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PREFACE
PREFACE
In issuing this volume of my Mathematical Puzzles, of which some have appeared in periodicals and others are given here for the first time, I must acknowledge the encouragement that I have received from many unknown correspondents, at home and abroad, who have expressed a desire to have the problems in a collected form, with some of the solutions given at greater length than is possible in magazines and newspapers. Though I have included a few old puzzles that have interested the world for gener
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ARITHMETICAL AND ALGEBRAICAL PROBLEMS.
ARITHMETICAL AND ALGEBRAICAL PROBLEMS.
"And what was he? Forsooth, a great arithmetician." Othello , I. i. The puzzles in this department are roughly thrown together in classes for the convenience of the reader. Some are very easy, others quite difficult. But they are not arranged in any order of difficulty—and this is intentional, for it is well that the solver should not be warned that a puzzle is just what it seems to be. It may, therefore, prove to be quite as simple as it looks, or it may contain some pitfall into which, through
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MONEY PUZZLES.
MONEY PUZZLES.
"Put not your trust in money, but put your money in trust." OLIVER WENDELL HOLMES. 1.—A POST-OFFICE PERPLEXITY. In every business of life we are occasionally perplexed by some chance question that for the moment staggers us. I quite pitied a young lady in a branch post-office when a gentleman entered and deposited a crown on the counter with this request: "Please give me some twopenny stamps, six times as many penny stamps, and make up the rest of the money in twopence-halfpenny stamps." For a m
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AGE AND KINSHIP PUZZLES.
AGE AND KINSHIP PUZZLES.
"The days of our years are threescore years and ten." — Psalm xc. 10. For centuries it has been a favourite method of propounding arithmetical puzzles to pose them in the form of questions as to the age of an individual. They generally lend themselves to very easy solution by the use of algebra, though often the difficulty lies in stating them correctly. They may be made very complex and may demand considerable ingenuity, but no general laws can well be laid down for their solution. The solver m
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CLOCK PUZZLES.
CLOCK PUZZLES.
"Look at the clock!" Ingoldsby Legends . In considering a few puzzles concerning clocks and watches, and the times recorded by their hands under given conditions, it is well that a particular convention should always be kept in mind. It is frequently the case that a solution requires the assumption that the hands can actually record a time involving a minute fraction of a second. Such a time, of course, cannot be really indicated. Is the puzzle, therefore, impossible of solution? The conclusion
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LOCOMOTION AND SPEED PUZZLES.
LOCOMOTION AND SPEED PUZZLES.
"The race is not to the swift."— Ecclesiastes ix. II. 67.—AVERAGE SPEED. In a recent motor ride it was found that we had gone at the rate of ten miles an hour, but we did the return journey over the same route, owing to the roads being more clear of traffic, at fifteen miles an hour. What was our average speed? Do not be too hasty in your answer to this simple little question, or it is pretty certain that you will be wrong. 68.—THE TWO TRAINS. I put this little question to a stationmaster, and h
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DIGITAL PUZZLES.
DIGITAL PUZZLES.
"Nine worthies were they called." DRYDEN: The Flower and the Leaf. I give these puzzles, dealing with the nine digits, a class to themselves, because I have always thought that they deserve more consideration than they usually receive. Beyond the mere trick of "casting out nines," very little seems to be generally known of the laws involved in these problems, and yet an acquaintance with the properties of the digits often supplies, among other uses, a certain number of arithmetical checks that a
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VARIOUS ARITHMETICAL AND ALGEBRAICAL PROBLEMS.
VARIOUS ARITHMETICAL AND ALGEBRAICAL PROBLEMS.
"Variety's the very spice of life, That gives it all its flavour." COWPER: The Task. 97.—THE SPOT ON THE TABLE. A boy, recently home from school, wished to give his father an exhibition of his precocity. He pushed a large circular table into the corner of the room, as shown in the illustration, so that it touched both walls, and he then pointed to a spot of ink on the extreme edge. "Here is a little puzzle for you, pater," said the youth. "That spot is exactly eight inches from one wall and nine
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GEOMETRICAL PROBLEMS.
GEOMETRICAL PROBLEMS.
"God geometrizes continually." PLATO. "There is no study," said Augustus de Morgan, "which presents so simple a beginning as that of geometry; there is none in which difficulties grow more rapidly as we proceed." This will be found when the reader comes to consider the following puzzles, though they are not arranged in strict order of difficulty. And the fact that they have interested and given pleasure to man for untold ages is no doubt due in some measure to the appeal they make to the eye as
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DISSECTION PUZZLES.
DISSECTION PUZZLES.
"Take him and cut him out in little stars." Romeo and Juliet , iii. 2. Puzzles have infinite variety, but perhaps there is no class more ancient than dissection, cutting-out, or superposition puzzles. They were certainly known to the Chinese several thousand years before the Christian era. And they are just as fascinating to-day as they can have been at any period of their history. It is supposed by those who have investigated the matter that the ancient Chinese philosophers used these puzzles a
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GREEK CROSS PUZZLES.
GREEK CROSS PUZZLES.
"To fret thy soul with crosses." SPENSER. "But, for my part, it was Greek to me." Julius Cæsar , i. 2. Many people are accustomed to consider the cross as a wholly Christian symbol. This is erroneous: it is of very great antiquity. The ancient Egyptians employed it as a sacred symbol, and on Greek sculptures we find representations of a cake (the supposed real origin of our hot cross buns) bearing a cross. Two such cakes were discovered at Herculaneum. Cecrops offered to Jupiter Olympus a sacred
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VARIOUS DISSECTION PUZZLES.
VARIOUS DISSECTION PUZZLES.
We will now consider a small miscellaneous selection of cutting-out puzzles, varying in degrees of difficulty. 146.—AN EASY DISSECTION PUZZLE. First, cut out a piece of paper or cardboard of the shape shown in the illustration. It will be seen at once that the proportions are simply those of a square attached to half of another similar square, divided diagonally. The puzzle is to cut it into four pieces all of precisely the same size and shape. 147.—AN EASY SQUARE PUZZLE. If you take a rectangul
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PATCHWORK PUZZLES.
PATCHWORK PUZZLES.
"Of shreds and patches."— Hamlet , iii. 4. 170.—THE CUSHION COVERS. The above represents a square of brocade. A lady wishes to cut it in four pieces so that two pieces will form one perfectly square cushion top, and the remaining two pieces another square cushion top. How is she to do it? Of course, she can only cut along the lines that divide the twenty-five squares, and the pattern must "match" properly without any irregularity whatever in the design of the material. There is only one way of d
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VARIOUS GEOMETRICAL PUZZLES.
VARIOUS GEOMETRICAL PUZZLES.
"So various are the tastes of men." MARK AKENSIDE. 178.—THE CARDBOARD BOX. This puzzle is not difficult, but it will be found entertaining to discover the simple rule for its solution. I have a rectangular cardboard box. The top has an area of 120 square inches, the side 96 square inches, and the end 80 square inches. What are the exact dimensions of the box? 179.—STEALING THE BELL-ROPES. Two men broke into a church tower one night to steal the bell-ropes. The two ropes passed through holes in t
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POINTS AND LINES PROBLEMS.
POINTS AND LINES PROBLEMS.
"Line upon line, line upon line; here a little and there a little."— Isa . xxviii. 10. What are known as "Points and Lines" puzzles are found very interesting by many people. The most familiar example, here given, to plant nine trees so that they shall form ten straight rows with three trees in every row, is attributed to Sir Isaac Newton, but the earliest collection of such puzzles is, I believe, in a rare little book that I possess—published in 1821— Rational Amusement for Winter Evenings , by
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MOVING COUNTER PROBLEMS.
MOVING COUNTER PROBLEMS.
"I cannot do't without counters." Winter's Tale , iv. 3. Puzzles of this class, except so far as they occur in connection with actual games, such as chess, seem to be a comparatively modern introduction. Mathematicians in recent times, notably Vandermonde and Reiss, have devoted some attention to them, but they do not appear to have been considered by the old writers. So far as games with counters are concerned, perhaps the most ancient and widely known in old times is "Nine Men's Morris" (known
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UNICURSAL AND ROUTE PROBLEMS.
UNICURSAL AND ROUTE PROBLEMS.
"I see them on their winding way." REGINALD HEBER. It is reasonable to suppose that from the earliest ages one man has asked another such questions as these: "Which is the nearest way home?" "Which is the easiest or pleasantest way?" "How can we find a way that will enable us to dodge the mastodon and the plesiosaurus?" "How can we get there without ever crossing the track of the enemy?" All these are elementary route problems, and they can be turned into good puzzles by the introduction of some
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COMBINATION AND GROUP PROBLEMS.
COMBINATION AND GROUP PROBLEMS.
"A combination and a form indeed." Hamlet , iii. 4. Various puzzles in this class might be termed problems in the "geometry of situation," but their solution really depends on the theory of combinations which, in its turn, is derived directly from the theory of permutations. It has seemed convenient to include here certain group puzzles and enumerations that might, perhaps, with equal reason have been placed elsewhere; but readers are again asked not to be too critical about the classification,
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CHESSBOARD PROBLEMS.
CHESSBOARD PROBLEMS.
"You and I will goe to the chesse." GREENE'S Groatsworth of Wit. During a heavy gale a chimney-pot was hurled through the air, and crashed upon the pavement just in front of a pedestrian. He quite calmly said, "I have no use for it: I do not smoke." Some readers, when they happen to see a puzzle represented on a chessboard with chess pieces, are apt to make the equally inconsequent remark, "I have no use for it: I do not play chess." This is largely a result of the common, but erroneous, notion
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THE CHESSBOARD.
THE CHESSBOARD.
"Good company's a chessboard." BYRON'S Don Juan , xiii. 89. A chessboard is essentially a square plane divided into sixty-four smaller squares by straight lines at right angles. Originally it was not chequered (that is, made with its rows and columns alternately black and white, or of any other two colours), and this improvement was introduced merely to help the eye in actual play. The utility of the chequers is unquestionable. For example, it facilitates the operation of the bishops, enabling u
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STATICAL CHESS PUZZLES.
STATICAL CHESS PUZZLES.
"They also serve who only stand and wait." MILTON. 295.—THE EIGHT ROOKS. It will be seen in the first diagram that every square on the board is either occupied or attacked by a rook, and that every rook is "guarded" (if they were alternately black and white rooks we should say "attacked") by another rook. Placing the eight rooks on any row or file obviously will have the same effect. In diagram 2 every square is again either occupied or attacked, but in this case every rook is unguarded. Now, in
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THE GUARDED CHESSBOARD.
THE GUARDED CHESSBOARD.
On an ordinary chessboard, 8 by 8, every square can be guarded—that is, either occupied or attacked—by 5 queens, the fewest possible. There are exactly 91 fundamentally different arrangements in which no queen attacks another queen. If every queen must attack (or be protected by) another queen, there are at fewest 41 arrangements, and I have recorded some 150 ways in which some of the queens are attacked and some not, but this last case is very difficult to enumerate exactly. On an ordinary ches
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DYNAMICAL CHESS PUZZLES.
DYNAMICAL CHESS PUZZLES.
"Push on—keep moving." THOS. MORTON: Cure for the Heartache . 320.—THE ROOK'S TOUR. The puzzle is to move the single rook over the whole board, so that it shall visit every square of the board once, and only once, and end its tour on the square from which it starts. You have to do this in as few moves as possible, and unless you are very careful you will take just one move too many. Of course, a square is regarded equally as "visited" whether you merely pass over it or make it a stopping-place,
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VARIOUS CHESS PUZZLES.
VARIOUS CHESS PUZZLES.
"Chesse-play is a good and wittie exercise of the minde for some kinde of men." Burton's Anatomy of Melancholy . 346.—SETTING THE BOARD. I have a single chessboard and a single set of chessmen. In how many different ways may the men be correctly set up for the beginning of a game? I find that most people slip at a particular point in making the calculation. 347.—COUNTING THE RECTANGLES. Can you say correctly just how many squares and other rectangles the chessboard contains? In other words, in h
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MEASURING, WEIGHING, AND PACKING PUZZLES.
MEASURING, WEIGHING, AND PACKING PUZZLES.
"Measure still for measure." Measure for Measure , v. 1. Apparently the first printed puzzle involving the measuring of a given quantity of liquid by pouring from one vessel to others of known capacity was that propounded by Niccola Fontana, better known as "Tartaglia" (the stammerer), 1500-1559. It consists in dividing 24 oz. of valuable balsam into three equal parts, the only measures available being vessels holding 5, 11, and 13 ounces respectively. There are many different solutions to this
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CROSSING RIVER PROBLEMS
CROSSING RIVER PROBLEMS
"My boat is on the shore." BYRON. This is another mediæval class of puzzles. Probably the earliest example was by Abbot Alcuin, who was born in Yorkshire in 735 and died at Tours in 804. And everybody knows the story of the man with the wolf, goat, and basket of cabbages whose boat would only take one of the three at a time with the man himself. His difficulties arose from his being unable to leave the wolf alone with the goat, or the goat alone with the cabbages. These puzzles were considered b
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PROBLEMS CONCERNING GAMES.
PROBLEMS CONCERNING GAMES.
"The little pleasure of the game." MATTHEW PRIOR. Every game lends itself to the propounding of a variety of puzzles. They can be made, as we have seen, out of the chessboard and the peculiar moves of the chess pieces. I will now give just a few examples of puzzles with playing cards and dominoes, and also go out of doors and consider one or two little posers in the cricket field, at the football match, and the horse race and motor-car race. 378.—DOMINOES IN PROGRESSION. It will be seen that I h
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PUZZLE GAMES.
PUZZLE GAMES.
"He that is beaten may be said To lie in honour's truckle bed." HUDIBRAS. It may be said generally that a game is a contest of skill for two or more persons, into which we enter either for amusement or to win a prize. A puzzle is something to be done or solved by the individual. For example, if it were possible for us so to master the complexities of the game of chess that we could be assured of always winning with the first or second move, as the case might be, or of always drawing, then it wou
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MAGIC SQUARE PROBLEMS.
MAGIC SQUARE PROBLEMS.
"By magic numbers." CONGREVE, The Mourning Bride. This is a very ancient branch of mathematical puzzledom, and it has an immense, though scattered, literature of its own. In their simple form of consecutive whole numbers arranged in a square so that every column, every row, and each of the two long diagonals shall add up alike, these magic squares offer three main lines of investigation: Construction, Enumeration, and Classification. Of recent years many ingenious methods have been devised for t
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SUBTRACTING, MULTIPLYING, AND DIVIDING MAGICS.
SUBTRACTING, MULTIPLYING, AND DIVIDING MAGICS.
Although the adding magic square is of such great antiquity, curiously enough the multiplying magic does not appear to have been mentioned until the end of the eighteenth century, when it was referred to slightly by one writer and then forgotten until I revived it in Tit-Bits in 1897. The dividing magic was apparently first discussed by me in The Weekly Dispatch in June 1898. The subtracting magic is here introduced for the first time. It will now be convenient to deal with all four kinds of mag
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MAGIC SQUARES OF PRIMES.
MAGIC SQUARES OF PRIMES.
The problem of constructing magic squares with prime numbers only was first discussed by myself in The Weekly Dispatch for 22nd July and 5th August 1900; but during the last three or four years it has received great attention from American mathematicians. First, they have sought to form these squares with the lowest possible constants. Thus, the first nine prime numbers, 1 to 23 inclusive, sum to 99, which (being divisible by 3) is theoretically a suitable series; yet it has been demonstrated th
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MAZES AND HOW TO THREAD THEM.
MAZES AND HOW TO THREAD THEM.
"In wandering mazes lost." Paradise Lost. The Old English word "maze," signifying a labyrinth, probably comes from the Scandinavian, but its origin is somewhat uncertain. The late Professor Skeat thought that the substantive was derived from the verb, and as in old times to be mazed or amazed was to be "lost in thought," the transition to a maze in whose tortuous windings we are lost is natural and easy. The word "labyrinth" is derived from a Greek word signifying the passages of a mine. The anc
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THE PARADOX PARTY.
THE PARADOX PARTY.
"Is not life itself a paradox?" C.L. DODGSON, Pillow Problems . "It is a wonderful age!" said Mr. Allgood, and everybody at the table turned towards him and assumed an attitude of expectancy. This was an ordinary Christmas dinner of the Allgood family, with a sprinkling of local friends. Nobody would have supposed that the above remark would lead, as it did, to a succession of curious puzzles and paradoxes, to which every member of the party contributed something of interest. The little symposiu
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UNCLASSIFIED PROBLEMS.
UNCLASSIFIED PROBLEMS.
"A snapper up of unconsidered trifles." Winter's Tale , iv. 2. 414.—WHO WAS FIRST? Anderson, Biggs, and Carpenter were staying together at a place by the seaside. One day they went out in a boat and were a mile at sea when a rifle was fired on shore in their direction. Why or by whom the shot was fired fortunately does not concern us, as no information on these points is obtainable, but from the facts I picked up we can get material for a curious little puzzle for the novice. It seems that Ander
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SOLUTIONS.
SOLUTIONS.
1.—A POST-OFFICE PERPLEXITY.— solution The young lady supplied 5 twopenny stamps, 30 penny stamps, and 8 twopence-halfpenny stamps, which delivery exactly fulfils the conditions and represents a cost of five shillings. 2.—YOUTHFUL PRECOCITY.— solution The price of the banana must have been one penny farthing. Thus, 960 bananas would cost £5, and 480 sixpences would buy 2,304 bananas. 3.—AT A CATTLE MARKET.— solution Jakes must have taken 7 animals to market, Hodge must have taken 11, and Durrant
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