Investigation

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Investigation

• By Peter Walter Brooke Nelson

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Introduction

• A staircase number is the number of cubes needed
  to make a staircase which has at least two steps
  with each step (other than the first) being one
  cube high.

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Starting the staircase at a number other than 1

• It is possible to start at a number other than
  one in staircase numbers, as long as each step
  after goes up by one. We discovered many other
  staircase numbers when taking this into account.
  For example, 7 is not a staircase number when
  starting at the number 1, but it is a staircase
  number if you put 3 cubes next to 4 cubes.

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• Staircases go up in sequence. For instance in
  the above graph the number of cubes needed to
  make an additional step goes up in a sequence.
  The sequence will always go up by one (as shown
  above) as each additional step can only be one
  cube high.

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It doesnt matter what number you start at, the
number of cubes needed to make an additional step
goes up by one. E.G. 5,6,7,8
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Patterns

• One of the patterns that was formed in our
  investigation was the triangular number pattern.
  Triangular numbers is where the number of cubes
  or dots makes a triangular pattern. For example.

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• Again if we look at the sequence in which the
  staircase numbers went up by we notice that it is
  still a difference of one. In addition the
  number of cubes needed to make an additional step
  goes up in sequence.

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Pattern of Odd and Even Numbers

• We discovered that the staircase numbers
  (starting at the number 1) go in a certain
  pattern of two odd numbers then two even numbers.
  For example, the first few staircase numbers are
  odd1, 3 then the next two numbers are even6,
  10. This pattern carries on for all of the
  staircase numbers-
• O15, 21, E28, 36,
• O45, 55, E66, 78.etc

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Pattern in last digit of each number

• There is a pattern involving the last digit in
  every number. The first 20 staircase numbers
  (starting at 1) have the last digits being-
  1,3,6,0,5,1,8,6,5,5,6,8,1,5,0,6,3,1,0,0.
• These numbers are then repeated as the last digit
  for the next 20 staircase numbers. The pattern
  furthermore continues for all of the staircase
  numbers.

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Reverse pattern

• We found a reverse pattern in the last digits of
  the staircase numbers in that the digits go-
  1,3,6,0,5,1,8,6,5.
• Then the next digits are the same but go in a
  reverse pattern- 5,6,8,1,5,0,6,3,1. However there
  are two 0s that are present at the end of the
  sequence which do not fit into this pattern.

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Fibonacci Sequence

• The concept involved in the Fibonacci Sequence
  (adding two numbers to get the next number) does
  not work in some investigations with staircase
  numbers. For example, when adding the staircase
  numbers 3 and 5, the result is 8, which is not a
  staircase number at all. The next staircase
  number is 6, proving that the Fibonacci Sequence
  doesnt work in this instance for staircase
  numbers, because 35 doesnt equal 6.

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Leonardos Leaps

• Leonardos leaps is where one investigates how
  many times someone can walk up a set of stairs by
  either taking one at a time or two at a time.

Leonardos leaps shows a common pattern known as
the Fibonacci sequence. The Fibonacci sequence
occurs where the last two numbers are added
together to make the next number in the sequence.
So if we look at the leaps made column we can
see a Fibonacci sequence occurring.
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Leonardos Leaps cont
Relationships If we look at the following table
we can see a pattern arising. If we add the
number of steps in a staircase to the number of
cubes needed to make the last staircase, it will
give us the number of cubes needed to make the
next staircase. At first we thought that this
pattern was a Fibonacci sequence. But upon
further investigation we found out that it wasnt
as the Fibonacci sequence requires the last two
number to be added together to produce the next
number as opposed to our pattern where the sum of
the last two numbers is added to the next number.
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Can all staircase numbers be evenly divided by a
certain number?

• We discovered that there is no single number
  which divides evenly into every staircase number,
  other than the number 1.
• We investigated and found out that there are
  three numbers which divide evenly into different
  staircase numbers, (when starting the staircase
  at one), including- 1, 3 and 5. For instance, at
  least one of these three numbers go evenly into
  the staircase numbers. For example- 78 (divisible
  by 3), 91(divisible by 1), and 105 (divisible by
  5.)

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All odd numbers are Staircase Numbers

• From the information that a staircase can start
  at any number, we discovered that all odd numbers
  are staircase numbers. We developed the formula
  of
• ( n1 ) -1 Staircase number
• 2
• Where n is an odd number. For example, starting
  with the odd number 15-
• (151) -1 7
• 2

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Scenario

• A boy had to build a staircase made up of cubes
  to reach his front door. He knew that he needed
  10 steps to get there, but how was he to know how
  many cubes were needed? The boy pondered for a
  while but then came up with the formula-
• c n1
• Where c is the amount of cubes needed for the
  next stair and n is the amount of cubes he is
  standing on at the moment.

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• But would this formula work? The boy investigated
  upon this formula and found-
• Starting with 1 cube c 11
• c2
• 2 cubes c21
• c 3
• 3 cubes c31
• c4

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• The boy continued this formula for the 10 stairs
  and added each amount of cubes to find that 55
  was the total amount of cubes needed to reach his
  door.
• 1234567891055
• The boy discovered that finding staircase numbers
  was another way to solve his problem, where he
  needed to find the 10th staircase number in order
  to discover how many cubes he needed. His result
  was also 55 cubes, as seen below, the staircase
  numbers in bold
• 123364105156217288369451055

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Another Formula Needed

• We investigated in aim to find how many cubes are
  needed to make 10 steps without knowing the
  previous amount of cubes needed beforehand. We
  were unsuccessful in this investigation as we
  found that knowledge of the prior numbers is
  essential in finding out the number of cubes
  needed for 10 steps.

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Not Staircase Numbers

• The following numbers are not staircase numbers
  between 1 and 20
• 1, 2, 4, 8, 16
• So if we look at the numbers which are not
  staircase numbers a pattern arises. Each number
  apart from one is doubled from the previous
  number. So double one is two, double two is
  four, etc, etc

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Pattern of Non Staircase Numbers

• So if we put this into the following pattern
• 20 1 26 64
• 21 2 27 128
• 22 4 28 256
• 23 8 29 512
• 24 16 210 1024
• 25 32
• Therefore each additional power of two is not a
  staircase number!

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Brooke Petes Breakthrough!

• Substitution
• Substitution We dont know if this is the
  correct term to use, and if its not well patent
  the idea and become extremely rich! We define
  substitution as taking away the cubes from the
  front of the staircase and putting them at the
  back of the staircase until we get a perfect
  staircase. Through this we discovered that all
  numbers are staircase numbers, except for numbers
  with the powers of two. For example

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E-Folio

• We decided that we would put this investigation
  in curriculum knowledge as we have learnt
  computer and investigation skills which we
  believe is vital to passing on to the next
  generation!
• THE END