Assignment 2 Investigation Seminars

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Assignment 2Investigation Seminars

• If nodes (points) are placed around the
  circumference of a circle, what is the maximum
  number of regions that can be formed.

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Investigation Process

• To establish what the task is asking
• What are nodes, regions and chords
• To look at logical reasoning
• Question were asked to find a solution
• Put pen to paper and start to experiment
• Table information found and look for any patterns
  that may lead to a solution
• Draw on conclusion

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What is this task wanting us to do?

• Nodes
• Chords
• Regions

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Logical Reasoning
A circle has 360 degrees if we place a node at
each degree join them with a straight line and
count the regions.

• Investigation was complicated
• Infinite amount of nodes
• Whole circle would be coloured

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Questions asked tofind Possible Solution

• Ten circles were drawn starting with one node, up
  to ten nodes. Results from the following
  questions were answered and the results were
  tabled.
• Are there any shapes being formed and what are
  they?
• Is there any correlation with the number of nodes
  and regions
• How many chords are drawn from each node?
• How many intersections are created with each
  additional node?

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Were there any shapes formed?

• There were many shapes formed and noticeably
  regular polygon shapes. Regular polygon shapes
  are hapes that have sides of equal length and the
  angles are equal, for example with 10 nodes a
  decagon was formed.

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Shapes formed
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Formula Found!!!!!

• Through trial and error the following formula was
  found
• Minus three from the total number of nodes around
  the circumference of the circle
• Multiply that number by ½ the number of corners
  formed when nodes are joined
• Then add that figure to the total number of nodes
• Which gives you the number of lines
• Example- a circle with 4 nodes the formula would
  look like this (4-3)2 4 6

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Problems found

• The number of lines had no correlation with the
  amount of regions formed.
• Placement of nodes varied the amount of regions
  formed.
• Back to the drawing board
• Tedious task to count the amount of regions and
  chords for example when there were ten nodes the
  amount of regions were 223.
• Numerical patterns were searched for in the table
  of results.

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Table of results
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Patterns Foundnodes and regions

• No. of regions doubled with every addition of a
  node
• Problem
• Pattern stoped after the fifth node.

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Possible Solution?nodes, regions, no. of chords
and no. of intersections
Maybe it has something to do with the amount of
chords and the amount of times they intersect???
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Possible Theory

• If you look at the amount of chords and
  intersections it would suggest that a formula may
  be possibly worked out if three chords or less
  intersect only once in the centre of a circle.

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Reassessment

• This investigation started with the focus on
  placing nodes around the circumference of a
  circle and joining them with straight lines.
  After looking at possible solutions and not
  finding an answer reassessment of the task led
  the investigation into a different direction.

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Conclusion

• If curved lines were drawn maybe it would be
  possible to have no more than three chords
  intersect more than once in the circle.
• A formula would still be required as infinite
  nodes could be drawn and it is a very tedious
  task counting all the chords, regions and
  intersections