ODDS N EVENS

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ODDS N EVENS

• By Michael Armstrong
• And
• Greg ONeill

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ODDS N EVENS

• Task
• Investigate number sequences formed according to
  the following two rules
• If a number is odd, the next number is 1 less
  than it
• If a number is even, the next number is half of
  it
• Example Sequence for 106
• 106, 53, 52, 26, 13, 12, 6, 3, 2, 1, 0.

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ODDS N EVENS

• Aspects of investigation
• Are there any sections of sequences that may be
  common to several sequences?
• How do the sequences end, and how often does this
  occurs?
• In what order do the endings occur?
• Investigate working backwards
• Consideration of consecutive starting numbers

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ODDS N EVENS

• Initially, we drew up a table of the patterns
  formed following the rules stated, from 0 to 19
  as starting numbers.
• The first thing that stood out on the table was
  the fact that every pattern, except 1 and 0,
  ended with the same three numbers.
• 2
• 1
• 0

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ODDS N EVENS

• Whilst we explored how the sequences ended
  (2,1,0) we also discovered that several
  consecutive starting numbers usually ended up
  following the same steps as one another within 1
  or 2 steps.
• To further test this, we randomly tested a group
  of consecutive starting numbers.

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ODDS N EVENS

• When looking at the numbers 120 through to 123 as
  consecutive starting numbers, it can be seen that
  they reach the same pattern within 2 moves of
  each other.
• Furthermore, when looking at the top table, it
  can also be seen that 60, 61 and 62 reach the
  same pattern within a step of each other.
• In testing 60 through to 63, we confirmed that
  the pattern where by 4 consecutive starting no.
  reached the same pattern within two steps.
• We also noticed that the first number in each
  group of four consecutive numbers was divisible
  by 4.
• We cross checked this with our initial table, and
  it proved to be correct.
• If you doubt us well prove it!!!!

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ODDS N EVENS

• WORKING BACKWARDS
• Working backwards proved to be our biggest
  problem during this investigation, as it resulted
  in infinite possibilities
• Starting 0, the only possibility, is 1
• 1 then goes to 2, as every odd number has to be
  doubled
• Once 2 is reached, there is the option of either
  doubling, or adding one, as an even number double
  will always equal an even number, though if you
  add one, the rule still applies as it would
  produce the next consecutive number, which would
  be odd.
• As a result of these options, every time an even
  number is presented whilst working backwards,
  there is always another way in which the pattern
  could go
• As a result of this, when attempting to
  illustrate these finding on a diagram, it became
  too crowded to follow after as few as 6 steps.

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ODDS N EVENS

• Other Discoveries Made
• We discovered some starting numbers led to
  patterns with far fewer steps then those on
  either side.
• All numbers that were divisible by 4 were the
  lowest in each group of four consecutive numbers.
• 20, 10, 5, 4, 2, 1, 0
• 21, 20, 10, 5, 4, 2, 1, 0
• 22, 11, 10, 5, 4, 2, 1, 0
• 23, 22, 11, 10, 5, 4, 2, 1, 0
• The more factors a starting number had, the fewer
  steps it took to reach zero
• 16 (factors 1,2, 4, 8, 16)
• 15, 14, 7, 6, 3, 2, 1, 0
• 16, 8, 4, 2, 1, 0
• 17, 16, 8, 4, 2, 1, 0
• A majority of prime numbers had more terms than
  the staring number on either side
• 78, 39, 38, 19, 18, 9, 8, 4, 2, 1, 0
• 79, 78, 39, 38, 19, 18, 9, 8, 4, 2, 1, 0
• 80, 40, 20, 10, 5, 4, 2, 1, 0

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ODDS N EVENS

• Other Discoveries Made
• When looking for any other patterns that may be
  produced, the only part that stood out was the
  first step for each starting number
• As can be seen on the table, the number is red
  doubled equals the next number in yellow.
• The red numbers also go up consecutively by one,
  and the yellow numbers go up by two
• Furthermore, it can also be seen that the numbers
  in yellow on the right are equal to the previous
  sequences even starting number.

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ODDS N EVENS

• Conclusion
• In conclusion, during this investigation, we
  discovered that there were infinite ways the
  sequences could be made, and as a result
  demonstrated that no matter what the starting
  number, the sequence would always work as long as
  you kept to the two rules stated
• We also discovered that whilst there were very
  few concrete patterns formed, all sequences end
  in the same way, and that many parts of various
  sequences can be shared.

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ODDS N EVENS
THE END!!!!
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Thanks for watching!!!!