Investigation Seminars

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Investigation Seminars

• Yvette Rodwell

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• 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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Polyominoes with 4 squares

• Consider the number of matches needed to make
  polyominoes with 4 squares.

13 matches
12 matches
13 matches
13 matches
Once the configurations of the squares began to
be repeated I realized that all possibilities
had been explored.
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Polyominoes with 6 squares

• Consider the number of matches needed to make
  polyominoes with 6
• squares.

19 matches
18 matches
18 matches
17 matches
19 matches

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• When the configurations of the squares began to
  be repeated
• I realized that all possibilities had been
  explored.

• Maximum number of matches used to make 6 boxes
  was 19.
• Minimum number of matches used was 17.

• The larger number of squares used increased the
  possibilities
• of different configurations.

• The more common sides less number of matches
  used.

Total of 19 matches used with 5 common interior
sides.
Total of 17 matches used with 7 common interior
sides.
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Formula
Number of boxes
x 4 (number of sides in a square)
number of matches used.
number of common sides
(n x 4) - c
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Examples
(n x 4) c (4 x 4) 4 12
(n x 4) - c (6 x 4) - 5 19
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Using a set number of matches

• Consider the number of polyominoes made using a
  set number of matches.
• Using 20 matches

6 squares - 19 matches used 1 left over
7 squares - all matches used
6 squares - 17 matches used 3 left over
7 squares - all matches used
7 squares - all matches used
7 squares - all matches used
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• For all matches to be used, there appeared to be
  a common pattern
• using 7 squares.

(This polyomino of 6 squares 17 matches)
(3 matches of a 7th square)

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• 7 was the maximum number of squares created
  using all 20 matches.
• 6 squares either left 1 or 3 matches.
• 5 squares left too many matches - leaving the
  possibility to complete a 7
• square configuration.

5 squares 16 matches 4 left over
7 squares - all matches used
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• Consider the possibilities by increasing the
  number of matches.
• Using 21 matches

7 squares - all matches used
7 squares - all matches used
7 squares - all matches used
7 squares - 20 matches used 1 left over
7 squares - 20 matches used 1 left over
By adding 1 match it created the possibility to
explore different configurations
12
n 1

• Consider how many matches are needed to change a
  polyomino with
• n squares to n 1 squares.

n 4 (4 x 4) - 3 13 matches
n 5 (5 x 4) - 5 15 matches plus 2 matches
n 5 (5 x 4) - 4 16 matches plus 3 matches
Where the extra square is placed determines the
number of matches required - therefore the
number of matches needed depends on the number of
common sides.
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Common Relationships

• What is the relationship between the number of
  squares, the number of matches needed to make it
  and the number of matches in the boundary?

• 7 squares
• 20 matches used
• 12 matches in the boundary
• 8 common sides

• 7 squares
• 21 matches used
• 14 matches in the boundary
• 7 common sides

The more common sides less matches used in total
14
Tetris
End