A ring was made with chalk on the floor of the hall, and divided into thirteen compartments, in which twelve discs of wood (called "frogs") were placed in the order shown in our illustration, one place being left vacant. The numbers 1 to 6 were painted white and the numbers 7 to 12 black. The puzzle was to get all the white numbers where the black ones were, and vice versa. The white frogs move round in one direction, and the black ones the opposite way. They may move in any order one step at a time, or jumping over one of the opposite colour to the place beyond, just as we play draughts to-day. The only other condition is that when all the frogs have changed sides, the 1 must be where the 12 now is and the 12 in the place now occupied by 1. The puzzle was to perform the feat in as few moves as possible. How many moves are necessary?
I will conclude in the words of the old writer: "These be some of the riddles which the monks of Riddlewell did set forth and expound each to the others in the merry days of the good Abbot David."
THE STRANGE ESCAPE OF THE KING'S JESTER.
A PUZZLING ADVENTURE.
At one time I was greatly in favour with the king, and his Majesty never seemed to weary of the companionship of the court fool. I had a gift for making riddles and quaint puzzles which ofttimes caused great sport; for albeit the king never found the right answer of one of these things in all his life, yet would he make merry at the bewilderment of those about him.
But let every cobbler stick unto his last; for when I did set out to learn the art of performing strange tricks in the magic, wherein the hand doth ever deceive the eye, the king was affrighted, and did accuse me of being a wizard, even commanding that I should be put to death. Luckily my wit did save my life. I begged that I might be slain by the royal hand and not by that of the executioner.
"By the saints," said his Majesty, "what difference can it make unto thee? But since it is thy wish, thou shalt have thy choice whether I kill thee or the executioner."
"Your Majesty," I answered, "I accept the choice that thou hast so graciously offered to me: I prefer that your Majesty should kill the executioner."
Yet is the life of a royal jester beset with great dangers, and the king having once gotten it into his royal head that I was a wizard, it was not long before I again fell into trouble, from which my wit did not a second time in a like way save me. I was cast into the dungeon to await my death. How, by the help of my gift in answering riddles and puzzles, I did escape from captivity I will now set forth; and in case it doth perplex any to know how some of the strange feats were performed, I will hereafter make the manner thereof plain to all.
49.—The Mysterious Rope.
My dungeon did not lie beneath the moat, but was in one of the most high parts of the castle. So stout was the door, and so well locked and secured withal, that escape that way was not to be found. By hard work I did, after many days, remove one of the bars from the narrow window, and was able to crush my body through the opening; but the distance to the courtyard below was so exceeding great that it was certain death to drop thereto. Yet by great good fortune did I find in the corner of the cell a rope that had been there left and lay hid in the great darkness. But this rope had not length enough, and to drop in safety from the end was nowise possible. Then did I remember how the wise man from Ireland did lengthen the blanket that was too short for him by cutting a yard off the bottom of the same and joining it on to the top. So I made haste to divide the rope in half and to tie the two parts thereof together again. It was then full long, and did reach the ground, and I went down in safety. How could this have been?
50.—The Underground Maze.
The only way out of the yard that I now was in was to descend a few stairs that led up into the centre (A) of an underground maze, through the winding of which I must pass before I could take my leave by the door (B). But I knew full well that in the great darkness of this dreadful place I might well wander for hours and yet return to the place from which I set out. How was I then to reach the door with certainty? With a plan of the maze it is but a simple matter to trace out the route, but how was the way to be found in the place itself in utter darkness?
51.—The Secret Lock.
When I did at last reach the door it was fast closed, and on sliding a panel set before a grating the light that came in thereby showed unto me that my passage was barred by the king's secret lock. Before the handle of the door might be turned, it was needful to place the hands of three several dials in their proper places. If you but knew the proper letter for each dial, the secret was of a truth to your hand; but as ten letters were upon the face of every dial, you might try nine hundred and ninety-nine times and only succeed on the thousandth attempt withal. If I was indeed to escape I must waste not a moment.
Now, once had I heard the learned monk who did invent the lock say that he feared that the king's servants, having such bad memories, would mayhap forget the right letters; so perchance, thought I, he had on this account devised some way to aid their memories. And what more natural than to make the letters form some word? I soon found a word that was English, made of three letters—one letter being on each of the three dials. After that I had pointed the hands properly to the letters the door opened and I passed out. What was the secret word?
52.—Crossing the Moat.
I was now face to face with the castle moat, which was, indeed, very wide and very deep. Alas! I could not swim, and my chance of escape seemed of a truth hopeless, as, doubtless, it would have been had I not espied a boat tied to the wall by a rope. But after I had got into it I did find that the oars had been taken away, and that there was nothing that I could use to row me across. When I had untied the rope and pushed off upon the water the boat lay quite still, there being no stream or current to help me. How, then, did I yet take the boat across the moat?
53.—The Royal Gardens.
It was now daylight, and still had I to pass through the royal gardens outside of the castle walls. These gardens had once been laid out by an old king's gardener, who had become bereft of his senses, but was allowed to amuse himself therein. They were square, and divided into 16 parts by high walls, as shown in the plan thereof, so that there were openings from one garden to another, but only two different ways of entrance. Now, it was needful that I enter at the gate A and leave by the other gate B; but as there were gardeners going and coming about their work, I had to slip with agility from one garden to another, so that I might not be seen, but escape unobserved. I did succeed in so doing, but afterwards remembered that I had of a truth entered every one of the 16 gardens once, and never more than once. This was, indeed, a curious thing. How might it have been done?
54.—Bridging the Ditch.
I now did truly think that at last was I a free man, but I had quite forgot that I must yet cross a deep ditch before I might get right away. This ditch was 10 feet wide, and I durst not attempt to jump it, as I had sprained an ankle in leaving the garden. Looking around for something to help me over my difficulty, I soon found eight narrow planks of wood lying together in a heap. With these alone, and the planks were each no more than 9 feet long, I did at last manage to make a bridge across the ditch. How was this done?
Being now free I did hasten to the house of a friend who provided me with a horse and a disguise, with which I soon succeeded in placing myself out of all fear of capture.
Through the goodly offices of divers persons at the king's court I did at length obtain the royal pardon, though, indeed, I was never restored to that full favour that was once my joy and pride.
Ofttimes have I been asked by many that do know me to set forth to them the strange manner of my escape, which more than one hath deemed to be of a truth wonderful, albeit the feat was nothing astonishing withal if we do but remember that from my youth upwards I had trained my wit to the making and answering of cunning enigmas. And I do hold that the study of such crafty matters is good, not alone for the pleasure that is created thereby, but because a man may never be sure that in some sudden and untoward difficulty that may beset him in passing through this life of ours such strange learning may not serve his ends greatly, and, mayhap, help him out of many difficulties.
I am now an aged man, and have not quite lost all my taste for quaint puzzles and conceits; but, of a truth, never have I found greater pleasure in making out the answers to any of these things than I had in mastering them that did enable me, as the king's jester in disgrace, to gain my freedom from the castle dungeon and so save my life.
THE SQUIRE'S CHRISTMAS PUZZLE PARTY
A fine specimen of the old English country gentleman was Squire Davidge, of Stoke Courcy Hall, in Somerset. When the last century was yet in its youth, there were few men in the west country more widely known and more generally respected and beloved than he. A born sportsman, his fame extended to Exmoor itself, where his daring and splendid riding in pursuit of the red deer had excited the admiration and envy of innumerable younger huntsmen. But it was in his own parish, and particularly in his own home, that his genial hospitality, generosity, and rare jovial humour made him the idol of his friends—and even of his relations, which sometimes means a good deal.
At Christmas it was always an open house at Stoke Courcy Hall, for if there was one thing more than another upon which Squire Davidge had very pronounced views, it was on the question of keeping up in a royal fashion the great festival of Yule-tide. "Hark ye, my lads," he would say to his sons: "our country will begin to fall on evil days if ever we grow indifferent to the claims of those Christmas festivities that have helped to win us the proud name of Merrie England." Therefore, when I say that Christmas at Stoke Courcy was kept up in the good old happy, rollicking, festive style that our grandfathers and great-grandfathers so dearly loved, it will be unnecessary for me to attempt a description. We have a faithful picture of these merry scenes in the Bracebridge Hall of Washington Irving. I must confine myself in this sketch to one special feature in the Squire's round of jollification during the season of peace and good will.
He took a curious and intelligent interest in puzzles of every kind, and there was always one night devoted to what was known as "Squire Davidge's Puzzle Party." Every guest was expected to come armed with some riddle or puzzle for the bewilderment and possible delectation of the company. The old gentleman always presented a new watch to the guest who was most successful in his answers. It is a pity that all the puzzles were not preserved; but I propose to present to my readers a few selected from a number that have passed down to a surviving member of the family, who has kindly allowed me to use them on this occasion. There are some very easy ones, a few that are moderately difficult, and one hard brain-racker, so all should be able to find something to their taste.
The little record is written in the neat angular hand of a young lady of that day, and the puzzles, the conditions of which I think it best to give mainly in my own words for the sake of greater clearness, appear to have been all propounded on one occasion.
55.—The Three Teacups.
One young lady—of whom our fair historian records with delightful inconsequence: "This Miss Charity Lockyer has since been married to a curate from Taunton Vale"—placed three empty teacups on a table, and challenged anybody to put ten lumps of sugar in them so that there would be an odd number of lumps in every cup. "One young man, who has been to Oxford University, and is studying the law, declared with some heat that, beyond a doubt, there was no possible way of doing it, and he offered to give proof of the fact to the company." It must have been interesting to see his face when he was shown Miss Charity's correct answer.
56.—The Eleven Pennies.
A guest asked some one to favour him with eleven pennies, and he passed the coins to the company, as depicted in our illustration. The writer says: "He then requested us to remove five coins from the eleven, add four coins and leave nine. We could not but think there must needs be ten pennies left. We were a good deal amused at the answer hereof."
57.—The Christmas Geese.
Squire Hembrow, from Weston Zoyland—wherever that may be—proposed the following little arithmetical puzzle, from which it is probable that several somewhat similar modern ones have been derived: Farmer Rouse sent his man to market with a flock of geese, telling him that he might sell all or any of them, as he considered best, for he was sure the man knew how to make a good bargain. This is the report that Jabez made, though I have taken it out of the old Somerset dialect, which might puzzle some readers in a way not desired. "Well, first of all I sold Mr. Jasper Tyler half of the flock and half a goose over; then I sold Farmer Avent a third of what remained and a third of a goose over; then I sold Widow Foster a quarter of what remained and three-quarters of a goose over; and as I was coming home, whom should I meet but Ned Collier: so we had a mug of cider together at the Barley Mow, where I sold him exactly a fifth of what I had left, and gave him a fifth of a goose over for the missus. These nineteen that I have brought back I couldn't get rid of at any price." Now, how many geese did Farmer Rouse send to market? My humane readers may be relieved to know that no goose was divided or put to any inconvenience whatever by the sales.
58.—The Chalked Numbers.
"We laughed greatly at a pretty jest on the part of Major Trenchard, a merry friend of the Squire's. With a piece of chalk he marked a different number on the backs of eight lads who were at the party." Then, it seems, he divided them in two groups, as shown in the illustration, 1, 2, 3, 4 being on one side, and 5, 7, 8, 9 on the other. It will be seen that the numbers of the left-hand group add up to 10, while the numbers in the other group add up to 29. The Major's puzzle was to rearrange the eight boys in two new groups, so that the four numbers in each group should add up alike. The Squire's niece asked if the 5 should not be a 6; but the Major explained that the numbers were quite correct if properly regarded.
59.—Tasting the Plum Puddings.
"Everybody, as I suppose, knows well that the number of different Christmas plum puddings that you taste will bring you the same number of lucky days in the new year. One of the guests (and his name has escaped my memory) brought with him a sheet of paper on which were drawn sixty-four puddings, and he said the puzzle was an allegory of a sort, and he intended to show how we might manage our pudding-tasting with as much dispatch as possible." I fail to fully understand this fanciful and rather overstrained view of the puzzle. But it would appear that the puddings were arranged regularly, as I have shown them in the illustration, and that to strike out a pudding was to indicate that it had been duly tasted. You have simply to put the point of your pencil on the pudding in the top corner, bearing a sprig of holly, and strike out all the sixty-four puddings through their centres in twenty-one straight strokes. You can go up or down or horizontally, but not diagonally or obliquely; and you must never strike out a pudding twice, as that would imply a second and unnecessary tasting of those indigestible dainties. But the peculiar part of the thing is that you are required to taste the pudding that is seen steaming hot at the end of your tenth stroke, and to taste the one decked with holly in the bottom row the very last of all.
60.—Under the Mistletoe Bough.
"At the party was a widower who has but lately come into these parts," says the record; "and, to be sure, he was an exceedingly melancholy man, for he did sit away from the company during the most part of the evening. We afterwards heard that he had been keeping a secret account of all the kisses that were given and received under the mistletoe bough. Truly, I would not have suffered any one to kiss me in that manner had I known that so unfair a watch was being kept. Other maids beside were in a like way shocked, as Betty Marchant has since told me." But it seems that the melancholy widower was merely collecting material for the following little osculatory problem.
The company consisted of the Squire and his wife and six other married couples, one widower and three widows, twelve bachelors and boys, and ten maidens and little girls. Now, everybody was found to have kissed everybody else, with the following exceptions and additions: No male, of course, kissed a male. No married man kissed a married woman, except his own wife. All the bachelors and boys kissed all the maidens and girls twice. The widower did not kiss anybody, and the widows did not kiss each other. The puzzle was to ascertain just how many kisses had been thus given under the mistletoe bough, assuming, as it is charitable to do, that every kiss was returned—the double act being counted as one kiss.
61.—The Silver Cubes.
The last extract that I will give is one that will, I think, interest those readers who may find some of the above puzzles too easy. It is a hard nut, and should only be attempted by those who flatter themselves that they possess strong intellectual teeth.
"Master Herbert Spearing, the son of a widow lady in our parish, proposed a puzzle in arithmetic that looks simple, but nobody present was able to solve it. Of a truth I did not venture to attempt it myself, after the young lawyer from Oxford, who they say is very learned in the mathematics and a great scholar, failed to show us the answer. He did assure us that he believed it could not be done, but I have since been told that it is possible, though, of a certainty, I may not vouch for it. Master Herbert brought with him two cubes of solid silver that belonged to his mother. He showed that, as they measured two inches every way, each contained eight cubic inches of silver, and therefore the two contained together sixteen cubic inches. That which he wanted to know was—'Could anybody give him exact dimensions for two cubes that should together contain just seventeen cubic inches of silver?'" Of course the cubes may be of different sizes.
The idea of a Christmas Puzzle Party, as devised by the old Squire, seems to have been excellent, and it might well be revived at the present day by people who are fond of puzzles and who have grown tired of Book Teas and similar recent introductions for the amusement of evening parties. Prizes could be awarded to the best solvers of the puzzles propounded by the guests.
Adventures of the Puzzle Club
When it recently became known that the bewildering mystery of the Prince and the Lost Balloon was really solved by the members of the Puzzle Club, the general public was quite unaware that any such club existed. The fact is that the members always deprecated publicity; but since they have been dragged into the light in connection with this celebrated case, so many absurd and untrue stories have become current respecting their doings that I have been permitted to publish a correct account of some of their more interesting achievements. It was, however, decided that the real names of the members should not be given.
The club was started a few years ago to bring together those interested in the solution of puzzles of all kinds, and it contains some of the profoundest mathematicians and some of the most subtle thinkers resident in London. These have done some excellent work of a high and dry kind. But the main body soon took to investigating the problems of real life that are perpetually cropping up.
It is only right to say that they take no interest in crimes as such, but only investigate a case when it possesses features of a distinctly puzzling character. They seek perplexity for its own sake—something to unravel. As often as not the circumstances are of no importance to anybody, but they just form a little puzzle in real life, and that is sufficient.
62.—The Ambiguous Photograph.
A good example of the lighter kind of problem that occasionally comes before them is that which is known amongst them by the name of "The Ambiguous Photograph." Though it is perplexing to the inexperienced, it is regarded in the club as quite a trivial thing. Yet it serves to show the close observation of these sharp-witted fellows. The original photograph hangs on the club wall, and has baffled every guest who has examined it. Yet any child should be able to solve the mystery. I will give the reader an opportunity of trying his wits at it.
Some of the members were one evening seated together in their clubhouse in the Adelphi. Those present were: Henry Melville, a barrister not overburdened with briefs, who was discussing a problem with Ernest Russell, a bearded man of middle age, who held some easy post in Somerset House, and was a Senior Wrangler and one of the most subtle thinkers of the club; Fred Wilson, a journalist of very buoyant spirits, who had more real capacity than one would at first suspect; John Macdonald, a Scotsman, whose record was that he had never solved a puzzle himself since the club was formed, though frequently he had put others on the track of a deep solution; Tim Churton, a bank clerk, full of cranky, unorthodox ideas as to perpetual motion; also Harold Tomkins, a prosperous accountant, remarkably familiar with the elegant branch of mathematics—the theory of numbers.
Suddenly Herbert Baynes entered the room, and everybody saw at once from his face that he had something interesting to communicate. Baynes was a man of private means, with no occupation.
"Here's a quaint little poser for you all," said Baynes. "I have received it to-day from Dovey."
Dovey was proprietor of one of the many private detective agencies that found it to their advantage to keep in touch with the club.
"Is it another of those easy cryptograms?" asked Wilson. "If so, I would suggest sending it upstairs to the billiard-marker."
"Don't be sarcastic, Wilson," said Melville. "Remember, we are indebted to Dovey for the great Railway Signal Problem that gave us all a week's amusement in the solving."
"If you fellows want to hear," resumed Baynes, "just try to keep quiet while I relate the amusing affair to you. You all know of the jealous little Yankee who married Lord Marksford two years ago? Lady Marksford and her husband have been in Paris for two or three months. Well, the poor creature soon got under the influence of the green-eyed monster, and formed the opinion that Lord Marksford was flirting with other ladies of his acquaintance.
"Now, she has actually put one of Dovey's spies on to that excellent husband of hers; and the myrmidon has been shadowing him about for a fortnight with a pocket camera. A few days ago he came to Lady Marksford in great glee. He had snapshotted his lordship while actually walking in the public streets with a lady who was not his wife."
"'What is the use of this at all?' asked the jealous woman.
"'Well, it is evidence, your ladyship, that your husband was walking with the lady. I know where she is staying, and in a few days shall have found out all about her.'
"'But, you stupid man,' cried her ladyship, in tones of great contempt, 'how can any one swear that this is his lordship, when the greater part of him, including his head and shoulders, is hidden from sight? And—and'—she scrutinized the photo carefully—'why, I guess it is impossible from this photograph to say whether the gentleman is walking with the lady or going in the opposite direction!'
"Thereupon she dismissed the detective in high dudgeon. Dovey has himself just returned from Paris, and got this account of the incident from her ladyship. He wants to justify his man, if possible, by showing that the photo does disclose which way the man is going. Here it is. See what you fellows can make of it."
Our illustration is a faithful drawing made from the original photograph. It will be seen that a slight but sudden summer shower is the real cause of the difficulty.
All agreed that Lady Marksford was right—that it is impossible to determine whether the man is walking with the lady or not.
"Her ladyship is wrong," said Baynes, after everybody had made a close scrutiny. "I find there is important evidence in the picture. Look at it carefully."
"Of course," said Melville, "we can tell nothing from the frock-coat. It may be the front or the tails. Blessed if I can say! Then he has his overcoat over his arm, but which way his arm goes it is impossible to see."
"How about the bend of the legs?" asked Churton.
"Bend! why, there isn't any bend," put in Wilson, as he glanced over the other's shoulder. "From the picture you might suspect that his lordship has no knees. The fellow took his snapshot just when the legs happened to be perfectly straight."
"I'm thinking that perhaps——" began Macdonald, adjusting his eye-glasses.
"Don't think, Mac," advised Wilson. "It might hurt you. Besides, it is no use you thinking that if the dog would kindly pass on things would be easy. He won't."
"The man's general pose seems to me to imply movement to the left," Tomkins thought.
"On the contrary," Melville declared, "it appears to me clearly to suggest movement to the right."
"Now, look here, you men," said Russell, whose opinions always carried respect in the club. "It strikes me that what we have to do is to consider the attitude of the lady rather than that of the man. Does her attention seem to be directed to somebody by her side?"
Everybody agreed that it was impossible to say.
"I've got it!" shouted Wilson. "Extraordinary that none of you have seen it. It is as clear as possible. It all came to me in a flash!"
"Well, what is it?" asked Baynes.
"Why, it is perfectly obvious. You see which way the dog is going—to the left. Very well. Now, Baynes, to whom does the dog belong?"
"To the detective!"
The laughter against Wilson that followed this announcement was simply boisterous, and so prolonged that Russell, who had at the time possession of the photo, seized the opportunity for making a most minute examination of it. In a few moments he held up his hands to invoke silence.
"Baynes is right," he said. "There is important evidence there which settles the matter with certainty. Assuming that the gentleman is really Lord Marksford—and the figure, so far as it is visible, is his—I have no hesitation myself in saying that—"
"Stop!" all the members shouted at once.
"Don't break the rules of the club, Russell, though Wilson did," said Melville. "Recollect that 'no member shall openly disclose his solution to a puzzle unless all present consent.'"
"You need not have been alarmed," explained Russell. "I was simply going to say that I have no hesitation in declaring that Lord Marksford is walking in one particular direction. In which direction I will tell you when you have all 'given it up.'"
63.—The Cornish Cliff Mystery.
Though the incident known in the Club as "The Cornish Cliff Mystery" has never been published, every one remembers the case with which it was connected—an embezzlement at Todd's Bank in Cornhill a few years ago. Lamson and Marsh, two of the firm's clerks, suddenly disappeared; and it was found that they had absconded with a very large sum of money. There was an exciting hunt for them by the police, who were so prompt in their action that it was impossible for the thieves to get out of the country. They were traced as far as Truro, and were known to be in hiding in Cornwall.
Just at this time it happened that Henry Melville and Fred Wilson were away together on a walking tour round the Cornish coast. Like most people, they were interested in the case; and one morning, while at breakfast at a little inn, they learnt that the absconding men had been tracked to that very neighbourhood, and that a strong cordon of police had been drawn round the district, making an escape very improbable. In fact, an inspector and a constable came into the inn to make some inquiries, and exchanged civilities with the two members of the Puzzle Club. A few references to some of the leading London detectives, and the production of a confidential letter Melville happened to have in his pocket from one of them, soon established complete confidence, and the inspector opened out.
He said that he had just been to examine a very important clue a quarter of a mile from there, and expressed the opinion that Messrs. Lamson and Marsh would never again be found alive. At the suggestion of Melville the four men walked along the road together.
"There is our stile in the distance," said the inspector. "This constable found beside it the pocket-book that I have shown you, containing the name of Marsh and some memoranda in his handwriting. It had evidently been dropped by accident. On looking over the stone stile he noticed the footprints of two men—which I have already proved from particulars previously supplied to the police to be those of the men we want—and I am sure you will agree that they point to only one possible conclusion."
Arrived at the spot, they left the hard road and got over the stile. The footprints of the two men were here very clearly impressed in the thin but soft soil, and they all took care not to trample on the tracks. They followed the prints closely, and found that they led straight to the edge of a cliff forming a sheer precipice, almost perpendicular, at the foot of which the sea, some two hundred feet below, was breaking among the boulders.
"Here, gentlemen, you see," said the inspector, "that the footprints lead straight to the edge of the cliff, where there is a good deal of trampling about, and there end. The soil has nowhere been disturbed for yards around, except by the footprints that you see. The conclusion is obvious."
"That, knowing they were unable to escape capture, they decided not to be taken alive, and threw themselves over the cliff?" asked Wilson.
"Exactly. Look to the right and the left, and you will find no footprints or other marks anywhere. Go round there to the left, and you will be satisfied that the most experienced mountaineer that ever lived could not make a descent, or even anywhere get over the edge of the cliff. There is no ledge or foothold within fifty feet."
"Utterly impossible," said Melville, after an inspection. "What do you propose to do?"
"I am going straight back to communicate the discovery to headquarters. We shall withdraw the cordon and search the coast for the dead bodies."
"Then you will make a fatal mistake," said Melville. "The men are alive and in hiding in the district. Just examine the prints again. Whose is the large foot?"
"That is Lamson's, and the small print is Marsh's. Lamson was a tall man, just over six feet, and Marsh was a little fellow."
"I thought as much," said Melville. "And yet you will find that Lamson takes a shorter stride than Marsh. Notice, also, the peculiarity that Marsh walks heavily on his heels, while Lamson treads more on his toes. Nothing remarkable in that? Perhaps not; but has it occurred to you that Lamson walked behind Marsh? Because you will find that he sometimes treads over Marsh's footsteps, though you will never find Marsh treading in the steps of the other."
"Do you suppose that the men walked backwards in their own footprints?" asked the inspector.
"No; that is impossible. No two men could walk backwards some two hundred yards in that way with such exactitude. You will not find a single place where they have missed the print by even an eighth of an inch. Quite impossible. Nor do I suppose that two men, hunted as they were, could have provided themselves with flying-machines, balloons, or even parachutes. They did not drop over the cliff."
Melville then explained how the men had got away. His account proved to be quite correct, for it will be remembered that they were caught, hiding under some straw in a barn, within two miles of the spot. How did they get away from the edge of the cliff?
64.—The Runaway Motor-Car.
The little affair of the "Runaway Motor-car" is a good illustration of how a knowledge of some branch of puzzledom may be put to unexpected use. A member of the Club, whose name I have at the moment of writing forgotten, came in one night and said that a friend of his was bicycling in Surrey on the previous day, when a motor-car came from behind, round a corner, at a terrific speed, caught one of his wheels, and sent him flying in the road. He was badly knocked about, and fractured his left arm, while his machine was wrecked. The motor-car was not stopped, and he had been unable to trace it.
There were two witnesses to the accident, which was beyond question the fault of the driver of the car. An old woman, a Mrs. Wadey, saw the whole thing, and tried to take the number of the car. She was positive as to the letters, which need not be given, and was certain also that the first figure was a 1. The other figures she failed to read on account of the speed and dust.
The other witness was the village simpleton, who just escapes being an arithmetical genius, but is excessively stupid in everything else.
He is always working out sums in his head; and all he could say was that there were five figures in the number, and that he found that when he multiplied the first two figures by the last three they made the same figures, only in different order—just as 24 multiplied by 651 makes 15,624 (the same five figures), in which case the number of the car would have been 24,651; and he knew there was no 0 in the number.
"It will be easy enough to find that car," said Russell. "The known facts are possibly sufficient to enable one to discover the exact number. You see, there must be a limit to the five-figure numbers having the peculiarity observed by the simpleton. And these are further limited by the fact that, as Mrs. Wadey states, the number began with the figure 1. We have therefore to find these numbers. It may conceivably happen that there is only one such number, in which case the thing is solved. But even if there are several cases, the owner of the actual car may easily be found.
"How will you manage that?" somebody asked.
"Surely," replied Russell, "the method is quite obvious. By the process of elimination. Every owner except the one in fault will be able to prove an alibi. Yet, merely guessing offhand, I think it quite probable that there is only one number that fits the case. We shall see."
Russell was right, for that very night he sent the number by post, with the result that the runaway car was at once traced, and its owner, who was himself driving, had to pay the cost of the damages resulting from his carelessness. What was the number of the car?
65.—The Mystery of Ravensdene Park.
The mystery of Ravensdene Park, which I will now present, was a tragic affair, as it involved the assassination of Mr. Cyril Hastings at his country house a short distance from London.
On February 17th, at 11 p.m., there was a heavy fall of snow, and though it lasted only half an hour, the ground was covered to a depth of several inches. Mr. Hastings had been spending the evening at the house of a neighbour, and left at midnight to walk home, taking the short route that lay through Ravensdene Park—that is, from D to A in the sketch-plan. But in the early morning he was found dead, at the point indicated by the star in our diagram, stabbed to the heart. All the seven gates were promptly closed, and the footprints in the snow examined. These were fortunately very distinct, and the police obtained the following facts:—
The footprints of Mr. Hastings were very clear, straight from D to the spot where he was found. There were the footprints of the Ravensdene butler—who retired to bed five minutes before midnight—from E to EE. There were the footprints of the gamekeeper from A to his lodge at AA. Other footprints showed that one individual had come in at gate B and left at gate BB, while another had entered by gate C and left at gate CC.
Only these five persons had entered the park since the fall of snow. Now, it was a very foggy night, and some of these pedestrians had consequently taken circuitous routes, but it was particularly noticed that no track ever crossed another track. Of this the police were absolutely certain, but they stupidly omitted to make a sketch of the various routes before the snow had melted and utterly effaced them.
The mystery was brought before the members of the Puzzle Club, who at once set themselves the task of solving it. Was it possible to discover who committed the crime? Was it the butler? Or the gamekeeper? Or the man who came in at B and went out at BB? Or the man who went in at C and left at CC? They provided themselves with diagrams—sketch-plans, like the one we have reproduced, which simplified the real form of Ravensdene Park without destroying the necessary conditions of the problem.
Our friends then proceeded to trace out the route of each person, in accordance with the positive statements of the police that we have given. It was soon evident that, as no path ever crossed another, some of the pedestrians must have lost their way considerably in the fog. But when the tracks were recorded in all possible ways, they had no difficulty in deciding on the assassin's route; and as the police luckily knew whose footprints this route represented, an arrest was made that led to the man's conviction.
Can our readers discover whether A, B, C, or E committed the deed? Just trace out the route of each of the four persons, and the key to the mystery will reveal itself.
66.—The Buried Treasure.
The problem of the Buried Treasure was of quite a different character. A young fellow named Dawkins, just home from Australia, was introduced to the club by one of the members, in order that he might relate an extraordinary stroke of luck that he had experienced "down under," as the circumstances involved the solution of a poser that could not fail to interest all lovers of puzzle problems. After the club dinner, Dawkins was asked to tell his story, which he did, to the following effect:—
"I have told you, gentlemen, that I was very much down on my luck. I had gone out to Australia to try to retrieve my fortunes, but had met with no success, and the future was looking very dark. I was, in fact, beginning to feel desperate. One hot summer day I happened to be seated in a Melbourne wineshop, when two fellows entered, and engaged in conversation. They thought I was asleep, but I assure you I was very wide awake.
"'If only I could find the right field,' said one man, 'the treasure would be mine; and as the original owner left no heir, I have as much right to it as anybody else.'
"'How would you proceed?' asked the other.
"'Well, it is like this: The document that fell into my hands states clearly that the field is square, and that the treasure is buried in it at a point exactly two furlongs from one corner, three furlongs from the next corner, and four furlongs from the next corner to that. You see, the worst of it is that nearly all the fields in the district are square; and I doubt whether there are two of exactly the same size. If only I knew the size of the field I could soon discover it, and, by taking these simple measurements, quickly secure the treasure.'
"'But you would not know which corner to start from, nor which direction to go to the next corner.'
"'My dear chap, that only means eight spots at the most to dig over; and as the paper says that the treasure is three feet deep, you bet that wouldn't take me long.'
"Now, gentlemen," continued Dawkins, "I happen to be a bit of a mathematician; and hearing the conversation, I saw at once that for a spot to be exactly two, three, and four furlongs from successive corners of a square, the square must be of a particular area. You can't get such measurements to meet at one point in any square you choose. They can only happen in a field of one size, and that is just what these men never suspected. I will leave you the puzzle of working out just what that area is.
"Well, when I found the size of the field, I was not long in discovering the field itself, for the man had let out the district in the conversation. And I did not need to make the eight digs, for, as luck would have it, the third spot I tried was the right one. The treasure was a substantial sum, for it has brought me home and enabled me to start in a business that already shows signs of being a particularly lucrative one. I often smile when I think of that poor fellow going about for the rest of his life saying: 'If only I knew the size of the field!' while he has placed the treasure safe in my own possession. I tried to find the man, to make him some compensation anonymously, but without success. Perhaps he stood in little need of the money, while it has saved me from ruin."
Could the reader have discovered the required area of the field from those details overheard in the wineshop? It is an elegant little puzzle, and furnishes another example of the practical utility, on unexpected occasions, of a knowledge of the art of problem-solving.
THE PROFESSOR'S PUZZLES
"Why, here is the Professor!" exclaimed Grigsby. "We'll make him show us some new puzzles."
It was Christmas Eve, and the club was nearly deserted. Only Grigsby, Hawkhurst, and myself, of all the members, seemed to be detained in town over the season of mirth and mince-pies. The man, however, who had just entered was a welcome addition to our number. "The Professor of Puzzles," as we had nicknamed him, was very popular at the club, and when, as on the present occasion, things got a little slow, his arrival was a positive blessing.
He was a man of middle age, cheery and kind-hearted, but inclined to be cynical. He had all his life dabbled in puzzles, problems, and enigmas of every kind, and what the Professor didn't know about these matters was admittedly not worth knowing. His puzzles always had a charm of their own, and this was mainly because he was so happy in dishing them up in palatable form.
"You are the man of all others that we were hoping would drop in," said Hawkhurst. "Have you got anything new?"
"I have always something new," was the reply, uttered with feigned conceit—for the Professor was really a modest man—"I'm simply glutted with ideas."
"Where do you get all your notions?" I asked.
"Everywhere, anywhere, during all my waking moments. Indeed, two or three of my best puzzles have come to me in my dreams."
"Then all the good ideas are not used up?"
"Certainly not. And all the old puzzles are capable of improvement, embellishment, and extension. Take, for example, magic squares. These were constructed in India before the Christian era, and introduced into Europe about the fourteenth century, when they were supposed to possess certain magical properties that I am afraid they have since lost. Any child can arrange the numbers one to nine in a square that will add up fifteen in eight ways; but you will see it can be developed into quite a new problem if you use coins instead of numbers."
67.—The Coinage Puzzle.
He made a rough diagram, and placed a crown and a florin in two of the divisions, as indicated in the illustration.
"Now," he continued, "place the fewest possible current English coins in the seven empty divisions, so that each of the three columns, three rows, and two diagonals shall add up fifteen shillings. Of course, no division may be without at least one coin, and no two divisions may contain the same value."
"But how can the coins affect the question?" asked Grigsby.
"That you will find out when you approach the solution."
"I shall do it with numbers first," said Hawkhurst, "and then substitute coins."
Five minutes later, however, he exclaimed, "Hang it all! I can't help getting the 2 in a corner. May the florin be moved from its present position?"
"Then I give it up."
But Grigsby and I decided that we would work at it another time, so the Professor showed Hawkhurst the solution privately, and then went on with his chat.
68.—The Postage Stamps Puzzles.
"Now, instead of coins we'll substitute postage-stamps. Take ten current English stamps, nine of them being all of different values, and the tenth a duplicate. Stick two of them in one division and one in each of the others, so that the square shall this time add up ninepence in the eight directions as before."
"Here you are!" cried Grigsby, after he had been scribbling for a few minutes on the back of an envelope.
The Professor smiled indulgently.
"Are you sure that there is a current English postage-stamp of the value of threepence-halfpenny?"
"For the life of me, I don't know. Isn't there?"
"That's just like the Professor," put in Hawkhurst. "There never was such a 'tricky' man. You never know when you have got to the bottom of his puzzles. Just when you make sure you have found a solution, he trips you up over some little point you never thought of."
"When you have done that," said the Professor, "here is a much better one for you. Stick English postage stamps so that every three divisions in a line shall add up alike, using as many stamps as you choose, so long as they are all of different values. It is a hard nut."
69.—The Frogs and Tumblers.
"What do you think of these?"
The Professor brought from his capacious pockets a number of frogs, snails, lizards, and other creatures of Japanese manufacture—very grotesque in form and brilliant in colour. While we were looking at them he asked the waiter to place sixty-four tumblers on the club table. When these had been brought and arranged in the form of a square, as shown in the illustration, he placed eight of the little green frogs on the glasses as shown.
"Now," he said, "you see these tumblers form eight horizontal and eight vertical lines, and if you look at them diagonally (both ways) there are twenty-six other lines. If you run your eye along all these forty-two lines, you will find no two frogs are anywhere in a line.
"The puzzle is this. Three of the frogs are supposed to jump from their present position to three vacant glasses, so that in their new relative positions still no two frogs shall be in a line. What are the jumps made?"
"I suppose——" began Hawkhurst.
"I know what you are going to ask," anticipated the Professor. "No; the frogs do not exchange positions, but each of the three jumps to a glass that was not previously occupied."
"But surely there must be scores of solutions?" I said.
"I shall be very glad if you can find them," replied the Professor with a dry smile. "I only know of one—or rather two, counting a reversal, which occurs in consequence of the position being symmetrical."
70.—Romeo and Juliet.
For some time we tried to make these little reptiles perform the feat allotted to them, and failed. The Professor, however, would not give away his solution, but said he would instead introduce to us a little thing that is childishly simple when you have once seen it, but cannot be mastered by everybody at the very first attempt.
"Waiter!" he called again. "Just take away these glasses, please, and bring the chessboards."
"I hope to goodness," exclaimed Grigsby, "you are not going to show us some of those awful chess problems of yours. 'White to mate Black in 427 moves without moving his pieces.' 'The bishop rooks the king, and pawns his Giuoco Piano in half a jiff.'"
"No, it is not chess. You see these two snails. They are Romeo and Juliet. Juliet is on her balcony, waiting the arrival of her love; but Romeo has been dining, and forgets, for the life of him, the number of her house. The squares represent sixty-four houses, and the amorous swain visits every house once and only once before reaching his beloved. Now, make him do this with the fewest possible turnings. The snail can move up, down, and across the board and through the diagonals. Mark his track with this piece of chalk."
"Seems easy enough," said Grigsby, running the chalk along the squares. "Look! that does it."
"Yes," said the Professor: "Romeo has got there, it is true, and visited every square once, and only once; but you have made him turn nineteen times, and that is not doing the trick in the fewest turns possible."
Hawkhurst, curiously enough, hit on the solution at once, and the Professor remarked that this was just one of those puzzles that a person might solve at a glance or not master in six months.
71.—Romeo's Second Journey.
"It was a sheer stroke of luck on your part, Hawkhurst," he added. "Here is a much easier puzzle, because it is capable of more systematic analysis; yet it may just happen that you will not do it in an hour. Put Romeo on a white square and make him crawl into every other white square once with the fewest possible turnings. This time a white square may be visited twice, but the snail must never pass a second time through the same corner of a square nor ever enter the black squares."
"May he leave the board for refreshments?" asked Grigsby.
"No; he is not allowed out until he has performed his feat."
72.—The Frogs who would a-wooing go.
While we were vainly attempting to solve this puzzle, the Professor arranged on the table ten of the frogs in two rows, as they will be found in the illustration.
"That seems entertaining," I said. "What is it?"
"It is a little puzzle I made a year ago, and a favourite with the few people who have seen it. It is called 'The Frogs who would a-wooing go.' Four of them are supposed to go a-wooing, and after the four have each made a jump upon the table, they are in such a position that they form five straight rows with four frogs in every row."
"What's that?" asked Hawkhurst. "I think I can do that." A few minutes later he exclaimed, "How's this?"
"They form only four rows instead of five, and you have moved six of them," explained the Professor.
"Hawkhurst," said Grigsby severely, "you are a duffer. I see the solution at a glance. Here you are! These two jump on their comrades' backs."
"No, no," admonished the Professor; "that is not allowed. I distinctly said that the jumps were to be made upon the table. Sometimes it passes the wit of man so to word the conditions of a problem that the quibbler will not persuade himself that he has found a flaw through which the solution may be mastered by a child of five."
After we had been vainly puzzling with these batrachian lovers for some time, the Professor revealed his secret.
The Professor gathered up his Japanese reptiles and wished us good-night with the usual seasonable compliments. We three who remained had one more pipe together, and then also left for our respective homes. Each believes that the other two racked their brains over Christmas in the determined attempt to master the Professor's puzzles; but when we next met at the club we were all unanimous in declaring that those puzzles which we had failed to solve "we really had not had time to look at," while those we had mastered after an enormous amount of labour "we had seen at the first glance directly we got home."
73.—The Game of Kayles.
Nearly all of our most popular games are of very ancient origin, though in many cases they have been considerably developed and improved. Kayles—derived from the French word quilles—was a great favourite in the fourteenth century, and was undoubtedly the parent of our modern game of ninepins. Kayle-pins were not confined in those days to any particular number, and they were generally made of a conical shape and set up in a straight row.
At first they were knocked down by a club that was thrown at them from a distance, which at once suggests the origin of the pastime of "shying for cocoanuts" that is to-day so popular on Bank Holidays on Hampstead Heath and elsewhere. Then the players introduced balls, as an improvement on the club.
In the illustration we get a picture of some of our fourteenth-century ancestors playing at kayle-pins in this manner.
Now, I will introduce to my readers a new game of parlour kayle-pins, that can be played across the table without any preparation whatever. You simply place in a straight row thirteen dominoes, chess-pawns, draughtsmen, counters, coins, or beans—anything will do—all close together, and then remove the second one as shown in the picture.
It is assumed that the ancient players had become so expert that they could always knock down any single kayle-pin, or any two kayle-pins that stood close together. They therefore altered the game, and it was agreed that the player who knocked down the last pin was the winner.
Therefore, in playing our table-game, all you have to do is to knock down with your fingers, or take away, any single kayle-pin or two adjoining kayle-pins, playing alternately until one of the two players makes the last capture, and so wins. I think it will be found a fascinating little game, and I will show the secret of winning.
Remember that the second kayle-pin must be removed before you begin to play, and that if you knock down two at once those two must be close together, because in the real game the ball could not do more than this.
74.—The Broken Chessboard.
There is a story of Prince Henry, son of William the Conqueror, afterwards Henry I., that is so frequently recorded in the old chronicles that it is doubtless authentic. The following version of the incident is taken from Hayward's Life of William the Conqueror, published in 1613:—
"Towards the end of his reigne he appointed his two sonnes Robert and Henry, with joynt authoritie, governours of Normandie; the one to suppresse either the insolence or levitie of the other. These went together to visit the French king lying at Constance: where, entertaining the time with varietie of disports, Henry played with Louis, then Daulphine of France, at chesse, and did win of him very much.
"Hereat Louis beganne to growe warme in words, and was therein little respected by Henry. The great impatience of the one and the small forbearance of the other did strike in the end such a heat between them that Louis threw the chessmen at Henry's face.
"Henry again stroke Louis with the chessboard, drew blood with the blowe, and had presently slain him upon the place had he not been stayed by his brother Robert.
"Hereupon they presently went to horse, and their spurres claimed so good haste as they recovered Pontoise, albeit they were sharply pursued by the French."
Now, tradition—on this point not trustworthy—says that the chessboard broke into the thirteen fragments shown in our illustration. It will be seen that there are twelve pieces, all different in shape, each containing five squares, and one little piece of four squares only.
We thus have all the sixty-four squares of the chess-board, and the puzzle is simply to cut them out and fit them together, so as to make a perfect board properly chequered. The pieces may be easily cut out of a sheet of "squared" paper, and, if mounted on cardboard, they will form a source of perpetual amusement in the home.
If you succeed in constructing the chessboard, but do not record the arrangement, you will find it just as puzzling the next time you feel disposed to attack it.
Prince Henry himself, with all his skill and learning, would have found it an amusing pastime.
75.—The Spider and the Fly.
Inside a rectangular room, measuring 30 feet in length and 12 feet in width and height, a spider is at a point on the middle of one of the end walls, 1 foot from the ceiling, as at A; and a fly is on the opposite wall, 1 foot from the floor in the centre, as shown at B. What is the shortest distance that the spider must crawl in order to reach the fly, which remains stationary? Of course the spider never drops or uses its web, but crawls fairly.
76.—The Perplexed Cellarman.
Here is a little puzzle culled from the traditions of an old monastery in the west of England. Abbot Francis, it seems, was a very worthy man; and his methods of equity extended to those little acts of charity for which he was noted for miles round.
The Abbot, moreover, had a fine taste in wines. On one occasion he sent for the cellarman, and complained that a particular bottling was not to his palate.
"Pray tell me, Brother John, how much of this wine thou didst bottle withal."
"A fair dozen in large bottles, my lord abbot, and the like in the small," replied the cellarman, "whereof five of each have been drunk in the refectory."
"So be it. There be three varlets waiting at the gate. Let the two dozen bottles be given unto them, both full and empty; and see that the dole be fairly made, so that no man receive more wine than another, nor any difference in bottles."
Poor John returned to his cellar, taking the three men with him, and then his task began to perplex him. Of full bottles he had seven large and seven small, and of empty bottles five large and five small, as shown in the illustration. How was he to make the required equitable division?
He divided the bottles into three groups in several ways that at first sight seemed to be quite fair, since two small bottles held just the same quantity of wine as one large one. But the large bottles themselves, when empty, were not worth two small ones.
Hence the abbot's order that each man must take away the same number of bottles of each size.
Finally, the cellarman had to consult one of the monks who was good at puzzles of this kind, and who showed him how the thing was done. Can you find out just how the distribution was made?
77.—Making a Flag.
A good dissection puzzle in so few as two pieces is rather a rarity, so perhaps the reader will be interested in the following. The diagram represents a piece of bunting, and it is required to cut it into two pieces (without any waste) that will fit together and form a perfectly square flag, with the four roses symmetrically placed. This would be easy enough if it were not for the four roses, as we should merely have to cut from A to B, and insert the piece at the bottom of the flag. But we are not allowed to cut through any of the roses, and therein lies the difficulty of the puzzle. Of course we make no allowance for "turnings."
78.—Catching the Hogs.
In the illustration Hendrick and Katruen are seen engaged in the exhilarating sport of attempting the capture of a couple of hogs.
Why did they fail?
Strange as it may seem, a complete answer is afforded in the little puzzle game that I will now explain.
Copy the simple diagram on a conveniently large sheet of cardboard or paper, and use four marked counters to represent the Dutchman, his wife, and the two hogs.
At the beginning of the game these must be placed on the squares on which they are shown. One player represents Hendrick and Katruen, and the other the hogs. The first player moves the Dutchman and his wife one square each in any direction (but not diagonally), and then the second player moves both pigs one square each (not diagonally); and so on, in turns, until Hendrick catches one hog and Katruen the other.
This you will find would be absurdly easy if the hogs moved first, but this is just what Dutch pigs will not do.
79.—The Thirty-one Game.
This is a game that used to be (and may be to this day, for aught I know) a favourite means of swindling employed by card-sharpers at racecourses and in railway carriages.
As, on its own merits, however, the game is particularly interesting, I will make no apology for presenting it to my readers.
The cardsharper lays down the twenty-four cards shown in the illustration, and invites the innocent wayfarer to try his luck or skill by seeing which of them can first score thirty-one, or drive his opponent beyond, in the following manner:—
One player turns down a card, say a 2, and counts "two"; the second player turns down a card, say a 5, and, adding this to the score, counts "seven"; the first player turns down another card, say a 1, and counts "eight"; and so the play proceeds alternately until one of them scores the "thirty-one," and so wins.
Now, the question is, in order to win, should you turn down the first card, or courteously request your opponent to do so? And how should you conduct your play? The reader will perhaps say: "Oh, that is easy enough. You must play first, and turn down a 3; then, whatever your opponent does, he cannot stop your making ten, or stop your making seventeen, twenty-four, and the winning thirty-one. You have only to secure these numbers to win."
But this is just that little knowledge which is such a dangerous thing, and it places you in the hands of the sharper.
You play 3, and the sharper plays 4 and counts "seven"; you play 3 and count "ten"; the sharper turns down 3 and scores "thirteen"; you play 4 and count "seventeen"; the sharper plays a 4 and counts "twenty-one"; you play 3 and make your "twenty-four."
  
   
   
   
   
    ]
Now the sharper plays the last 4 and scores "twenty-eight." You look in vain for another 3 with which to win, for they are all turned down! So you are compelled either to let him make the "thirty-one" or to go yourself beyond, and so lose the game.
You thus see that your method of certainly winning breaks down utterly, by what may be called the "method of exhaustion." I will give the key to the game, showing how you may always win; but I will not here say whether you must play first or second: you may like to find it out for yourself.
80.—The Chinese Railways.
Our illustration shows the plan of a Chinese city protected by pentagonal fortifications. Five European Powers were scheming and clamouring for a concession to run a railway to the place; and at last one of the Emperor's more brilliant advisers said, "Let every one of them have a concession!" So the Celestial Government officials were kept busy arranging the details. The letters in the diagram show the different nationalities, and indicate not only just where each line must enter the city, but also where the station belonging to that line must be located. As it was agreed that the line of one company must never cross the line of another, the representatives of the various countries concerned were engaged so many weeks in trying to find a solution to the problem, that in the meantime a change in the Chinese Government was brought about, and the whole scheme fell through. Take your pencil and trace out the route for the line A to A, B to B, C to C, and so on, without ever allowing one line to cross another or pass through another company's station.
81.—The Eight Clowns.
This illustration represents a troupe of clowns I once saw on the Continent. Each clown bore one of the numbers 1 to 9 on his body. After going through the usual tumbling, juggling, and other antics, they generally concluded with a few curious little numerical tricks, one of which was the rapid formation of a number of magic squares. It occurred to me that if clown No. 1 failed to appear (as happens in the illustration), this last item of their performance might not be so easy. The reader is asked to discover how these eight clowns may arrange themselves in the form of a square (one place being vacant), so that every one of the three columns, three rows, and each of the two diagonals shall add up the same. The vacant place may be at any part of the square, but it is No. 1 that must be absent.
82.—The Wizard's Arithmetic.
Once upon a time a knight went to consult a certain famous wizard. The interview had to do with an affair of the heart; but after the man of magic had foretold the most favourable issues, and concocted a love-potion that was certain to help his visitor's cause, the conversation drifted on to occult subjects generally.
"And art thou learned also in the magic of numbers?" asked the knight. "Show me but one sample of thy wit in these matters."
The old wizard took five blocks bearing numbers, and placed them on a shelf, apparently at random, so that they stood in the order 41096, as shown in our illustration. He then took in his hands an 8 and a 3, and held them together to form the number 83.
"Sir Knight, tell me," said the wizard, "canst thou multiply one number into the other in thy mind?"
"Nay, of a truth," the good knight replied. "I should need to set out upon the task with pen and scrip."
"Yet mark ye how right easy a thing it is to a man learned in the lore of far Araby, who knoweth all the magic that is hid in the philosophy of numbers!"
The wizard simply placed the 3 next to the 4 on the shelf, and the 8 at the other end. It will be found that this gives the answer quite correctly—3410968. Very curious, is it not? How many other two-figure multipliers can you find that will produce the same effect? You may place just as many blocks as you like on the shelf, bearing any figures you choose.
83.—The Ribbon Problem.
If we take the ribbon by the ends and pull it out straight, we have the number 0588235294117647. This number has the peculiarity that, if we multiply it by any one of the numbers, 2, 3, 4, 5, 6, 7, 8, or 9, we get exactly the same number in the circle, starting from a different place. For example, multiply by 4, and the product is 2352941176470588, which starts from the dart in the circle. So, if we multiply by 3, we get the same result starting from the star. Now, the puzzle is to place a different arrangement of figures on the ribbon that will produce similar results when so multiplied; only the 0 and the 7 appearing at the ends of the ribbon must not be removed.
84.—The Japanese Ladies and the Carpet.
Three Japanese ladies possessed a square ancestral carpet of considerable intrinsic value, but treasured also as an interesting heirloom in the family. They decided to cut it up and make three square rugs of it, so that each should possess a share in her own house.
One lady suggested that the simplest way would be for her to take a smaller share than the other two, because then the carpet need not be cut into more than four pieces.
There are three easy ways of doing this, which I will leave the reader for the present the amusement of finding for himself, merely saying that if you suppose the carpet to be nine square feet, then one lady may take a piece two feet square whole, another a two feet square in two pieces, and the third a square foot whole.
But this generous offer would not for a moment be entertained by the other two sisters, who insisted that the square carpet should be so cut that each should get a square mat of exactly the same size.
Now, according to the best Western authorities, they would have found it necessary to cut the carpet into seven pieces; but a correspondent in Tokio assures me that the legend is that they did it in as few as six pieces, and he wants to know whether such a thing is possible.
Yes; it can be done.
Can you cut out the six pieces that will form three square mats of equal size?
85.—Captain Longbow and the Bears.
That eminent and more or less veracious traveller Captain Longbow has a great grievance with the public. He claims that during a recent expedition in Arctic regions he actually reached the North Pole, but cannot induce anybody to believe him. Of course, the difficulty in such cases is to produce proof, but he avers that future travellers, when they succeed in accomplishing the same feat, will find evidence on the spot. He says that when he got there he saw a bear going round and round the top of the pole (which he declares is a pole), evidently perplexed by the peculiar fact that no matter in what direction he looked it was always due south. Captain Longbow put an end to the bear's meditations by shooting him, and afterwards impaling him, in the manner shown in the illustration, as the evidence for future travellers to which I have alluded.
When the Captain got one hundred miles south on his return journey he had a little experience that is somewhat puzzling. He was surprised one morning, on looking down from an elevation, to see no fewer than eleven bears in his immediate vicinity. But what astonished him more than anything else was the curious fact that they had so placed themselves that there were seven rows of bears, with four bears in every row. Whether or not this was the result of pure accident he cannot say, but such a thing might have happened. If the reader tries to make eleven dots on a sheet of paper so that there shall be seven rows of dots with four dots in every row, he will find some difficulty; but the captain's alleged grouping of the bears is quite possible. Can you discover how they were arranged?
86.—The English Tour.
This puzzle has to do with railway routes, and in these days of much travelling should prove useful. The map of England shows twenty-four towns, connected by a system of railways. A resident at the town marked A at the top of the map proposes to visit every one of the towns once and only once, and to finish up his tour at Z. This would be easy enough if he were able to cut across country by road, as well as by rail, but he is not. How does he perform the feat? Take your pencil and, starting from A, pass from town to town, making a dot in the towns you have visited, and see if you can end at Z.
87.—The Chifu-Chemulpo Puzzle.
Here is a puzzle that was once on sale in the London shops. It represents a military train—an engine and eight cars. The puzzle is to reverse the cars, so that they shall be in the order 8, 7, 6, 5, 4, 3, 2, 1, instead of 1, 2, 3, 4, 5, 6, 7, 8, with the engine left, as at first, on the side track. Do this in the fewest possible moves. Every time the engine or a car is moved from the main to the side track, or vice versa, it counts a move for each car or engine passed over one of the points. Moves along the main track are not counted. With 8 at the extremity, as shown, there is just room to pass 7 on to the side track, run 8 up to 6, and bring down 7 again; or you can put as many as five cars, or four and the engine, on the siding at the same time. The cars move without the aid of the engine. The purchaser is invited to "try to do it in 20 moves." How many do you require?
88.—The Eccentric Market-woman.
Mrs. Covey, who keeps a little poultry farm in Surrey, is one of the most eccentric women I ever met. Her manner of doing business is always original, and sometimes quite weird and wonderful. She was once found explaining to a few of her choice friends how she had disposed of her day's eggs. She had evidently got the idea from an old puzzle with which we are all familiar; but as it is an improvement on it, I have no hesitation in presenting it to my readers. She related that she had that day taken a certain number of eggs to market. She sold half of them to one customer, and gave him half an egg over. She next sold a third of what she had left, and gave a third of an egg over. She then sold a fourth of the remainder, and gave a fourth of an egg over. Finally, she disposed of a fifth of the remainder, and gave a fifth of an egg over. Then what she had left she divided equally among thirteen of her friends. And, strange to say, she had not throughout all these transactions broken a single egg. Now, the puzzle is to find the smallest possible number of eggs that Mrs. Covey could have taken to market. Can you say how many?
89.—The Primrose Puzzle.
Select the name of any flower that you think suitable, and that contains eight letters. Touch one of the primroses with your pencil and jump over one of the adjoining flowers to another, on which you mark the first letter of your word. Then touch another vacant flower, and again jump over one in another direction, and write down the second letter. Continue this (taking the letters in their proper order) until all the letters have been written down, and the original word can be correctly read round the garland. You must always touch an unoccupied flower, but the flower jumped over may be occupied or not. The name of a tree may also be selected. Only English words may be used.
90.—The Round Table.
Seven friends, named Adams, Brooks, Cater, Dobson, Edwards, Fry, and Green, were spending fifteen days together at the seaside, and they had a round breakfast table at the hotel all to themselves. It was agreed that no man should ever sit down twice with the same two neighbours. As they can be seated, under these conditions, in just fifteen ways, the plan was quite practicable. But could the reader have prepared an arrangement for every sitting? The hotel proprietor was asked to draw up a scheme, but he miserably failed.
91.—The Five Tea Tins.
Sometimes people will speak of mere counting as one of the simplest operations in the world; but on occasions, as I shall show, it is far from easy. Sometimes the labour can be diminished by the use of little artifices; sometimes it is practically impossible to make the required enumeration without having a very clear head indeed. An ordinary child, buying twelve postage stamps, will almost instinctively say, when he sees there are four along one side and three along the other, "Four times three are twelve;" while his tiny brother will count them all in rows, "1, 2, 3, 4," etc. If the child's mother has occasion to add up the numbers 1, 2, 3, up to 50, she will most probably make a long addition sum of the fifty numbers; while her husband, more used to arithmetical operations, will see at a glance that by joining the numbers at the extremes there are 25 pairs of 51; therefore, 25x51=1,275. But his smart son of twenty may go one better and say, "Why multiply by 25? Just add two 0's to the 51 and divide by 4, and there you are!"
A tea merchant has five tin tea boxes of cubical shape, which he keeps on his counter in a row, as shown in our illustration. Every box has a picture on each of its six sides, so there are thirty pictures in all; but one picture on No. 1 is repeated on No. 4, and two other pictures on No. 4 are repeated on No. 3. There are, therefore, only twenty-seven different pictures. The owner always keeps No. 1 at one end of the row, and never allows Nos. 3 and 5 to be put side by side.
The tradesman's customer, having obtained this information, thinks it a good puzzle to work out in how many ways the boxes may be arranged on the counter so that the order of the five pictures in front shall never be twice alike. He found the making of the count a tough little nut. Can you work out the answer without getting your brain into a tangle? Of course, two similar pictures may be in a row, as it is all a question of their order.
92.—The Four Porkers.
The four pigs are so placed, each in a separate sty, that although every one of the thirty-six sties is in a straight line (either horizontally, vertically, or diagonally), with at least one of the pigs, yet no pig is in line with another. In how many different ways may the four pigs be placed to fulfil these conditions? If you turn this page round you get three more arrangements, and if you turn it round in front of a mirror you get four more. These are not to be counted as different arrangements.
93.—The Number Blocks.
The children in the illustration have found that a large number of very interesting and instructive puzzles may be made out of number blocks; that is, blocks bearing the ten digits or Arabic figures—1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. The particular puzzle that they have been amusing themselves with is to divide the blocks into two groups of five, and then so arrange them in the form of two multiplication sums that one product shall be the same as the other. The number of possible solutions is very considerable, but they have hit on that arrangement that gives the smallest possible product. Thus, 3,485 multiplied by 2 is 6,970, and 6,970 multiplied by 1 is the same. You will find it quite impossible to get any smaller result.
Now, my puzzle is to find the largest possible result. Divide the blocks into any two groups of five that you like, and arrange them to form two multiplication sums that shall produce the same product and the largest amount possible. That is all, and yet it is a nut that requires some cracking. Of course, fractions are not allowed, nor any tricks whatever. The puzzle is quite interesting enough in the simple form in which I have given it. Perhaps it should be added that the multipliers may contain two figures.
94.—Foxes and Geese.
Here is a little puzzle of the moving counters class that my readers will probably find entertaining. Make a diagram of any convenient size similar to that shown in our illustration, and provide six counters—three marked to represent foxes and three to represent geese. Place the geese on the discs 1, 2, and 3, and the foxes on the discs numbered 10, 11, and 12.
Now the puzzle is this. By moving one at a time, fox and goose alternately, along a straight line from one disc to the next one, try to get the foxes on 1, 2, and 3, and the geese on 10, 11, and 12—that is, make them exchange places—in the fewest possible moves.
But you must be careful never to let a fox and goose get within reach of each other, or there will be trouble. This rule, you will find, prevents you moving the fox from 11 on the first move, as on either 4 or 6 he would be within reach of a goose. It also prevents your moving a fox from 10 to 9, or from 12 to 7. If you play 10 to 5, then your next move may be 2 to 9 with a goose, which you could not have played if the fox had not previously gone from 10. It is perhaps unnecessary to say that only one fox or one goose can be on a disc at the same time. Now, what is the smallest number of moves necessary to make the foxes and geese change places?
95.—Robinson Crusoe's Table.
Here is a curious extract from Robinson Crusoe's diary. It is not to be found in the modern editions of the Adventures, and is omitted in the old. This has always seemed to me to be a pity.
"The third day in the morning, the wind having abated during the night, I went down to the shore hoping to find a typewriter and other useful things washed up from the wreck of the ship; but all that fell in my way was a piece of timber with many holes in it. My man Friday had many times said that we stood sadly in need of a square table for our afternoon tea, and I bethought me how this piece of wood might be used for that purpose. And since during the long time that Friday had now been with me I was not wanting to lay a foundation of useful knowledge in his mind, I told him that it was my wish to make the table from the timber I had found, without there being any holes in the top thereof.
"Friday was sadly put to it to say how this might be, more especially as I said it should consist of no more than two pieces joined together; but I taught him how it could be done in such a way that the table might be as large as was possible, though, to be sure, I was amused when he said, 'My nation do much better: they stop up holes, so pieces sugars not fall through.'"
Now, the illustration gives the exact proportion of the piece of wood with the positions of the fifteen holes. How did Robinson Crusoe make the largest possible square table-top in two pieces, so that it should not have any holes in it?
96.—The Fifteen Orchards.
In the county of Devon, where the cider comes from, fifteen of the inhabitants of a village are imbued with an excellent spirit of friendly rivalry, and a few years ago they decided to settle by actual experiment a little difference of opinion as to the cultivation of apple trees. Some said they want plenty of light and air, while others stoutly maintained that they ought to be planted pretty closely, in order that they might get shade and protection from cold winds. So they agreed to plant a lot of young trees, a different number in each orchard, in order to compare results.
One man had a single tree in his field, another had two trees, another had three trees, another had four trees, another five, and so on, the last man having as many as fifteen trees in his little orchard. Last year a very curious result was found to have come about. Each of the fifteen individuals discovered that every tree in his own orchard bore exactly the same number of apples. But, what was stranger still, on comparing notes they found that the total gathered in every allotment was almost the same. In fact, if the man with eleven trees had given one apple to the man who had seven trees, and the man with fourteen trees had given three each to the men with nine and thirteen trees, they would all have had exactly the same.
Now, the puzzle is to discover how many apples each would have had (the same in every case) if that little distribution had been carried out. It is quite easy if you set to work in the right way.
97.—The Perplexed Plumber.
When I paid a visit to Peckham recently I found everybody asking, "What has happened to Sam Solders, the plumber?" He seemed to be in a bad way, and his wife was seriously anxious about the state of his mind. As he had fitted up a hot-water apparatus for me some years ago which did not lead to an explosion for at least three months (and then only damaged the complexion of one of the cook's followers), I had considerable regard for him.