Mystifying Mathematical Puzzles: Golden Spheres, Squared Eggs and Other Brainteasers
256
Mystifying Mathematical Puzzles: Golden Spheres, Squared Eggs and Other Brainteasers
256Paperback
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Product Details
| ISBN-13: | 9780486825076 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 05/16/2018 |
| Series: | Dover Puzzle Books: Math Puzzles |
| Pages: | 256 |
| Product dimensions: | 5.50(w) x 8.00(h) x (d) |
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CHAPTER 1
geometric dissections
The volume will get off to a gentle start with a mathematical recreation that requires almost no work with numbers, and, as will be shown in other chapters, it is hardly alone in occupying such an apparently contradictory position. Geometric dissections involve the cutting up of any geometrical figure in some specified manner. For example, rectangles and squares can be dissected into any number of smaller, unequal squares or smaller, unequal rectangles. A similar dissection is that of a square or an obtuse-angled triangle, a triangle in which one angle is more than 90°, into a minimum number of acute-angled triangles, those whose angles are all less than 90°.
Until 1938, it was believed that the dissection of a square into smaller, unequal squares had no solution. In that year, R. L. Brooks, C. A. B. Smith, A. H. Stone, and W. T. Tutte, members of the Trinity Mathematical Society of Trinity College, England, succeeded in finding solutions. A typical solution to this problem is shown in Figure 20.
While this kind of diversion is quite interesting and affords the practitioner much amusement, this chapter will deal mainly with another type of geometric dissection: the conversion of one figure to another by the straightforward method of cutting one into a finite number of pieces and rearranging them to form the other. There is a certain aesthetic and mathematical satisfaction in such a procedure. The new figure, of course, has the same area as the original, and the sense of accomplishment is heightened when the transformation is performed by a dissection involving the least possible number of pieces.
It has been known for some time that any rectilinear plane figure, that is, any polygon, can be transformed into any other rectilinear plane figure of equal area by cutting it into some finite number of pieces. Any polygon, A, can be cut into triangles by drawing diagonals from one vertex to each of the other vertexes, and these triangles can be transformed into rectangles such that all have bases with the same length. These rectangles can be joined together to form a larger rectangle, A'. Any other polygon, B, with the same area as the first, but of a different shape, can also be divided into triangles by drawing diagonals from vertex to vertex. Again, these triangles can be transformed into rectangles having the same base as those formed from the triangles of polygon A, and another large rectangle, B', can be assembled by combining these smaller triangles. The two large rectangles A' and B* will be congruent; that is, one can be superimposed on the other and the two will coincide at all points. Now, A' can be subdivided into the same number of rectangles as there were triangles in polygon B. If these rectangles are made the same size as those that resulted from B, they can be transformed to triangles of the same size as those originally derived from it. Combining these triangles into B completes the transformation from A to B. Unfortunately, this proof offers no assistance whatever if it is desired to find the minimum number of pieces required for the transformation from A to B.
No calculations were needed for this transformation, but geometric dissections do sometimes require mathematical knowledge, which is usually used merely to establish that a given dissection of one figure does indeed produce the new one. For instance, when a rectangle-to-square transformation is made, calculations of lengths and areas should be made to assure that a true square results instead of a rectangle that is only very close to being a square. Eyesight is not to be trusted.
This field has a very long history. For example, a dissection in which a square is transformed into two smaller, unequal squares is used in an ancient proof of the Pythagorean theorem — a geometrical theorem stating that in the case of right-angled triangles, the square on the hypotenuse, or side opposite the right angle, is equal to the sum of the squares on the other 2 sides — utilizing geometric dissection, as shown in Figure 1. The pieces forming the large square on the hypotenuse are used to form the 2 smaller squares on the other sides of the triangle.
Transformation of squares to regular hexagons, six-sided polygons, or heptagons, seven-sided polygons, were known at the beginning of the nineteenth century, whereas rectangle-to-square dissections were described by the French mathematician Jean Etienne Montucla (1725–1799) at least a century earlier. The other two most-noted workers in this field, Sam Loyd (1844–1911), America's greatest puzzlist, and Henry E. Dudeney (1847–1930), one of England's greatest creators and collectors of recreational mathematics, made their contributions to geometric dissections during the last quarter of the nineteenth century and the first quarter of the twentieth century. Since the 1920's there have been a number of isolated discoveries, but now, and probably for many years to come, one man leads the research in this fascinating diversion. Harry Lindgren, an examiner for the patent office in Canberra, Australia, has devoted years to its study, and he has broken more records for minimum-piece dissections than anyone else. His book, Geometric Dissections, is the only one devoted solely to the subject. This classic has been revised by Greg Frederickson, retitled Recreational Problems in Geometric Dissections and How to Solve Them, and published by Dover Publications, Inc., in 1972.
The simplest possible rectangle-to-square dissection is the conversion of a 1 X 4 rectangle to a square, as shown in Figure 2. If the length-to-width ratio of the rectangle is not exactly 4 to 1, a two-piece dissection using a straight cut is not possible.
Certain other rectangles can be cut into only two pieces and converted into squares if the "step-cut" technique is utilized. A 9 X 4 rectangle can be cut into two pieces and transformed into a 6 X 6 square as in Figure 3. The rises in this two-step dissection are equal to 2 units, the treads are equal to 3. A 16 X 9 rectangle can be transformed into a 12 X 12 square using a three-step dissection (Figure 4) and a 25 X 16 rectangle transformed into a 20 X 20 square using a four-step procedure (Figure 5).
From these examples, the following rule can be easily derived: an n-step, two-piece dissection of a rectangle to a square can be made if the dimensions of the rectangle are in the ratio (n + 1)2 to n2 where this ratio is equal to or less than 4 to 1. The dimensions of the sides of the resulting square will be n2 + n. From this comes the conclusion: as n approaches infinity, the ratio [n + 1)2 to n2 approaches 1. The2-piece, nearly infinite-step dissection of such a rectangle into the corresponding 1 X 1 square is shown in Figure 6. What appears to be a diagonal line in the figure is, of course, a nearly infinite number of steps.
There are an infinite number of rectangles whose length-to-width ratio is less than 4 to 1, depending on the value chosen for n, that can be dissected into two pieces and transformed into squares. However, three or more pieces may be required for the transformation of rectangles with length-to-width ratios greater than 4 to 1 or where the ratios are between (n + 1)2 to n2 and (n + 2)2 to (n + 1)2, where n is any positive number.
A 5 X 2 retangle can be dissected into three or four pieces in a variety of ways to yield a [square root of 10] x [square root of 10] square, as shown in Figure 7.
Rectangles whose length-to-width ratios are less than 4 to 1, but not with a ratio of the form (n + 1)2 to n2, where n is an integer, can be transformed into squares in three pieces by using the general construction shown in Figure 8 (x/y is less than 4). As the length-to-width ratio becomes larger, more pieces may be required. For example, see Figure 9.
The subject of rectangle-to-square dissections has hardly been exhausted, only some idea of the scope of the subject has been given. Figures and dissections, of somewhat more complexity, will now be studied.
Geometric dissections are not a trial-and-error recreation. Admittedly, some beautiful results have come from painstaking labor or brilliant strokes of genius. However, the majority of minimal-piece dissections have resulted from the use of certain standard procedures. One such technique will be illustrated by showing the dissection of an equilateral triangle, in which all three sides are equal, into four pieces to form a square.
A strip of squares is drawn (Figure 10a), as is a strip of equilateral triangles, each having the same area as one of the squares (Figure 106). If these two strips are superimposed so that the edges of one strip pass through congruent points of the other, the desired dissection can then be constructed. Figure 10c shows the method and the result.
More sophisticated methods can yield some marvelous dissections, including curved figures, the conversion of a figure into two or more smaller similar ones, or the combination of several small similar figures into a larger, but different, one. This last example is illustrated by Figure 11, in which a square has been dissected and reassembled to form two pentagons.
Obviously, not all dissections can be solved by using systematic or standard techniques; some problems may require completely novel approaches. Standard techniques usually will point the way toward minimalpiece solutions, but only in a very few exceptional cases can it be proven that a given dissection is indeed minimal. Lindgren has shown that an apparent minimal-piece dissection can often be improved.
Those who wish to indulge in dissections of their own should follow these hints:
1. Decide on a fixed area for all polygons. Approximately 4 square inches is a convenient area, but any one will do so long as the polygons are large enough that small pieces can be included in the dissection without confusion. The area of each regular polygon will, of course, have to be calculated, and the lengths of the sides of the different polygons will have to be determined.
2. Make accurate black-ink drawings on tracing paper, so that the figure can be turned over and seen in reverse. The lines should not be heavy.
3. Keep all drawings, even those that appear to be quite useless. It can be exasperating to have to redraw a difficult figure when one remembers having thrown the same design away a few days ago.
I have indicated that standard dissection techniques usually produce the best results, but nonstandard techniques can also produce marvelous dissections. Sam Loyd's dissection of a square into five pieces that can be rearranged into five other geometric figures is shown in Figure 12.
Also, polygons that are not standard in shape can be dissected — for example, the letters of the alphabet. Figure 13 shows how to cut CUT in two places and rearrange the pieces to form a square. The squat H in Figure 14 can be cut into four identical pieces — hardly startling, but it leads to the H2 problem: Take the same H, make only one cut, and rearrange the pieces to form a square. The solution is shown in Figure 15. The H is first folded along the dotted line. Then a single cut is made, slicing through both sides of the figure. The five pieces that result form the square.
This five-piece dissection is minimal and can be derived from a tessellation technique as shown in Figure 16. A tessellation is a tilelike pattern produced on a plane by the repetition of a given figure. In Figure 16, the two tessellation patterns are the repeated squares and the repeated H's. The same method can be used in another five-piece dissection to transform the H into one large Greek cross (Figure 17), or, using six pieces, into two smaller and equal Greek crosses (Figure 18).
A given polygon can be dissected into noncongruent pieces that can be arranged to produce the original polygon in two different ways. Such dissections of a triangle and a pentagon are shown in Figure 19. These interesting dissections were found by superimposing two different strips of triangle or pentagon elements. They are not minimal-piece solutions, by the way. Although the problem would be a bit more difficult, it might be worth the effort to find dissections of a given figure that could be arranged in three or more different ways to produce the original shape.
Arrangements usually referred to as "squaring the square" are those in which a square is dissected into a number of smaller, unequal ones. Until recently, the best — that is, the one having the least number of smaller squares — was the one found by an English student of the subject, T. H. Willcocks; it is shown in Figure 20. Each of the numbers gives the lengths of the sides of the squares in which they are placed; the whole square is 175 X 175.
More recently, in 1978, the lowest possible order of perfect square was discovered by A. J. W. Duijvestijn of Twente University of Technology in the Netherlands. Employing a sophisticated computer program Duijvestijn obtained an order 21 perfect square with 112 units on a side. A reproduction of this perfect square can be found on page 86 of the June 1978 issue of Scientific American.
One of the many interesting results of the research into squaring the square was the discovery of a rectangle that could be dissected into unequal squares, which could then be arranged in another way to form the original rectangle. This dissection is shown in Figure 21.
CHAPTER 2chessboard placement problems
The game of chess has entertained mankind for centuries. When men first began to play it is unknown, as is the place where it originated. It is general historical opinion that chess was played in India as early as the sixth century A.D. Problems associated with the various pieces and their placement, unrelated to any actual or hypothetical game, have probably entertained puzzlists for an equal length of time.
For example: What is the maximum number of chess pieces of a given type that can be placed on a chessboard, on which no other type of piece is put, so that no piece can be captured by another? Someone may ask, how many white queens can be placed on a board so that no queen can attack another? In this case, the answer is that, since white queens cannot capture white queens, the number that can be placed is limited only by the manual dexterity of the person who piles them on the board. Such questions and answers will be ignored.
Assuming queens to be belligerent toward each other but still limited in their movements to those allowed in chess, we can ask: What is the maximum number of queens that can be placed on a chessboard so that no piece can be captured by another? The queen can move an unlimited number of squares horizontally, vertically, or diagonally. Since a chess board has eight horizontal rows and eight vertical columns, nine or more queens cannot be placed without some mutual attack, for there would obviously be at least two queens in a single row or column.
The maximum number of queens (()) fulfilling the stated conditions is 8. There are 12 distinct solutions to this problem, and 80 others can be derived from these solutions by simple rotation or reflection. Rotation, turning the board 90, 180, or 270° in either direction, and reflection, reflecting the board in a mirror placed along each of the four sides of the board, of the 12 distinct solutions will produce 84 more. The last board in Figure 22 yields only 3 different solutions since only the 90° rotation is different from the original; the 270° rotation duplicates the 90° rotation; the right and left mirror images are identical, as are the top and bottom mirror images. Hence, there is a total of 12 4* 80, or 92, solutions to the problem of the queens.
The same question could be asked of other chess pieces. A rook (R) can move any number of squares horizontally or vertically, but it cannot move diagonally; a bishop (B) can move any number of squares diagonally; a knight (Kt) can move only from one corner of a 2 X 3 rectangle to the corner diagonally opposite; a king (Kg) can move only one square horizontally, vertically, or diagonally.
The maximum number of rooks that can be placed on a chessboard without any one capturing another can be calculated readily. Eight rooks could be placed along one of the main diagonals, and no mutual capture would be possible (see Figure 23). A ninth rook would necessarily have to be placed in a row or a column that was already occupied. Hence, the maximum number of rooks is 8.
(Continues…)
Excerpted from "Mystifying Mathematical Puzzles"
by .
Copyright © 2018 Joseph S. Madachy.
Excerpted by permission of Dover Publications, Inc..
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Table of Contents
1. GEOMETRIC DISSECTIONS2. CHESSBOARD PLACEMENT PROBLEMS
3. FUN WITH PAPER
4. MAGIC AND ANTIMAGIC SQUARES
5. PUZZLES AND PROBLEMS
6. NUMBER RECREATIONS
7. ALPHAMETICS
8. CONGLOMERATE
GLOSSARY
BIBLIOGRAPHY
INDEX