EEM111 OPENENDED INVESTIGATION

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EEM111 OPENENDED INVESTIGATION

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Title: EEM111 OPENENDED INVESTIGATION

1
EEM111OPEN-ENDED
INVESTIGATION

The MatchstickProblem

NICHOLAS MORGAN, LEANNE McCARTHY
2
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.

3
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.

4
The Numbers

The following chart shows the initial results of
my investigation.

5
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.

6
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

7
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 !!

8
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

9
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.

10
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

11
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 !!
12
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