### Archimedes* Puzzle (Square)

```Archimedes’ Puzzle
(Square/Box)
Morgan Allen
History
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Also known as Stomachion
World's oldest puzzle - 2200 years old
Traces its roots back to the Ancient Greece
An ingenious inventor Archimedes of Syracuse
(287 B.C. - 212 B.C.)
• Was discovered accidentally, in 1846, but then
again lay in obscurity for over a century.
What is it?
• The puzzle consists of 14 pieces of various
polygonal shapes originally forming a square.
• Either 3, 4, or 5 sided
• Objects of the puzzle are to rearrange the
pieces to form its original square or to form
interesting shapes
• People, animals, geometrical shapes, etc.
Activity
• Begin with the 12 X 12 square
• Cut the square into these 14 pieces
• Put them back into the square
Activity
•Make the triangle
Hint
Colorful Hint
Other shapes?
Find Area
Lattice Points:
Interior- blue
Boundary- red
Pick’s theorem (1899)
• Area = I +B/2 – 1
• I = number of interior lattice points
• B = number of boundary lattice points
Figure 1: Interior points: 7
Boundary points: 12
7 + 12/2 – 1 = 12
Figure 8: Interior: 18
Boundary: 14
18 + 14/2 – 1 =24
Solutions – notice anything?
1: 12
2: 12
3: 12
4: 6
5: 6
6: 3
7: 21
8: 24
9: 12
10: 3
11: 6
12: 6
13: 9
14: 12
Why use it?
• Creation of diff. objects by rearranging pieces
– animals, plants: an elephant, a tree, a barking dog,
a ship, a sword, a tower
• It exercises and develops memory skills in
young children by having them put the pieces
back in their box.
• Plays into a math class, fun, interesting
Fun Fact
• How many solutions?
Questions
• What grade levels can this be used and which
activity would go with it
• Discussed areas all multiple of 3…
Does it matter that the square is 12 X 12?
References
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http://www.math.nyu.edu/~crorres/Archimedes/Stomachion/construction.html
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http://www.math.nyu.edu/~crorres/Archimedes/Stomachion/Pick.html
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http://www.gamepuzzles.com/536solt.htm
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http://illuminations.nctm.org/LessonDetail.aspx?id=L720
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http://www.puzzles.com/puzzleplayground/Stomachion/Stomachion.htm
The End
Archimedes’ Puzzle
(Square/Box)
Morgan Allen
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