Polyominoes using Matches

1
Polyominoes using Matches

• Polyominoes are made up by a number of squares
  connected by common sides

2
Analysing the Question

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

3
Where to start.

• Definition of a Polyomino
• A plane shape made up of squares of the same
  size, each square being connected to at least one
  of the others by a common side or edge.
• Examples.
• Domino
  Petomino
• Triomino
• Tetromino

4
Drawing

• I Firstly started out making the polyominoes with
  matches, which I thought would be a small task
  but this wasnt the case.
• I found that there are different types of
  polyominoes, free polyominoes which are different
  under translation, rotation and reflection and
  there is fixed polyominoes which must be
  different only under translation.
• I then went to drawing them with pen and paper so
  I could see all the different options on the one
  page. This was a small task until I reached six
  squares so I decided to research the
  configurations on the internet as I was curious
  of how large the configurations would get, out of
  this research I formed this table in excel for
  configurations on free polyominoes.

5
Researched these configurations
6
Formula
S1
S2
S3

• 1match (3 x n) total number matches
• ntotal number of squares.
• So in this example, start with S1 and want to add
  two squares the equation will look like this
• 1 match ( 3 x 3) 10 so ten is the total no.
  matches
• The lines represent the matches
• But this formula doesnt take into account for
  the different shaped polyominoes, that dont
  need to add 3 matches.

7
Hint sheet

• Gave new options to explore
• Consider the number of matches needed to make
  polyominoes with 4 squares
• Consider polyominoes with various numbers of
  squares
• Consider the number of matches in the boundaries
  of the polyominoes
• Relationships between the variables like number
  squares, number of matches, number in boundaries
  and number in interiors.

8
new leads

• Hint one discover number matches needed for a
  poly with 4 squares
• Only five different ways the matches can be
  arranged differently without repeating, in the
  form of rotating and mirroring etc
• Out of all five of these configurations, the
  total number of matches needed varied. Four
  configs have the total of 13matches while one
  gave 12matches as total.
• In using my previous formula 1(3x4)13 total.
• This change in answers leads to the next idea.

9
Minimums and Maximums

• With the formula now challenged, I decided on a
  new path.
• Look at the minimum and maximum number of matches
  in the boundaries of polyominoes up ten squares

10
Patterns forming
From the table recorded a pattern was shown, as
the squares increase the minimum, every second
number (odd) increases by 2. So no increase than
2 Where as the maximum number of the boundary
matches increases by 2 matches every time the
squares increase. So from showing this pattern a
number could be predicted.
Number of squares
11
Conclusion

• In working and researching polyominoes I have
  found them quiet interesting and a wide range of
  possibilities you could work with them.
• This powerpoint presentation I will be placing
  under the curriculum knowledge content area in
  my efolio