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Archimedes’ Puzzle (Square/Box) Morgan Allen History • • • • 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 – Basic to Advanced • Discussed areas all multiple of 3… Does it matter that the square is 12 X 12? References • http://www.math.nyu.edu/~crorres/Archimedes/Stomachion/construction.html • http://www.math.nyu.edu/~crorres/Archimedes/Stomachion/Pick.html • http://www.gamepuzzles.com/536solt.htm • http://illuminations.nctm.org/LessonDetail.aspx?id=L720 • http://www.puzzles.com/puzzleplayground/Stomachion/Stomachion.htm The End Archimedes’ Puzzle (Square/Box) Morgan Allen