EEM111 OPENENDED INVESTIGATION

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EEM111OPEN-ENDED
INVESTIGATION

• The MatchstickProblem

NICHOLAS MORGAN, LEANNE McCARTHY
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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 4 squares
• Investigate the number of matches needed to make
  others.

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Some Simple Terms

• In the presentation I will be referring to
  Blocks. This is when 4 squares are connected in
  such a way that they make one larger square.
• In this polyomino, there are 4 blocks
• It should also be noted thatthe number of blocks
  isalso equal to the number ofenclosed vertices
• WHY?As these are found at thecentre of the
  blocks.

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The Numbers

• The following chart shows the initial results of
  my investigation.

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The Relationship

• From these numbers, a number of equations can be
  seen to hold true.
• If a polyomino has no blocks than the equation
  is m 3s 1
• With 1 block m 3s
• With 2 blocks m 3s 1
• With 3 blocks m 3s 2
• To test these, simply substitute the no. of
  squares, s, and the no. of matches, m, and see
  whether the left-hand side does equal the
  right-hand side.
• From this it can be seen that by changing the
  number of blocks, the variant is that of the
  number following the constant 3s.
• As the no. of blocks b increases by 1, the
  variant v decreases by 1. Therefore the equation
  m 3s (1-b) can than be made.

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Exterior Matches / Boundary

• The following chart shows the number of exterior
  matches, e, when a polyomino has s amount of
  squares and b amount of blocks

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The Relationship

• Again, it can be seen that a pattern is
  occurring. In this case, as the number of squares
  (s) increases by 1, the number of exterior
  matches (e) increases by 2 starting (with 0
  block, b) at 4. This allows the following
  equations to hold true.
• With 0 block e 2s 2With 1 block e
  2sWith 2 blocks e 2s 2With 3 blocks
  e 2s 4
• Again, a simple substitution show that these hold
  true
• In this case, the constant is 2s as the
  changing b value has no effect on it. As b
  increases by 1, variant v decreases by 2.
• By adding b into the equation, an overall
  equation of e 2s (2
  2b) can be formulated !!

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Interior Matches

• The following chart shows the number of interior
  matches, i, when a polyomino has s amount of
  squares and b amount of blocks

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The Relationship

• Again, a pattern can be seen to occur. In this
  case, as the number of squares (s) increase by 1,
  the number of interior matches (i) also increase
  by 1 starting (with 0 block b) at 0. This then
  allows the following equations to hold true
• With 0 block i s 1With 1 block i
  sWith 2 blocks i s 1With 3 blocks i s
  2
• In these instances the constant is that of s, as
  it is not effected by the value of b.
• The formula i s (b 1) is then
  created.
• While it may be convenient to have the exterior
  and interior matches formulae , there is a more
  practical way in finding the number when both
  need to be found. This is seen through the
  realisation that m e i , and so by using
  this, only one of the formulas is needed to find
  the other.

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Putting it to the test A worked
example

• The following shape has 11 squares (s) and 2
  block (b). Find
• 1. The number of exterior matches e Use
  equation e 2s (2 2b) e 2(11)
  (2 2(2)) e 22 2
  20
• 2. The number of interior matches i Use
  equation i s (b 1) i (11)
  ((2) 1) i 11 1
  12
• 3. The total amount of matches m Use
  equation m 3s (1 b) m 3(11)
  (1 (2)) m 33 - 1
  32
• OR m e I
  20 12 32

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Squares vs. Max No. of blocks

• Being an open-ended investigation, I sought out
  to see whether a relationship or pattern occurs
  when looking at the max number of blocks b, with
  a given amount of squares s.
• The following Table shows the raw data.

It is seen here that as number of squares
increases by 1 (starting at 4 as that is where
the first block appears) the max b has the
following pattern in terms of the value staying
the same and increasing occurs Same,
increase, same, increase2, same, increase2,
same, increase3, same, increase3, same,
increase4,same and this pattern continues so
that for every two same b, the number of
increasesincreases. If the no. of squares is
that of a square number, than it can also be seen
to be located at the 1st of the new increased b
value !!
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Final Thought

• In looking at an open-ended investigation, the
  number of different possible formulae patterns
  which could be found all depend on
• the individuals / groups interest
  in the project the amount of
  effort put in by these people
• the mathematical abilities
  of a student the creative
  abilities as well as the amount
  of time to complete it