LEAP FROG INVESTIGATION

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LEAP FROG INVESTIGATION
By Gillian Stewart and Stephanie Nicholls
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Investigation
On a row of five squares, two red counters and
two blue counters are placed like this The
counters can slide one place (either left or
right) into a vacant space or they can hop over a
counter, again only into a vacant space. Can you
use a combination of these moves to finish with
the blue counters where the red counters were and
the red counters where the blue counters
were? What would the least number of moves
needed? What if there were more counters?
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The first problem
6.
1. 2. 3. 4.
7.
8.
9.
5.
4 counters 8 moves
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Prediction.
After solving the first problem, and realising
that with 4 counters the least number of moves
was 8, our prediction was The least amount of
moves will be double the amount of counters. For
example, 4 counters 8 moves 6 counters 12
moves
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After making our initial prediction we
experimented with 6 counters on 7 squares. We
found that it took 15 moves, therefore making our
prediction incorrect.
We then decided to simplify the problem by
reducing the number of squares to 3, and
discovered that it only took 3 moves.
3.
1.
2.
4.
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New Prediction As we can see from this table,
the number of moves increases by 5, then 7.
Therefore we predict that by having 9 squares and
8 counters, the number of moves will be 15 9,
which is 24.
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Testing our prediction We tested out our
prediction first by using 9 squares, and then 11
squares. The answers proved our prediction
correct.
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Strategies and patterns
Through trial and error we realised that we can
not move a blue counter directly behind another
blue counter, or a red counter behind another red
counter.
Through using the two types of moves, slides and
hops, we discovered patterns emerging
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• The amount of sliding moves equals the number
  of counters.
• For example, the pattern of slides and hops for 4
  counters is
• S H S H H S H S
• From this above example we can also see that the
  pattern is a palindrome (i.e. is read the same
  forwards as it is backwards).

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Examples of patterns
Slide and hop pattern S H S HH S HHH S HH S H
S S H S 2(H) S 3(H) S 2(H) S H S
Slide and hop pattern S H S HH S HHH S HHHH S
HHH S HH S H S S H S 2(H) S 3(H) S 4(H) S 3(H) S
2(H) S H S
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Comparison of Slides and Hops
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Formula We discovered a pattern by comparing the
number of squares to the number of moves. We
developed a formula which allows us to calculate
the number of moves for any amount of counters or
squares. This formula is Total number
total counters (blue counters)2 of
moves OR Total number total
counters (red counters)2 of moves Where X
total moves, C total counters, B ½ counters,
blue or red i.e. X C B²
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Application of Formula
X C B²
Counters 8 X 8 4²
24 From previous results we can
see that our formula is correct and can be
applied to any number.