= 1540n
54 (55) (56) = 27720
6
27720 x g= 27720g
54 (55) (56) = 27720
6
This makes the formula 1540n+ 27720g+ 27720.
As I am unable to fully explain how I generated the formula for finding 'g' and the added numbers in the formula, I will use another method to find the formula for them.
The 'g' numbers and the added numbers in the formula are all tetrahedral numbers. Tetrahedral numbers are the sum of consecutive triangular numbers; therefore I will be able to use summation of a series for the formula.
Triangle number formula= (s) (s+1)
2
s= step stair size
g=
The values above and below the sigma symbol represent the highest and lowest values that 'r' takes. 'r' takes all values between these extreme values.
's' was previously used in the triangle number formula but I have now changed it to 'r' as this is the commonly used letter when using summation of a series. I have also changed the lettering because 's' represents step stair size and if 's' was used above and below the sigma symbol the highest value would always equal 0 making the formula incorrect.
is used as it is the triangle number formula and the triangle number are what need to be added to make the required tetrahedral number.
is used as the lowest value of r as it is the first triangle number so must be used as the start of every series.
is used as the highest value because we always need the tetrahedral number that is one less than the step stair size e.g.
Step Stair Size
'g' Number
Tetrahedral Ranking
3
4
2nd
4
0
3rd
5
20
4th
6
35
5th
I shall now test the formula using a 3 step stair:
The added numbers at the end of the formula are always the same as the 'g' numbers making them tetrahedral numbers as well so they must be worked out in the same way:
Added numbers=
For example using a three step stair:
Therefore the formula for working out any sized step stair's total on any sized grid is:
I can now use this general formula to find out any sized stair on any sized grid, for example a 55 step stair on a 100 x 100 grid:
(55) (56) = 1540
2
540 x n= 1540n
27720 x g= 27720g
This makes the formula 1540n+ 27720g+ 27720.
