GEOMETRIC DISSECTIONS 1. Introduction A geometric

GEOMETRIC DISSECTIONS
JAPHETH WOOD
Abstract. We explore the theme of area with some classic geometric dissections.
1. Introduction
A geometric dissection is a decomposition of one shape into finitely many pieces that can form a different
shape of the same area. There is a long history, going back at least as far as the ancient Greeks. In fact,
Euclid Proposition 14 (Book II) shows a (straight edge and compass) construction of a square of equal area
to a given rectangle.
Sam Loyd (1841–1911) was a great American puzzlist, perhaps most famous for popularizing the fifteen
puzzle, as well as the tangram puzzle. Several of the problems in this selection are taken from Martin
Gardner’s Mathematical Puzzles of Sam Loyd [4]. Gardner edited the problems from Loyd’s Cyclopedia of
Puzzles [5] which is now in the public domain and available for download.
Other problems here are from Henry Ernest Dudeney (1857–1930), a British puzzlist and mathematician.
A contemporary of Samuel Loyd, Dudeney discovered a wonderful 4 piece dissection of an equilateral triangle
into a square, found later in these notes.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
Claudi Alsina and Roger B. Nelsen, Math Made Visual, The Mathematical Association of America, 2006.
William Dunham, Journey Through Genius, Penguin, 1991.
Aaron J. Friedland, Puzzles in Math & Logic, Dover Publications, 1970.
Martin Gardner, Mathematical Puzzles of Sam Loyd, Dover Publications, 1959.
Samuel Loyd, Cyclopedia of Puzzles, 1914, www.mathpuzzle.com/loyd.
Roger B. Nelsen, Proofs Without Words, The Mathematical Association of America, 1993.
Geoffrey Mott-Smith, Mathematical Puzzles, Dover Publications, 1954.
Date: March 13, 2010.
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2. The Battle of the Four Oaks (Sam Loyd, Cyclopedia of Puzzles, 1914)
Divide the Field Into Four Identical Parts, Each Containing a Tree. The town of Four Oaks derives
its name from the fact that one of the early settlers, who owned a large tract of land, left it to four sons
with the stipulation that they “divide it into equal portions, as indicated by the positions of four ancient
oaks which had always served as landmarks.”
The sons were unable to divide the land amicably, since the four trees really furnished no clue to guide
them, so they went to law over it and squandered the entire estate in what was known as the “battle of the
four oaks.” The person who told me this story thought it might form the groundwork for a good puzzle,
which it has done, so far as suggesting a theme is concerned.
The picture represents a square field with four ancient oak trees, equal distance apart, in a row from the
center to one side of the field. The property was left to four sons who were instructed to divide the field into
four pieces, each of the same shape and size, and so that each piece of land would contain one of the trees.
The puzzle is an impromptu one, gotten up on the spur of the moment, so it is really not very difficult.
Nevertheless it is safe to say that everyone will not hit upon the best possible answer.
Divide the field into four pieces, each
of the same shape and size, and so
that each piece of land contains one
of the trees.
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3. A Battle Royal (Sam Loyd, Cyclopedia of Puzzles, 1914)
Rearrange the Eight Pieces to Form a Perfect Checkerboard. In the history of France is told an
amusing story of how the Dauphin saved himself from an impending checkmate, while playing chess with the
Duke of Burgundy, by smashing the chess board into eight pieces over the Duke’s head. It is a story often
quoted by chess writers to prove that it is not always politic to play to win, and has given rise to a strong
line of attack in the game known as the King’s gambit.
The smashing of the chess board into eight pieces was the feature which always struck my youthful fancy
because it might possibly contain the elements of an important problem. The restriction to eight pieces does
not give scope for a great difficulty or variety, but not feeling at liberty to depart from historical accuracy,
I shall give our puzzlists, a simple little problem suitable for summer weather. Show how to put the eight
pieces together to form a perfect 8x8 checkerboard.
The puzzle is a simple one, given to teach a valuable rule which should be followed in the construction of
puzzles of this kind. By given no two pieces the same shape, other ways of doing the puzzle are prevented
and the feat is much more difficult of accomplishment.
Rearrange the Eight Pieces to Form a Perfect Checkerboard.
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4. Dissecting the Chessboard (Aaron J. Friedland, Puzzles in Math & Logic, 1970.)
A chessboard may easily be divided into four pieces of the same size and shape, such that each piece
contains the same number of black and white squares. However, the reader is asked to perform the dissection
in such a way that each piece contains more than twice as many squares of one color than of the other. The
cuts are to be made in the usual manner along the edges of the squares and not cutting through any square.
Dissect the Chessboard into Four Pieces. Each Piece Must Contain More Than
Twice as Many Squares of One Color as the Other.
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5. The Patch Quilt Puzzle (Sam Loyd, Cyclopedia of Puzzles, 1914)
Into how few squares, containing one or more pieces of patchwork, can the quilt be divided?
The sketch represents the members of the “Willing Workers” society overwhelming their good parson with
a token of love and esteem in the shape of a beautiful patchwork quilt. Every member contributed one
perfectly square piece of patchwork consisting of one or more small squares.
Any lady would have resigned if her particular piece of work was tampered with or omitted, so it became
a matter of considerable study to find out how to unite all the squares, of various sized, together to form one
large square quilt. Incidentally it may be mentioned that since every member contributed one square piece
of patch quilt, you will know just how many members there were when you discover into how few square
pieces the quilt can be divided. It is a simple puzzle which will give considerable scope of ingenuity and
patience.
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Some Classics
6. Triangles into Rectangles
Find a way to cut any triangle into as few pieces as possible that can be rearranged into a rectangle of
equal area. Here are some example triangles for you to try.
7. The Kitchen Linoleum [7], p.41
Mr. Houseman wishes to lay down linoleum on the floor of his kitchen, which is exactly 12 feet square. He
has a piece of linoleum just sufficient for the purpose, in the form of a rectangle 16 feet by 9 feet. Obviously
he will have to cut this piece to make it fit, but he doesn’t want to cut it into any more parts than necessary.
Fortunately, the linoleum is uniformly brown, without pattern, so that he can cut it in any manner he pleases
without spoiling its appearance.
What is the least number of pieces into which the linoleum can be cut to solve Mr. Houseman’s problem?
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8. The Haberdasher Problem
The famous British puzzlist H. E. Dudeney discovered a fabulous method to dissect an equilateral triangle
into four pieces that rearrange into a square. Can you discover your own dissection before you check the
solution sheet?
9. The Lune of Hippocrates
In a slight generalization of geometric dissection, Hippocrates (470 BCE–410 BCE) show that the shaded
lune in the diagram has area equal to the shaded square by adding and subtracting shapes. Can you?
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10. Solution Sheet
(5) Triangles into Rectangles
(1) The Battle of the Four Oaks
(2) A Battle Royal
(6) The Kitchen Linoleum
(3) Dissecting the Chessboard
(7) The Haberdasher Problem
(4) The Patch Quilt Puzzle
(8) The Lune of Hippocrates
We’ll leave you this beautiful result!
New York Math Circle
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