Leapfrog Investigation

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Leapfrog Investigation

• Adam Ross Brett Klotz
  Clay Starr

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The rules for the leapfrog investigation are
simple1. The counters can slide one spot
(either right or left) into a vacant space.2.
They can also hop over another counter but only
into a vacant space.3. The red counters have to
end up where the black counters began.4. The
black counters have to end up where the red
counters began.
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The first aspect that we decided to investigate
was the minimum number of moves required to
complete the task.After having many attempts at
this we found that we kept coming up with
different answers.We therefore decided that we
needed to add another rule to the task.This
would be that the black counters could only move
right and the red counters could only move
left.
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We were then able to work out, by trial and
error, that the minimum number of moves required
to complete the task was eight.Below is an
example of the eight moves required to complete
the task.
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The next aspect that we decided to investigate
was the minimum number of moves required when
more counters were added.We added a black
counter and a red counter to each side, and used
the same pattern we used for four counters, to
complete the task with six counters.The
following is an example of how leapfrog works
with six counters.
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After this we repeated the process with eight
counters and ten counters to try and find some
patterns in what we were doing. The following
is a table of the results that we found.
Number of counters
Number of moves
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Pattern Analysis

• This part of the investigation began to look at
  any patterns that may have occurred.
• The first obstacle encountered was developing an
  effective way of recording the moves taken.
• We needed a way to search for any patterns and
  to compare results when different counters were
  used.
• Two methods were tried.

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First Method Drawing The Moves

• In the first method we drew where the counters
  moved to and labeled the lines with the moves.

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First Method Drawing The Moves

• This method turned out to be very hard to read
  and made interpreting and comparing patterns near
  impossible.

I cant read this mess! Lets try something new!
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Second Method Recording The Moves

• In the second method we recorded the moves in
  tables
• This enabled us to clearly see the moves and look
  for any patterns

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Looking for the Patterns

• After discovering an effective way of recording
  the moves we then started looked for any patterns
  using the results from 2 to 10 counters.
• We discovered
• Adding the previous amount of turns and
  counters, then adding three, will give me the
  minimum amount of moves for the next amount of
  counters.
• Example I want to find out how many moves it
  will take me for 8 counters.
• 6 counters 15 moves
• 6 15 3 24 moves
• The minimum amount of moves it will take for
  8 counters is 24

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Is there an easier way?

• After discovering this pattern we started to
  wonder if it is possible to find a formula that
  does not require knowing the previous amount of
  moves.
• We wondered what it might look like if we made a
  graph of the points that we had already found.
• If it made a straight line we could use the
  formula for a straight line to help us to make a
  formula for amount of moves.

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(No Transcript)
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Looking for the Formula

• Unfortunately it did not form a straight line.
• At this stage of the investigation we may have
  been able to figure out what the line actually
  was i.e. an exponential etc. and use the equation
  for that.
• However we did not have enough knowledge on the
  subject, or enough time, to be able to research
  this possibility in depth.
• A better formula would eventually be found.

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Adding Spaces?????

• Adding a spare space to the leapfrog
  investigation opens up a new dimension to this
  task.
• The same strategy was used to attempt this
  problem that was used to initially attempt the
  minimum amount of moves for a single space.
• Once this had been stated we went through a
  trial and error process to find the amount of
  moves required for each additional square added.

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TWO SPACES
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THREE SPACES
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FOUR SPACES
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Comparison of Spaces to Moves
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FORMULA

• 6 (n x 2)
• n Number of spaces
• Example for 7 spaces
• 6 (7 x 2)
• 6 (14) 20
• Examples for 10 spaces
• 6 (10 x 2)26
• Check answers

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FACTS

• No matter what the number of vacant spaces there
  is always an even amount of moves.
• Each time a space is added it only increases by 2
  moves.
• To get the shortest amount of moves no counters
  are required to move backwards.
  

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In this presentation we have addressed

• Minimum amount of moves required and the problems
  we faced
• What happened when additional counters were added
• The process of linking a formula or equation to
  this leapfrog problem
• Trying to find a pattern through the use of
  graphs
• And finally the differences in adding an extra
  space and discovering a formula to apply to this
  task

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Conclusion

• Throughout this assignment we have learnt the
  importance of completing an open ended task.
• The different thought processes used and how our
  imagination was allowed to explore this task.
• An open ended task allows the student to think
  outside the box. It provides them with the skills
  to attempt problems from various angles which
  they can take into the problems in their
  profession and everyday life.