Matches'

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Bek Salmon EMM111 Maths Investigation.
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Matches
SCORE
4 2 3
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Matches.

• Polyominoes are made up of a number of squares
  connected by common sides. Thirteen matches were
  used to make this one with four squares.
  Investigate the numbers of matches needed to make
  others.

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Focus questions.

• What effect does shape have on a polyomino?
• What can affect the amount of matches used?
• Are there any patterns to be found?

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The first step was to experiment with
polyominoes. At first the process was simple
exploration, and no patterns seemed to emerge.

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• 4

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5
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14
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It was clear that some sort of system had to be
implemented. When keeping the number of matches
constant, no definite patterns emerged. However,
when keeping the number of squares the
same, for every Polyonimo group, the total amount
of matches needed went up one by one according
to the arrangement . For example a polyonimo
group of six squares could be made from 17, 18,
or 19 matches, depending on the number of sides
that were shared. The more shared sides there
were, the less matches were needed, per amount of
squares.
Matches 19
1 2 3 4 5 6
Matches17
Matches 18
1 2 3 4 5
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• 2
• 4
• 5 6

Shared sides6
Shared sides7
Shared sides5
It then became evident that the sum of a)the
total number of matches needed, and b)the matches
that formed the shared sides, equalled 4 times
n)the amount of the total squares, or ab4n.
Using the first example above, the equation would
be 1774x6, and the answer is 24.
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This equation (ab4n) worked when applied to all
polyomino groups. See examples below.
a) 15
b) 5 n)5
Matches 23
Squares 8.
a) 13 b)3 n)4
23932 8x432.
Shared sides 9
15520, 5x420.
a)18 b)6 n)6
13316 4x416
a)10 b) 2 n) 3
10212 3x412.
18624 6x424.
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I formed a prediction, which was that if four
matches made a square, that in any polyomino
shape, there would be 3 matches per square plus
two.I tested this theory extensively.It proved
to be almost conclusively wrong in every case.
(so much for my theory).
However, it paved the way for a second theory
which was that in every polyomino that used the
maximum number of matches, there would be 3
matches per square plus one. This one worked.
22 matches (maximum amount for 7 squares)
Formations of seven squares
This could be written as an equation where
ssqaures mmatches M3s1
7x321 21122 M22
20 matches
21 matches
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A third pattern occurred when the border sides of
the polyominoes were examined. For every
polyonimo, there would be a number of border
sides (see below), and the number of border sides
would again depend on the number of shared sides.
The borders didnt initially show any pattern.
But when taking the highest possible number of
border sides for each polyonimo group, the
numbers formed a pattern.
Border sides12.
The matches increase by three every time, And
the borders increase by two.
This formula could be applied to all polyonimoes,
eliminating the need to draw them by hand.
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Conclusions.

• My maths investigation involved lots of trial and
  error, and forced me to think more deeply than I
  normally would in mathematics.
• I managed to answer my focus questions, although
  not as conclusively as I had hoped.
• The shape of a polyomino affects the amount of
  matches needed to make it, as well as varying the
  boundary matches and shared sides.
• The amount of matches used, depends on the number
  of squares and shape of a polyomino.
• There were three patterns that I could find, two
  of which could be applied as an equation.
• Below is proof that this assignment is embedded
  in my E-portfolio.