Leanne McCarthy presents Matches

1
Leanne McCarthy presents 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.

2

• After drawing polyominoes of the same number of
  squares in different positions, I made my 1st
  prediction.
• Polyominoes that have boxes inside them use one
  less match then those that do not have boxes.
• Polyominoes of different shapes without a box
  inside the shape use an equal amount of matches.
• I checked this prediction with 6 squares. It
  proved to be correct.
• A new prediction occurred. If 2 boxes were inside
  a shape the number of matches would be two less
  than the those without boxes.

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17
17
18
19
19
19
19
19
19
3
I checked this prediction for 7 squares and this
proved to be successful. It is here I noticed
that the numbers of matches was moving up by
threes.
I decided on this formula n x 3 1 polyominoes
without a box n x 3 squares with one box n x 3
-1 squares with two boxes
22
12
17
4
As the squares grew larger the formula changed
if there was more than one box such as 8 squ was
n x 3-2 m. 10 squ was n x 3 -3 m. I made a
prediction increase the last digit by one.This
gave the correct amount of m but only on even
numbers odd numbers repeat the previous formula
e.g.11squ is n x 3 -3 m . Predict that add 1
for every second or even number. Therefore say 12
polys m n x 3 -4 this works so check 24 polys m
n x 3 -10 62 works for some polys but not for
blocks of 24 polyominoes.
58
62
5
It is obvious very quickly how many vertices
are enclosed within the boundary of the squares
that build a polyominoes. n x 4 v.
4
4
4
4
4
4
4
4
4
4
6
This chart represents a pattern that emerged.
Every odd number of squares went up 2 matches,
even squares went up 3 matches. Outside vertices
also were the same for even and odd squares.
7
I tried a different angle, looking at the
previous number of matches to the current number
of squares. e.g 13-5 8, this did not work in
any formulas. Then I tried the current answer
5-16 11. The original difference appeared to be
n x 2 1 e.g 5 to 11. Looking for a way to
predict the number of matches for larger
polyominoes. If I multiplied the number of
squares by 2 1,then added this to the original
number of squares. For e. g. (5 x 2) 1 5 m.
I tried this for 8 squares. It works for 8,
checked larger numbers. This formula does not
work for polyominoes that have boxes inside
them.
8
Different strategy -This pattern emerged if I
times the number of squares by 4 and then
subtracted the previous number of squares the
correct number of matches resulted for the
polyominoes without boxes inside them. Leading to
this formula of which I will use for a prediction.
n x 4 - (n -1) This was not useful for the
1sts square or the square that contained a box or
two boxes.
9
I decided to put the numbers in a column and I
could see a pattern with 17 squares and 45
matches. 21 squares had exactly 10 more matches.
Then I saw that squares 4 matches 10.
Therefore 12 4 32 10. Formula n 4 m
10 and also n 2 M 5. Formula n 2 m 5
10
We can see several patterns here. 1. The outside
vertices from 3 squares are 6-4-6-4. Every even
number is 4 every odd is 6. 2. Boarders move up
by two repeating themselves e.g. 8,8. Every odd
square the boarders are 5, every even squares
the boarder is 4.