Staircases

1
Staircases

• A staircase number is the number of cubes needed
  to make a staircase which has at least two steps
  with each step being one cube high.
• INVESTIGATE!

2
Constraints and Ideas

• Constraints
• Each step being one cube high
• A staircase consists of at least 2 steps.
• Ideas
• Staircases with steps ranging from 2 steps,
  altering the number of steps and the number of
  cubes in the first step.
• 3-D staircases

3
Staircases with 2 steps
Using blocks, constructed staircase containing
two steps. With a variable of the number of cubes
present in the first step.

6
5
4
3
2
1
no. of cubes is 1st step

13
11
9
7
5
3
Total no. of cubes
4
Staircases with 3 steps
The same process was used though this time with
three steps making up the staircase.

6
5
4
3
2
1
no. of cubes is 1st step

21
18
15
12
9
6
Total no. of cubes
5
Staircases with More than 3 steps
The same process was used up until there being 6
steps in the staircase and a clear pattern was
beginning to form.
6
5
4
3
2
No. of step in the staircase
No. of cubes in first step
21
15
10
6
3
1
27
20
14
9
5
2
33
25
18
12
7
3
39
30
22
15
9
4
45
35
26
18
11
5
51
40
30
21
13
6
57
45
34
24
15
7
6
Patterns

• I was able to identify that each set of
  staircases (i.e. those with the
• same number of steps) presented staircase numbers
  that formed an
• arithmetic series.

An arithmetic series is when there is a common
difference between each number in the series.
For example, the series representing staircases
of 3 steps 6, 9, 12, 15, 18, 21, 24 . There is
a common difference of 3 between each of the
terms. The same was noticed with the series of
numbers representing staircases of 6 steps 21,
27, 33,39, 45, 51, 57 Where the common
difference is six.
7
Patterns cont.
Used to find term in a series once the first term
is known.

• Arithmetic Series formulas
• Tn a (n 1)d
• Sn n/2(a Tn) n/22a (n 1)d
• Where a the first term in the series
• d the common difference
• T term
• n number of term within the series
• S sum

Used to find the first term in series involving a
large number of steps, for example 15 steps.
In this instance this term stands for the number
of cubes in the first step on the staircase
8
Using Arithmetic Series Formulas.

• For example
• Find the number of cubes required to form a
  staircase that contains
• 100 steps, with the first step being made up of
  100 cubes.
• Sn n/22a (n 1)d
• Where a 1 , n 100, and d 1
• Therefore, S100 50299
• S100 5050
• Therefore if there are 100 steps in a staircase
  and the first step is made up
• of 1 cube there are a total of 5050 cubes in the
  stair case.
• Tn a (n 1)d
• Where a 5050, n 100, and d 100
• Therefore T100 5050 99(100)
• T100 14950
• Therefore a staircase of 100 steps, with the
  first step containing 100 cubes,
• contains a total of 14950 cubes.

9
3-D staircase
Do 3-D staircases present a different pattern?
Restricted to the area formed by a cube so that
staircase are regular and consistent in
shape. Therefore the number of base cubes in a
3-D staircase are the squares of odd numbers.
This process was continued and for the 3 by 3
square the values of 10, 19, 28, 37, 46, 55, etc
were calculated.
10
3-D Staircases cont.
The same process was used for 5 by 5 squares, 7
by 7 squares and 9 by 9 squares.
9 x 9
7 x 7
5 x 5
3 x 3
No. of step in the staircase
No. of cubes in first set of steps
165
84
35
10
1
246
133
60
19
2
327
182
85
28
3
408
231
110
37
4
489
280
135
46
5
570
329
160
55
6
651
378
185
64
7
11
3-D staircase patterns

• An arithmetic series is formed for each sequence
  of
• calculations, as there is a common difference.
  This
• common difference is relative to the size of the
  square
• base. i.e. 9 for 3 x 3, 25 for 5 x 5, 49 for 7 x
  7, etc.
• As set of arithmetic series, a value in the
  series can be
• calculated using
• Sn n/2(a Tn) n/22a (n 1)d.
• Though, as the 3-D staircase is not linear the
  first
• value cannot be calculated using the formula
• Tn a (n 1)d.

12
Conclusion

• Staircase numbers are numbers that can be
  arranged in a number of arithmetic series, in
  which the staircases contain the number of steps.
• Though this is using the constraints stated at
  the beginning of the investigation process.
• Though I am sure with more time and persistence a
  number of ideas, involving staircase numbers
  could be investigated.