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Staircases

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!. Constraints and Ideas. Constraints: Each step being one cube high A staircase consists of at least 2 steps. Ideas:

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Staircases

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  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. no. of cubes is 1st step 1 2 3 4 5 6 … Total no. of cubes 3 5 7 9 11 13 …

  4. Staircases with 3 steps The same process was used though this time with three steps making up the staircase. no. of cubes is 1st step 1 2 3 4 5 6 … Total no. of cubes 6 9 12 15 18 21 …

  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. No. of step in the staircase 2 3 4 5 6 No. of cubes in first step 1 3 6 10 15 21 2 5 9 14 20 27 3 7 12 18 25 33 4 9 15 22 30 39 5 11 18 26 35 45 6 13 21 30 40 51 7 15 24 34 45 57

  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/2[2a + (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/2[2a + (n – 1)d] Where a =1 , n = 100, and d = 1 Therefore, S100 = 50[2+99] 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. No. of step in the staircase 3 x 3 5 x 5 7 x 7 9 x 9 No. of cubes in first set of steps 1 10 35 84 165 2 19 60 133 246 3 28 85 182 327 4 37 110 231 408 5 46 135 280 489 6 55 160 329 570 7 64 185 378 651

  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/2[2a + (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.

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