I only realized recently that an edition of College Mathematics Journal dedicated to Martin Gardener is freely available for all rather than just. BY. Henry Ernest Dudeney. EDITED BY MARTIN GARDNER, EDITOR OF. THE MATHEMATICAL GAMES DEPARTMENT,. Scientific American. “Mathematical Games” from to recounts. 25 years of amusing puzzles and serious discoveries by Martin Gardner. MARTIN GARDNER continues to.

Martin Gardner Pdf

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Also by Martin Gardner from. The Mathematical Association of America. Eighteenth Street, N.W.. Washington, D. C. () Riddles of the. MARTIN GARDNER. Mathematical Puzzles. A NEW SELECTION: from Origami to Recreational Logic, from Digital Roots and Dudeney Puzzles to the Diabolic. Anxiety & Depression Workbook For Dummies® Trademarks: Wiley, the Wiley Publishing logo, For Dummies, the Dummies Man.

For me God is a "Wholly Other" transcendent intelligence, impossible for us to understand. He or she is somehow responsible for our universe and capable of providing, how I have no inkling, an afterlife. I believe in a personal God, and I believe in an afterlife, and I believe in prayer, but I don't believe in any established religion. This is called philosophical theism. Philosophical theism is entirely emotional. As Kant said, he destroyed pure reason to make room for faith.

While eschewing systematic religious doctrine, he retained a belief in God, asserting that this belief cannot be confirmed or disconfirmed by reason or science. He stated that while he would expect tests on the efficacy of prayers to be negative, he would not rule out a priori the possibility that as yet unknown paranormal forces may allow prayers to influence the physical world. Buckley, Jr. In some cases, he attacked prominent religious figures such as Mary Baker Eddy on the grounds that their claims are unsupportable.

His semi-autobiographical novel The Flight of Peter Fromm depicts a traditionally Protestant Christian man struggling with his faith, examining 20th century scholarship and intellectual movements and ultimately rejecting Christianity while remaining a theist.

In this regard, he said, he was an adherent of the " New Mysterianism ". His annotated version of Alice's Adventures in Wonderland and Through the Looking Glass , explaining the many mathematical riddles, wordplay, and literary references found in the Alice books, was first published as The Annotated Alice Clarkson Potter, Sequels were published with new annotations as More Annotated Alice Random House, , and finally as The Annotated Alice: The Definitive Edition Norton, , combining notes from the earlier editions and new material.

He was a perennial fan of the Oz books written by L. Frank Baum , [] and in he published Visitors from Oz , based on the characters in Baum's various Oz books. Frank Baum Memorial Award.

It's very mysterious! First he asks Lady Muriel for three of her handkerchiefs.

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Two are placed together and held up by their top corners. The top edges a r e sewn together, then one handkerchief is given a half-twist and the bottom edges a r e similarly joined. The result is of course a Moebius surface with a single edge consisting of four handkerchief edges. The third handkerchief likewise has four edges that also form a closed loop.

If these four edges a r e now sewn to the four edges of the Moebius surface, the professor explains, the result will be a closed, edgeless surface that is like that of a sphere except that it will have only one side. But i t will take time. I'll sew it up after tea. So you have all the wealth of the world in that leetle Purse! I t cannot be done without self-inter- section of the surface, but the proposed construction does give a valuable insight into the structure of the projective plane. Adinirers of Count Alfred Korzybski, who founded general semantics, are fond of saying that "the map is not the territory.

To increase accuracy, map makers gradually expanded the scale of their maps, first to six yards to the mile, then yards. We actually malde a map of the country, on the scale of a mile to the mile! So we now use the country itself, aLs its own map, and I assure you it does nearly as well. On December 19, , he wrote: Actually there is no limit to the number of right triangles that can be found with integral sides and equal areas, but beyond three triangles the areas a r e never less than six-digit numbers.

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Carroll came very close to finding three such triangles, as we will explain in the answer section. There is one answer in which the area involved, although greater than the area of each triangle cited by Carroll, is still less than 1, Who lies and who tells the t r u t h? Of several unusual word games invented by Carroll, the soli- taire game of Doublets became the most popular in his day, partly because of prize competitions sponsored by the English magazine Vanity Fair.

The idea is to take two appropriate words of the same length, then change one to the other by a series of inter- mediate words, each differing by only one letter from the word preceding. Proper names must not be used for the linking words, and the words should be common enough to be found in the aver- age abridged dictionary. For readers who enjoy word pu: It will be interesting to see if any readers succeed in making the changes with fewer links.

The Doublets are: Like so many mathematicians, Carroll enjoyed all sorts of wordplay: William Ewart Gladstone - Wild agitator! Means well , writing acrostic verses on the names of little: His letters to his child friends were filled with this sort of thing.

Can anyone discover i t? Carroll's writings abound in puns, though they incline lto be clever rather than outrageous. He once defined a "sillygism" a s the combining of two prim misses to yield a delusion. His virtu- osity in mathematical punning reached its highest point in a pamphlet of political satire entitled Dynamics of a Parti-cle. I t opens with the following definitions: Plain Anger is the inclination of two voters to one another, who meet together, but whose viewl3 are not in the same direction.

When a Proctor, meeting another Proctor, makes the votes on one side equal to those on the other, the feeling entertained by each side is called Right Anger. When two parties, coming together, feel a Right Anger, each is said to be Complementary to the other though, strictly speaking, this is very seldom the case. Obtuse Anger is that which is greater than Right Anger.

Pi stands for the salary of Benjamin Jowett, professor of Greek and translator of Plato, whom many suspected of harboring un- orthcldox religious views.

The tract satirizes the failure of Oxford officials to agree on Professor Jowett's salary. The following passage, in which J stands for Jowett, will convey the pamphlet's flavor: There are many Carrollian references in James Joyce's Finnegans W a k e , including one slightly blasphe- mous reference to Carroll himself: There are other interesting Carroll-Nabokov links.

Like Carroll, Nabokov is fond of chess one of his novels, T h e Defense, is about a monomaniacal chess player and Humbert Humbert, the narrator of Lolita, resembles Carroll in his enthu- siasm for little girls. One must hasten to add that Carroll would surely have been shocked by Lolita.

Doldgson considered himself a happy man, but there is a gentle undertow of sadness that runs beneath much of his nonsense: Y e t w h a t are all such gaieties t o m e W h o s e thoughts are full o f indices and surds? Green, revised edi- tion, Oxford Press, pages Doublet problems appear in scores of old and new puzzle books. Dmitri Borgmann, on page of his recent Language on Vaca- tion Scribner's, , calls them "word ladders" and points out that the ideal word ladder is one in which the two words have no identical letters a t the same positions, and the change is ac- complished with the same number of steps as there are letters in each word.

I t is not surprising to find Doublets turning up under the name of "word golf" in Nabokov's Pale Fire. Solutions to the first two are provided by Mary McCarthy iin her remarkable review of the novel New Republic, June 4, Miss McCarthy adds some new Doublets of her own, based on the words in the novel's title. Good, Basic Books, , pages , finds a strikin. In each case the area is Henry Ernest Dudeney, in the answer to problem in his Canterburg Puzzles, gives a formula by which such triangle triplets can be easily found.

Calrroll's truth-and-lie problem has only one answer that does not lead to a logical contradiction: A and C lie; B speaks the truth. The problem yields easily to the propositional calculus by taking the word "says" a s the logical connective called equiva- lence.

Without drawing on symbolic logic one can simply list the eighl; possible combinations of lying and truth-telling for the three men, then explore each combination, eliminating those that lead to logical contradictions. Cstrroll's solutions to the six Doublets a r e: After Carroll's answers to his Doublets appeared in Scientific American, a large number of readers sent in shorter answers.

Cohen, Scott Gallagher, Lawrence Jase: Lord, Sidney J. Osborn and H. Percival and Richard D. Thurston independently found: Bancroft, Robert Bauman, Frederick J. Hooven, Arthur H.

Lord, Mrs. Henry A. Morss, Sidney J. Percival for this six-stepper: Thurston right: Morss, Richard D. Thurston, and H. The object is to find your w a y out of the central space. Paths cross over and under one another, but a r e occasionally blocked by single-line barriers. When a pair of scissors is brought into play, a wealth of interesting new possibilities open up, many of which serve to dramatize basic and important theo- rems of plane geometry in curious ways.

For example, consider the well-known theorem which states that the sum of the interior angles of any triangle is a straight angle an angle of degrees.

Cut a triangle from a sheet of paper.

Calculus Made Easy By Silvanus P. Thompson and Martin Gardner

Put a dot near the vertex of each angle, snip off the cor- ners, and you will find that the three dotted angles always fit together neatly to form a straight angle [see Fig. Try it with the corners of a quadrilateral. The figure may be of any shape, inclulding concave forms such as the one shown in Figure 17b.

The four snipped angles always join to form a perigon: If we extend the sides of any convex polygon a s shown in Figure 17c, the dotted angles are called exterior angles. Regardless of how many sides the polygon may have, if its ex- terior angles are cut out and joined, they also will add up to degrees.

If two or more sides of a polygon intersect, we have what is sometimes called a crossed polygon. Rule the star as irregularly a s you please you may even include the degenerate forms shown in Figure 18, in w'hich one or two points of the star fail t o extend beyond the body , dot the five corners, cut out the s t a r and trim off the cor- ners.

You may be surprised to find that, as in the case of the tri- angle, the points of any pentagram join to form a straight angle. This theorem can be confirmed by another quaint empirical tech- nique that might be called the sliding-match method. Draw a large pentagram, then place a match alongside one of the lines as shown in the top illustration of Figure Slide the match up until its head touches the top vertex, then swing its tail to the left until the match is alongside the other line.

The match has now alterled its orientation on the plane by an angle equal to the angle a t the top corner of the star. Slide the match down to the next corner and do the same thing. Continue sliding the match around the star, repeating this procedure a t each vertex.

When the match is back to its original position, it will be upside down, having made a clockwise rotation of exactly degrees. This rotation is clearly the sum of the pentagram's five angles. The sliding-match method can be used for confirming all of the theorems mentioned, as well as for finding new ones. It is a handy device for measuring the angles of any type of polygon, including the s t a r forms and the helter-skelter crossed varieties.

Since the match must return to its starting position either pointing the same way or in the opposite direction, it follows providing the match has always rotated in the same direction that the sum of the traversed angles must be a multiple of a straight angle. If the match rotates in both directions during its trip, as is often the case with crossed polygons, we cannot obtain a sum of the angles, although other theorems can be stated. For instance, a match slid around the perimeter of the crossed octagon in Figure 19 will rotate clockwise a t the angles marked A, and the same distance counterclockwise a t the angles marked B.

Thus we cannot arrive a t the sum of the eight angles, but we can say that the sum of the four A angles equals the sum of the four B angles. This can be easily verified by the scissors method or by a formal geometrical proof.

The familiar Pythagorean theorem lends itself to many elegant scissors-and-paper demonstrations. Construct squares on the two legs of any right triangle [see Fig. Divide the larger square or either square if they are the same size into four identical parts by ruling two lines through the center, a t right angles to each other and with one line parallel to the triangle's hypotenuse. You will find that all five pieces can be shifted in position, without changing their orientation on the plane, to form one large square shown by brokien lines on the hypotenuse.

Paper Cutting 63 Perigal discovered this dissection in about , but dicl not publish it until He was so delighted with it that he had the diagram printed on his business card, and gave away hundreds of puzzles consisting of the five pieces. Someone who hats not seen the diagram will have considerable difficulty fitting the pieces together, first to make two squares, then one large square.

It is amusing to learn from Perigal's obituary, in the no- tices of the Royal Astronomical Society of London, that his "imain astronomical aim in life" was to convince others, "especially young men not hardened in the opposite belief," that it was a FIG.

Escott discovered this dissection of a regular hexagram to a square. He wrote pamphlets, built models and even composed poems to prove his point, "bearing with heroic cheeirfulness the continual disappointment of finding none of them of any avail. It has been proved that any polygon can be cut into a finite number of pieces that will form any other polygon of the same area, but of course such dissections have little interest un- less the number of pieces is small enough to make the change startling.

Who would imagine, for example, that the regular hex- agram, or six-pointed Star of David, could be cut [see Fig. The regular pentagram cannot be dissected into a square with less than eight pieces.

Harry Lindgren, of the Australian patent office, is per- haps the world's leading expert on dissections of this type. In Figure 22 we see his beautiful six-piece dissection of a regular dodecagon to a square.

A quite different class of paper-cutting recreation, more fa- miliar to magicians than mathematicians, involves folding a sheet of paper several times, giving it a single straight cut, then open- ing up one or both of the folded pieces to reveal some sort of surprising result. Loe, which deals almost entirely with such stunts. The book explains how to fold a sheet so that a single cut will produce any desired letter of the alphabet, various types of stars and crosses, and such complex patterns a s a circular chain of stars, a star within a star, and so on.

An unusual single-cut trick that is familiar to Arneri- can magicians is known as the bicolor cut. A square of t: The cut separates the red squares from the black and simulta- neously cuts out each individual square.

With a sheet of onionskin paper the thin paper makes it possible to see outlines through several thicknesses it is not difficult to devise a method for this trick, as well as methods for single-cutting simple geometrical figures; but more complicated designs- the swastika foir in- stance - present formidable problems. An old paper-cutting stunt, of unknown origin, is illustrated in Figure I t is usually presented with a story about two contem- porary political leaders, one admired, the other hated.

Both men die and approach the gates of heaven. The Bad Guy naturally lacks the necessary sheet of paper authorizing his admittance. He seeks the aid of the Good Guy, standing just behind him. GG folds his sheet of paper as shown in a,b,c,d and e, then cuts it allong the indicated dotted line.

Paper Cutting 67 the rest to BG. Saint Peter opens the BG's pieces, arranges them to form "Hell" as shown a t bottom left, and sends him off.

VVhen Saint Peter opens the paper presented by the GG, he finds i t in the shape of the cross shown a t bottom right. It is obviously impossible to fold a sheet flat in such a way that a straight cut will produce curved figures, but if a sheet is rolled into a cone, plane slices through i t will leave edges in the form of circles, ellipses, parabolas or hyperbolas, depending on the angle of the cut.

These of course are the conic sections studied by the Greeks. Less well known is the fact that a sine curve cam be quickly produced by wrapping a sheet of paper many times around a cylindrical candle, then cutting diagonally through both paper and candle.

When unrolled, each half of the paper will have a cut edge in the form of a sine curve, or sinusoid, one o: The trick is also useful to the housewife who wants to put a rippling edge on a sheet of shelf paper. Here are two fascinating cut-and-fold problems, both involving cubes. The first is easy; the second, not so easy. What is the shortest strip of paper one inch wide that can be folded to make all six sides of a one-inch cube?

A square of paper three inches wide is black on one side and white on the other. Rule the square into nine one-inch squares. By cutting only along the ruled lines, is it possible to cut a pat- tern that will fold along the ruled lines into a cube that is all lblack on the outside?

The pattern must be a single piece, and no cuts or folds a r e permitted that are not along the lines that divide the sheet into squares.

The reader may enjoy working out some of them, if only to see how much simpleir and more intuitively evident the sliding-match proofs are. Perigal first published his Pythagorean dissection in Messenger of Mathematics, Vol. The elegant hexagram-to-square dissection was discovered by Edward Brind Escott, an insurance company actuary who lived in 0;ak Park, Illinois, and who died in He was an expert on number theory, contributing frequently to many different mathe- matical journals.

His hexagram dissection is given by Henry Ernest Dudeney as the solution to problem in Modern Puzzles For more about Lindgren's remarkable dissections, see the Mathematical Games department of Scientific American, Novem- ber , and Lindgren's book on dissections listed in the Biblj.

A method of folding is de- picted in Figure If the strip is black on one side, eight inches are necessary for folding an all-black cube. A way of doing this is shown in Recreational Mathematics Magazine, February: L, page The three-inch-square sheet, black on one side only, can be cut and folded into an all-black cube in many different ways.

This cannot be accomplished with a pattern of less than eight unit squares, but the missing square inch may be in any posi. Figure 25 shows how a pattern with the missing square in the center can be folded into the black cube. In all solutions, the cuts have a total length of five units. If the entire sheet is used for the pattern, the length of the cut lines can be reduced to four. Pattern is black on underside.

When we play games, or even when we watch them being played by others, we pass from the incomprehensible uni- verse of given reality into a neat little man-made world, where everything is clear, purposive and easy to understand. Competi- tion adds to the intrinsic charm of games by making them excit- ing, while betting and crowd intoxication add, in their turn, to the thrills of competition.

Such games are as old as civilization and ;as varied as the wings of butterflies. Fantastic amounts of ment a1energy have been expended on them, considering the fact that until quite recently they had no value whatever beyond that of relaxing and refreshing the mind. Today they have suddenly become important in computer theory.

Chess-playing and checker- playing machines that profit from experience may be the fore- runners of electronic minds capable of developing powers as yet unim. Board Games 71 FIG. Relief dates from B. The earliest records of mathematical board games are found in the a r t of ancient Egypt, but they convey little information be- cause of the Egyptian convention of showing scenes only in profile [see Fig.

Some games involving boards have been found in Egyptian tombs [Fig. A bit more is known about Greek and Roman board games, but it was not until the 13th century A. Like biological organisms, games evolve and proliferate new species. A few simple games, such as ticktacktoe, may remain unchanged for centuries; others flourish for a time, then vanish completely. The outstanding example of a dinosaur diversion is Rithmomachy. It traces back a t least to the 12th century, and as late as the 17th century it was mentioned by Robert Burton, in The Anatomy of Melancholy, as a popular English game.

Many learned treatises were written about it, but no one plays it today except a few mathematicians and medievalists. In the U. Both have long and fascinating his- tories, with unexpected mutations in rules from time to time and place to place.

Today the American checkers is identical with the English "draughts," but in other countries there are wide varia- tions. The so-called Polish checkers actually invented in France is now the dominant form of the game throughout most of Europe. It is played on a ten-by-ten board, each side having twenty men that capture backward as well as forward.

Crowned pieces called queens instead of kings move like the bishop in chess, and in making a jump can land on any vacant cell beyond the captured piece. I n the French-speaking provinces of Canada, and in parts of India, Polish checkers is played on a twelve-by-twelve board. German checkers Damenspiel resembles Polish checkers, but i t is usually played on the English eight-by-eight board. A simi- lar form of this "minor Polish" game, as it is sometimes c,alled, is popular in the U.

Spanish and Italian variants also are closer to the English. Turkish check- ers duma is also played on an eight-by-eight board, but each side has sixteen men that occupy the second and third rows att the outset.

Pieces move and jump forward and sideways, but not diagonally, and there are other radical departures from both the English and the Polish forms. Chess likewise has varied enormously in its rules, tracing back ultimately to a n unknown origin in India, probably in the sixth century A.

True, there is today an international chess that is standardized, but there are still many excellent non-European forms of the game that obviously share a common origin with international chess. Shogi is played on a nine-by-nine board, with twenty men on each side, arranged a t the start on the first three rows.

The game is won, as in Western chess, by check- mating a piece that moves exactly like the king. An interesting feature of the game is that captured pieces can be returned to the board to be used by the captor.

Chinese chess tse'ung k'i also ends with the checkmate of a piece that moves like the king in Western chess, but the rules are quite different from those of the Japanese game. Martian chess "jetan" , explained by Edgar Rice Burroughs in the appendix to his novel The Chessmen of Mars, is an amusing variant, played on a ten-by-ten board with unusual pieces and novel rules. For example, the princess which corresponds rough- ly to our king has the privilege of one "escape move" per game that permits her to flee a n unlimited distance in any direction.

In addition to these regional variants of chess, modern players, momlentarily bored with the orthodox game, have invented a weird assortment of games known a s fairy chess. Among the many fairy-chess games that can be played on the standard board a r e: Dozens of strange new pieces have been introduced, such as the chancellor combining the moves of rook and knight , the centaur combining bishop and knight and even neuter pieces e.

In Lewis Padgett's science-fiction novel The Fairy Chessmen a war is won by a mathematician who makes a hobby of fairy chess. His mind, accustomed to breaking rules, is elastic enough to cope with a n equation too bizarre for his more brilliant but rnore orthodox colleagues. An amusing species of fairy chess that is quite old, but still provides a delightful interlude between more serious games, is played as follows. One player sets up his sixteen men in the usual way, but his opponent has only one piece, called the maharajah.

A queein may be used for this piece, but its moves combine those of queen and knight. I t is placed a t the outset on any free square not threatened by a pawn ; then the other side makes the first move.

The ]maharajah loses if he is captured, and wins if he checkmates the king. Pawns a r e not permitted to be replaced by queens or other pieces if they advance to the last row. Without this proviso it is easy to defeat the maharajah simply by advancing the rook pawns until they can be queened. With three queens and two rooks in play, the game is easily won. Even with this proviso, it might be thought that the maharajah has a poor chance of winning, but his mobility is so great that if he moves swiftly and aggressively, he of-ten checkmates early in the game.

At other times he can sweep the board clean of pieces and then force the lone king into a corner checkmate.

Hundreds of games have been invented that are played on a standard chessboard but have nothing in common with either chess or checkers.

One of the best, in my opinion, is the now- forgotten game of "reversi. A crude set can be made by coloring one side of a sheet of cardboard, then cutting out small circles; a better set can be constructed by download- ing inexpensive checkers or poker chips and gluing the pieces into red-black pairs.

It is worth the trouble, because the game can be an exciting one for every member of the family. Reversi starts with an empty board. One player has 32 pieces turned red-side up ; the other has 32 turned black-side up. Players alternate in placing a single man on the board in conformity with the following rules: The first four men must be placed on the four central squares. Experience has shown that it is better for the first player to place his second man above, below, or to the side of his first piece an example is shown in Figure 2 8 , rather than diagonally adjacent, but this is not obligatory.

By the same token, it is wise for the second player not to play diagonally opposite his oppo- nent's first move, especially if his opponent is a novice. This gives the first player a chance to make the inferior diagonal move on his second play.

Between experts, the game always begins with the pattern shown in Figure After the four central squares are filled, players continue placing single pieces. Each must be placed so that it is adjacent to a hostile piece, orthogonally or diagonally. Moreover, it must also be placed so that it is in direct line with another piece of the same color, and with one or more enemy pieces and no vacant cells in between. Numbers are for reference only. The enemy pieces are considered captured, but instead of being re- moved they are turned over, or "reversed," so that they becom4 friendly pieces.

They are, so to speak, "brainwashed" so that they join their captors. Pieces remain fixed throughout the game, but may be reversed any number of times. If the placing of a piece simultaneously captures more than one chain of enemy pieces, the pieces in both chains are reversed. Pieces are captured only by the placing of a hostile piece.

Chains that become flanked a t both ends as a result of other causes are not captured. Board Games 77 FIG. If a player cannot move, he loses his turn. He continues to lose his t u r n until a legal move becomes possible for him. The game ends when all 64 squares are filled, or when neither player can move either because he has no legal move or because his counters are gone. The winner is the person with the ]most pieces on the board.

Two examples will clarify the rules: In Figure 28, black plays only on cells 43, 44, 45 and In each case he captures and re- verses a single piece. In Figure 29, if red plays on cell 22 11e is compelled to reverse six pieces: Dramatic reversals of color are character- istic of this unusual game, and it is often difficult to say who has the better game until the last few plays a r e made. The player with the fewest pieces frequently has a strong positional advantage.

Some pointers for beginners: If possible, confine early play to the central sixteen squares, and t r y especially to occupy cells 19, 22, 43 and The first player forced outside this area is usually placed a t a disadvantage. Outside the central sixteen squares, the most valuable cells to occupy are the corners of the board. For this reason it is unwise to play on cells 10, 15, 50 or 55, because this gives your opponent a chance to take the corner cells.

Next to the corners, the most desirable cells a r e those that a r e next but one to the corners 3, 6, 17, 24, 41, 48, 59 and Avoid giving your opponent a chance to occupy these cells. Deeper rules of strategy will occur to any player who advances beyond the novice stage. Little in the way of analysis has been published about reversi ; it is hard to say who, if either player, has the advantage on even a board a s small a s four-by-four.

Here is a problem some readers may enjoy trying to solve. Is it possible for a game tc, occur in which a player, before his tenth move, wins by removing all the enemy pieces from the board? Mollett, both claimed to be the sole inventor of reversi. Each called the other a fraud. In the late 's, whenathegame was enormously popular in England, rival handbooks and rival firms for the manufacture of equipment were authorized by the two claimants.

Regardless of who invented it, reversi is a game that combines complexity of structure with rules of delightful simplicity, and a game that does not deserve oblivion. Bell's Board and Table G a m e s can always be won by the player with conven- tional pieces if he plays circumspectly. Richard A. Blue, Dennis A. Keen, William Knight and Wallace Smith all sent strategies against which the maharajah could not save himself, but the most efficient line of play came from William E.

If Rudge's strategy is bug-free, as it seems to be, the maharajah can always be captured in 25 moves or less. The strategy is independent of the moves made by M thc ma- harajah except for three possible moves.

Only the moves ad the offense a r e listed: P-QR4 diagonal, permitting the fol- 2. P - QR5 lowing move. P - QR6 P - QR7 N - KR3 KR - QR6 7. N-KB4 R-K7 8. B-Q3 M is forced to retreat to his 9. Castles first row. Q-KR5 KR-K6 N-QB3 B - KN7 QN- Q5 This move need be made on- P - QN4 P - QB3 M is now forced to move to This move is made only if his first or second row. M is on his KN1.

P - KR3 The move forces captured on the next move. This inter- change may be necessary if M blocks a pawn. Moves 1 5 and 22 are stalling moves, required only when M is on the squares indi- cated.

Move 23 is required only if M must be forced over to the queen's side of the board. Not much is known about the early history of reversi. I t seems to have first appeared in London in as "The Game of Annex- ation," played on a cross-shaped board. Articles about it ran: Reversi, and games derived from it, have been sold in more recent years, in the United States, under a variety of names.

Tryne Products brought out reversi, about , as a game called "Las Vegas Backfire. A fixed cover for each cell can be turned to make the cell red, blue or white neutral , thus eliminating the need for pieces.

The answer is yes. In my Scien- tific American column I gave what I believed then to be the shortest possible reversi game corresponding to the "fool's matt? I had found the game in an old reversi handbook. But two readers discovered shorter games. D,, H. Peregrine, of Jesus College, Oxford, sent the following six-mover: These features can be understood without models, but if the reader can obtain a supply of 30 or more spheres, he will find them a n excellent aid to understanding.

Table-tennis balls a r e per: They can be coated with rubber cement, allowed to dry, then stuck together to make rigid models. First let us make a brief two-dimensional foray. If we arrange spheres in square formation [see Fig.

If we form a triangle [see Fig. These a r e the simplest examples of what the ancients 15 FIG.

F o r example, it takes only a glance a t Figure 30, left, to see t h a t the sum of any number of consecutive positive integers, be- ginning with 1, is a triangular number.

A glance a t Figure 30, right, shows that square numbers are formed by the addition of consecutive odd integers, beginning with 1. Figure 31 rnakes immediately evident an interesting theorem known to the ancient Pythagoreans: Every square number is the sum of two conLsecu- tive triangular numbers.

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The algebraic proof is simple. Are there numbers that are simultaneously square and triangular? Yes, there a r e infinitely many of them. The smallest not count- ing 1, which belongs to any figurate series is 36; then the aeries continues: I t is not so easy to devise a formula for the nth term of this series.

Three-dimensional analogies of the plane-figurate numbers a r e obtained by piling spheres in pyramids. Three-sided pyramids, the base and sides of which are equilateral triangles, are models of what a r e called the tetrahedral numbers.

Four-sided pyramids, with square bases and equilateral triangles for sides i. Just as a square can be divided by a straight line into two consecutive triangles, so can a square pyramid be divided by a plane into two consecutive tetra- hedral pyramids. If you build a model of a pyramidal number, the bottom layer has to be kept from rolling apart.

This can be done by placing rulers or other strips of wood along the sides. For example, in making a courthouse monu- ment out of cannon balls, what is the smallest number of balls that can first be arranged on the ground as a square, then piled in a square pyramid? The surprising thing about the answer 4, is that i t is the onlv answer. The proof of this is difficult, and was not achieved until Another example: A grocer is dis- playing oranges in two tetrahedral pyramids.

By putting together the oranges in both pyramids he is able to make one large tetra- hedral pyramid.

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What is the smallest number of oranges he can have? If the two small pyramids are the same size, the unique answer is If they are different sizes, what is the answer?

Imagine now that we have a very large box, say a crate for a piano, which we wish to fill with as many golf balls as we can. What packing procedure should we use? First we form a layer packed as shown by the unshaded circles with light gray circum- ferences in Figure The second layer is formed by placing balls in alternate hollows as indicated by the shaded circles with black rims.

I n making the third layer we have a choice of two different procedures: We place each ball on a hollow A that is directly above a ball in the first layer. If we continue in this way, placing the balls of each layer directly over those in the next layer but one, we pro- duce a structure called hexagonal close-packing. We place each ball in a hollow B, directly above a hollow in the first layer. If we follow this procedure for each layer each ball will be directly above a ball in the third layer beneath i t , the result is known as cubic close-packing.

In forming the layers of a close-packing we can switch back and forth whenever we please from hexagonal to cubic packing to produce various hybrid forms of close-packing. Is this the largest density obtainable?

No denser packing is known, but in an article published in on the relation of close-packing to froth H. Coxeter, of the University of Toronto, made the startling suggestion that perhaps the densest packing has not yet been found. I t is true that no more than twelve balls can be placed so that all of them touch a central sphere, but a thirteenth ball can almost be added.

The large lee- way here in the spacing of the twelve balls, in contrast to the complete absence of leeway in the close-packing of circles on a plane, suggests that there might be some form of irregular pack- ing that would be denser than. As a result of FIG. Scott, of the University of To- ronto, recently made some experiments in random packing by pouring large numbers of steel balls into spherical flasks, then weighing them to obtain the density.

He found that stable ran- dom-packings had a density that varied from about. So if there is a packing denser than.

Assuming that close-packing is the closest packing, readers may like to test their packing prowess on this exceedingly tricky little problem. The interior of a rectangular box is ten inches on each side and five inches deep. What is the largest number of steel spheres one inch in diameter that can be packed in this space? If close-packed circles on a plane expand uniformly until they fill the interstices between them, the result is the familiar hex- agonal tiling of bathroom floors.

This explains why the pattern is so common in nature: What happens when closely packed spheres expand uniformly in a closed vessel, or are subjected to uniform pressure from without? Cubic close- packing transforms each sphere into a rhombic dodecahedron [see Fig. Hexagonal close-packing turns each ball into a trapezo-rhombic dodecahedron [see Fig. If this figure is sliced in half along the gray plane and one half is rotated 60 degrees, it becomes a rhombic dodecahedron.

In the English physiologist Stephen Hales wrote in his book Vegetable Staticks that he had poured some fresh peas into a pot, compressed them and had obtained "pretty regular dodeca- hedrons. Matzke, a botanist a t Columbia University, repeated the experiment.

In ex- periments reported in Matzke compressed lead shot and found that if the spheres had been cubic close-packed, rhornbic dodecahedrons were formed; but if they had been randomly packed, irregular fourteen-faced bodies predominated. These re- sults have important bearing, Matzke has pointed out, on the study of such structures as foam, and living cells in undifferen- tiated tissues.

What is the loosest packing; that is, what rigid structure will have the lowest possible density? For the structure to be rigid, each sphere must touch a t least four others, and the contact points must not be all in one hemisphere or all on one equator of the sphere. In his Geometry and the Imagination, first published in Germany in , David Hilbert describes what was then be- lieved to be the loosest packing: In the following year, however, two Dutch mathematicians, Heinrich Heesch and Fritz Laves, published the details of a much looser packing with a density of only.

Whether there are still looser packings is another intriguing question that, like the question of the closest packing, remains undecided. Large spheres a r e first packed as shown on left, then each sphere is replaced by three smaller spheres t o obtain the packing shown on right.

It has a density of. Watson in Messenger of Mathematics, new series, Vol. This had been conjectured a s early as by the French mathematician Edouard Lucas. Henry Ernest Dudeney makes the same guess in his answer to problem , Amusements i n Mathematics There is a large literature on numbers that a r e both triangular and square. Highlights a r e cited in a n editorial note t o problem E, American Mathematical Monthly, February , page , and the following formula for the nth square triangular number is given: The question of the densest possible regular packing of spheres has been solved for all spaces up through eight dimensions.

In 3-space, the ques- tion is answered by the regular close-packings described earlier, which have a density of.

But, a s Constance Reid notes in her Introduction to Higher Mathematics , when 9-space is considered, the problem takes one of those sudden, mysterious turns that so often occur in the geometries of higher Euclidean spaces. So f a r a s I know, no one yet knows how to regularly close- pack hyperspheres in 9-space. Nine-space is also the turning point for the related problem of how many congruent spheres can be made to touch another sphere of the same size.

I t was not until that K. Schiitte and B. I n four-dimensions i t has been proved that 24 hyperspheres can touch a 25th sphere, and for spaces of 5, 6, 7 and 8 dimen- sions, the maximum number of hyperspheres is known to be 40, 72, and respectively. But in 9-space, the problem remains unsolved. This is a tetrahedral number that can be split into two smaller tetrahedral numbers: The edges of the three pyramids are 8,14 and A box ten inches square and five inches deep can be close-packed with one-inch-diameter steel balls in a surprising variety of ways, each accommodating a different number of balls.

The maximum number, , is obtained as follows: Turn the box on its side and form the first layer by making a row of five, then a row of four, then of five, and so on. I t is possible to make eleven rows six rows of five each, five rows of four each , accommodating 50 balls and leaving a space of more than.

The second layer also will take eleven rows, alternating four and five balls to a row, but this time the layer begins and ends with four-ball rows, so that the number of balls in the layer is only The last row of four balls will project.

Twelve layers with a total height of 9. Bzit piercing eyes looked out f r o m t h e m a s k , inexorable, cold, and enigmatic. The English mathematician Augustus de Morgan once wrote of pi a s "this mysterious 3.

The surprising answer is six divided by the square of pi. I t is pi's connection with the circle, however, that has made it the most familiar member of the infinite class of transcendental numbers. What is a transcendental number?

I t is described a s a n irra- tional number that is not the root of an algebraic equation that has rational coefficients.

The square root of two is irrational, but it is a n "algebraic irrational" because i t is a root of the equ: P i cannot be expressed a s the root of such a n equation, but only as the limit of some type of infinite process.

The decimal form of pi, like that of all irrational numbers, is endless and nonrepeating. The most remarkable was recorded in the fifth century A. We can obtain this fraction by a kind of numerological hocus-pocus. Write the first three odd integers in pairs: I t is hard to believe, but this gives pi to a n accuracy of six decimal places.

There are also roots that come close to pi. The square root of 10 3. More numerology: A cube with a volume of 31 cubic inches would have a n edge that differed from pi by less than a thousandth of a n inch.

And the sum of the square root of 2 and the square root of 3 is 3. Ea,rly attempts to find a n exact value for pi were closely linked with attempts to solve the classic problem of squaring the circle. Is it possible to construct a square, using only a compass and a straightedge, that is exactly equal in area to the area of a given circle? If pi could be expressed as a rational fraction or as the root of a first- or second-degree equation, then it would be pos- sible, with compass and straightedge, to construct a straight line exactly equal to the circumference of a circle.

The squaring of the circle would quickly follow. We have only to construct a rectangle with one side equal to the circle's radius and the other equal to half the circumference.

This rectangle has a n area equal to that of the circle, and there are simple procedures for converting the rectangle to a square of the same area. Conversely, if the circle couldl be squared, a means would exist for constructing a line seg- ment exactly equal to pi. However, there a r e ironclad proofs that pi is transcendental and that no straight line of transcendental length can be constructed with compass and straightedge. There a r e hundreds of approximate constructions of pi, of which one of the most accurate is based on the Chinese astron- omer's fraction mentioned earlier.

I n a quadrant of unit radius draw the lines shown in Figure 35 so that bc is of the radius, dg is , de is parallel to ac, and d f is parallel to be. The distance f g is easily shown to be or. Circle squarers who thought they had discovered an exact value for pi are legion, but none has excelled the English philosopher Thomas Hobbes in combining height of intellect with depth of ignorance. Educated Englishmen were not taught mathematics in Hobbes's day, and it was not until he was 40 that he looked into Euclid.

When he read a statement of the Pythagorean theorem, he first exclaimed: For the rest of his long life Hobbes pursued geometry with all the ardor of a man in love. In , a t the age of 67, he published in Latin a book titlecl De corpore Concerning B o d y that included a n ingenious method of squaring the circle. The method was a n excellent ap- proximation, but Hobbes believed that i t was exact. John Wallis, a distinguished English mathematician and cryptographer, ex- posed Hobbes's errors in a pamphlet, and thus began one of the longest, funniest and most profitless verbal duels ever to engage two ]brilliant minds.

I t lasted almost a quarter of a century, each man writing with skillful sarcasm and barbed invective. I trust the reader will forgive me if I shorten the endless 17th-century titles. Wallis replied with Due Correc- tion for M r. Hobbes in School Discipline for n o t saqing his Lessons right. Hobbes countered with M a r k s o f the Absurd Geometry, Rurc! Hobbes's Points.

Several pamphlets later meztnwhile Hobbes had anonymously published in Paris a n ab- surd method of duplicating the cube Hobbes wrote: Hobbes has been always f a r from provoking any man," Hobbes wrote in one of his later attacks on Wallis [as a matter of fact, in social relations Hobbes was excessively timid], "though, when he is provoked, you find his pen a s sharp a s yours.

I have done. I have considered you now, but will not again. His first, and one of his best, is shown in Figure Inside a unit square, draw arcs AC and BD.

These are quarter arcs of circles with unit radii. Draw line FS, extending it until it meets the side of square a t T. BT, Hobbes asserted, is exactly equal to a r c BF. This gives pi s value of 3. One of the philosopher's major difficulties was his inability t o believe that points, lines and surfaces could be regarded in the abstract a s having less than three dimensions. Excudebat 7. Sumptibuc Andraa Crook,. The Transcendental Number Pi 97 Authors, "in spite of all the reasonings of the geometricians on this side of it, with a firm conviction that its superficies had both depth and thickness.

Although the circle cannot be squared, figures bounded by cir- cular arcs often can be; this fact still arouses false hopes in many a circle squarer. An interesting example is shown in Figure How many square units does this figure contain?

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The upper half is bounded by three quarter-arcs of a circle the same size. How quickly can the reader give, down to the last decimal, the exact length of the side of a square that has the same area as this figure? This can be done, of course, by using any infinite expression that converges on pi. Wallis him- self discovered one of the simplest: The upper terms of these fractions a r e even numbers in se- quence, taken in pairs.

Note the fortuitous resemblance of the first five lower terms to the digits in the Chinese astronomer's fraction! A few decades later the German philosopher Gottfried Wilhelm von Leibniz found another beautiful formula: The most indefatigable of pi computers was the English mathe- matician William Shanks. Over a year period he managed to calculate pi to decimals. Alas, poor Shanks made an error on his th decimal, and all the rest a r e wrong.

This was not dis- covered until , so Shanks's decimals are still found in many current books. In the electronic computer ENIAC was used for 70 machine hours to calculate pi to more than 2, decimals; later another computer carried it to more than 3, decimals in 13 minutes. By , a computer in England and an- other in France had computed pi to 10, decimal places.

One of the strangest aspects of Shanks's decimals was the fact that they seemed to snub the number 7. Each digit appeared about 70 times in the first decimals, just as i t should, except 7, which appeared a mere 51 times.

Tlhe in- tuitionist school of mathematics, which maintains that you cannot say of a statement that it is "either true or false" unless there is a known way by which it can be both verified and refuted, has always used as its stock example: The new figures for pi show not only the expected number of triplets for each digit, but also several runs of and one unexpected So f a r pi has passed all statistical tests for randomness.

This is disconcerting to those who feel that a curve so simple and beauti- ful as the circle should have a less-disheveled ratio between the way around and the way across, but most mathematicians believe that no pattern or order of any sort will ever be found in pi's decimal expansion.

Of course the digits are not random in the sense that they represent pi, but then in this sense neither are the million random digits that have been published by the Rand Corporation of California. They too represent a single number, and an integer a t that. If it is true that the digits in pi are random, perhaps we are justi- fied in stating a paradox somewhat similar to the assertion that if a group of monkeys pound long enough on typewriters, they will eventually type all the plays of Shakespeare.

Stephen Barr has pointed out that if you set no limit to the accuracy with. One bar is taken as unity. The other dliffers from unity by a fraction that is expressed as a very long decimal. This decimal codes the Britannica by the simple process of assign- ing a different number excluding zero as a digit in the number to every word and mark of punctuation in the language.

Zero is used to separate the code numbers. Obviously the entire Britan- nica can now be coded as a single, but almost inconceivably long, number. P u t a decimal point in front of this number, add 11, and you have the length of the second of Barr's bars.

Where does pi come i n? The work was done by Daniel Shanks no relation to William Shanks ; just amother of those strange numerological coincidences that dog the history of pi and John W. Wrench, Jr. The running time was one minute more than eight hours, then an additional 42 min- utes. Computing pi to a few thousand decimals is now a popular device for testing a new computer or training new programmers. Davis in his book The Lore of Large Numbers , "is reduced to a gargle that helps computing machines clear their throats.

I n anticipation of this, Dr. Matrix, the famous numerologist, has sent me a letter asking that I put on record his prediction that the uiillionth digit of pi will be found to be 5. His calculation is based on the third book of the King James Bible, chapter 14, verse 16 it mentions the number 7, and the seventh word has five letters , combined with some obscure calculations involving Euler's constant and the transcendental number e.

Norman Gridgeman, of Ottawa, wrote to point out that Barr's bars can be reduced to a single bar with a scratch on it. The scratch divides the bar into two lengths, the ratio of which codes the Britannica in the manner previously described.

If we draw the broken squares shown in the illustration, it is obvious that segments A, B, C will fit into spaces A', B', C' to form two squares with a combined area of square inches. Figure 39 shows how the vase can be "squared" by cutting it into as few as three parts that will form a ten-inch square.Transformation e is not really a change, but mathematicians call it a "transforma- tion" anyway, in the same sense that a null or empty class is called a class. How does it work? Skip to main content.

By the same token, it is wise for the second player not to play diagonally opposite his oppo- nent's first move, especially if his opponent is a novice.

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Board Games 77 FIG. Mathe- matician" Scientific American, April , let us consider some of the Reverend Dodgson's more obscure excursions into the game and puzzle field. John Wallis, a distinguished English mathematician and cryptographer, ex- posed Hobbes's errors in a pamphlet, and thus began one of the longest, funniest and most profitless verbal duels ever to engage two ]brilliant minds.

Obviously the entire Britan- nica can now be coded as a single, but almost inconceivably long, number. Move five times, take the eight. There a r e only two such values:

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Look over my other posts. One of my extra-curricular activities is tumbling. I am fond of unexpectedly .