The simple person will at once suggest that we might take our diagonal first, say an exact foot, and then construct our square. Yes, you can do this, but then you can never say exactly what is the length of the side. You can have it which way you like, but you cannot have it both ways.
All my readers know what a magic square is. The numbers 1 to 9 can be arranged in a square of nine cells, so that all the columns and rows and each of the diagonals will add up It is quite easy; and there is only one way of doing it, for we do not count as different the arrangements obtained by merely turning round the square and reflecting it in a mirror.
Now if we wish to make a magic square of the 16 numbers, 1 to 16, there are just different ways of doing it, again not counting reversals and reflections. This has been finally proved of recent years. But how many magic squares may be formed with the 25 numbers, 1 to 25, nobody knows, and we shall have to extend our knowledge in certain directions before we can hope to solve the puzzle.
But it is surprising to find that exactly , such squares may be formed of one particular restricted kind only—the bordered square, in which the inner square of nine cells is itself magic. And I have shown how this number may be at once doubled by merely converting every bordered square—by a simple rule—into a non-bordered one. Then vain attempts have been made to construct a magic square by what is called a "knight's tour" over the chess-board, numbering each square that the knight visits in succession, 1, 2, 3, 4, etc.
But it is not certain that it cannot be done. Though the contents of the present volume are in the main entirely original, some very few old friends will be found; but these will not, I trust, prove unwelcome in the new dress that they have [Pg 22] received. The puzzles are of every degree of difficulty, and so varied in character that perhaps it is not too much to hope that every true puzzle lover will find ample material to interest—and possibly instruct.
In some cases I have dealt with the methods of solution at considerable length, but at other times I have reluctantly felt obliged to restrict myself to giving the bare answers. Had the full solutions and proofs been given in the case of every puzzle, either half the problems would have had to be omitted, or the size of the book greatly increased.
And the plan that I have adopted has its advantages, for it leaves scope for the mathematical enthusiast to work out his own analysis. Even in those cases where I have given a general formula for the solution of a puzzle, he will find great interest in verifying it for himself. This we all know was the origin of the immortal Canterbury Tales of our great fourteenth-century poet, Geoffrey Chaucer.
Unfortunately, the tales were never completed, and perhaps that is why the quaint and curious "Canterbury Puzzles," devised and propounded by the same body of pilgrims, were not also recorded by the poet's pen. This is greatly to be regretted, since Chaucer, who, as Leland tells us, was an "ingenious mathematician" and the author of a learned treatise on the astrolabe, was peculiarly fitted for the propounding of problems.
In presenting for the first time some of these old-world posers, I will not stop to explain the singular manner in which they came into my possession, but proceed at once, without unnecessary preamble, to give my readers an opportunity of solving them and testing their quality.
There are certainly far more difficult puzzles extant, but difficulty and interest are two qualities of puzzledom that do not necessarily go together. The Reve was a wily man and something of a scholar. As Chaucer tells us, "There was no auditor could of him win," and "there could no man bring him in arrear. When the pilgrims were stopping at a wayside tavern, a number of cheeses of varying sizes caught his alert eye; and calling for four stools, he told the company that he would show them a puzzle of his own that would keep them amused during their rest.
He then placed eight cheeses of graduating sizes on one of the end stools, the smallest cheese being at the top, as clearly shown in the illustration. And yet it is withal full easy, for all that I do desire is that, by the moving of one cheese at a time from one stool unto another, ye shall remove all the cheeses to the stool at the other end without ever putting any cheese on one that is smaller than itself.
To him that will perform this feat in the least number of moves that be possible will I give a draught of the best that our good host can provide. The gentle Pardoner, "that straight was come from the court of Rome," begged to be excused; but the company would not spare him. Blame my lack of knowledge of such matters if it be not to your liking. He produced the accompanying plan, and said that it represented sixty-four towns through which he had to pass [Pg 26] during some of his pilgrimages, and the lines connecting them were roads.
He explained that the puzzle was to start from the large black town and visit all the other towns once, and once only, in fifteen straight pilgrimages. Try to trace the route in fifteen straight lines with your pencil. You may end where you like, but note that the omission of a little road at the bottom is intentional, as it seems that it was impossible to go that way. The Miller next took the company aside and showed them nine sacks of flour that were standing as depicted in the sketch.
And mark ye, my lords and masters, that there be single sacks on the outside, pairs next unto them, and three together in the middle thereof.
By Saint Benedict, it doth so happen that if we do but multiply the pair, 28, by the single one, 7, the answer is , which is of a truth the number shown by the sacks in the middle. Yet it be not true that the other pair, 34, when so multiplied by its neighbour, 5, will also make Wherefore I do beg you, gentle sirs, so to place anew the nine sacks with as little trouble as possible that each pair when thus multiplied by its single neighbour shall make the number in the middle.
This worthy man was, as Chaucer tells us, "a very perfect, gentle knight," and "In many a noble army had he been: At [Pg 27] mortal battles had he been fifteen. When this knight was called on to propound a puzzle, he said to the company, "This riddle a wight did ask of me when that I fought with the lord of Palatine against the heathen in Turkey. In thy hand take a piece of chalk and learn how many perfect squares thou canst make with one of the eighty-seven roses at each corner thereof.
The frolicsome Wife of Bath, when called upon to favour the company, protested that she had no aptitude for such things, but that her fourth husband had had a liking for them, and she [Pg 28] remembered one of his riddles that might be new to her fellow pilgrims: Perhaps no puzzle of the whole collection caused more jollity or was found more entertaining than that produced by the Host of [Pg 29] the "Tabard," who accompanied the party all the way.
He called the pilgrims together and spoke as follows: And yet methinks it is but a simple matter when the doing of it is made clear. Here be a cask of fine London ale, and in my hands do I hold two measures—one of five pints, and the other of three pints.
Pray show how it is possible for me to put a true pint into each of the measures. It is a knotty little problem and a fascinating one. A good many persons to-day will find it by no means an easy task. Yet it can be done. The silent and thoughtful Clerk of Oxenford, of whom it is recorded that "Every farthing that his friends e'er lent, In books and learning was it always spent," was prevailed upon to give his companions a puzzle.
He said, "Ofttimes of late have I given much thought to the study of those strange talismans to ward off the plague and such evils that are yclept magic squares, and the secret of such things is very deep and the number of such squares [Pg 30] truly great.
But the small riddle that I did make yester eve for the purpose of this company is not so hard that any may not find it out with a little patience. It will be found that this is a just sufficiently easy puzzle for most people's tastes. Then came forward the Tapiser, who was, of course, a maker of tapestry, and must not be confounded with a tapster, who draws and sells ale. He produced a beautiful piece of tapestry, worked in a simple chequered pattern, as shown in the diagram.
The Carpenter produced the carved wooden pillar that he is seen holding in the illustration, wherein the knight is propounding his knotty problem to the goodly company No. Some few days gone he did bring unto me a piece of wood that had three feet in length, one foot in breadth and one foot in depth, and did desire that it be carved and made into the pillar that you do now behold.
Also did he promise certain payment for every cubic inch of wood cut away by the carving thereof. Of a truth I have therefore cut away one cubic foot which is to say one-third of the three cubic feet of the block; but this scholar withal doth hold that payment may not thus be fairly made by weight, since the heart of the block may be heavier, or perchance may be more light, than the outside. How then may I with ease satisfy the scholar as to the quantity of wood that hath been cut away?
Chaucer says of the Squire's Yeoman, who formed one of his party of pilgrims, "A forester was he truly as I guess," and tells us that "His arrows drooped not with feathers low, And in his hand he bare a mighty bow. Selecting nine good arrows, he said, "Mark ye, good sirs, how that I shall shoot these nine arrows in such manner that each of them shall lodge in the middle of one of the squares that be upon the sign of the 'Chequers,' and yet of a truth shall no arrow be in line with any other arrow.
Then the Yeoman said: Remove three of the arrows each to one of its neighbouring squares, so that the nine shall yet be so placed that none thereof may be in line with another. These games, such as cards and the game of chess, do they cunningly hide from the abbot's eye by putting them away in holes [Pg 33] that they have cut out of the very hearts of great books that be upon their shelves. Shall the nun therefore be greatly blamed if she do likewise?
I will show a little riddle game that we do sometimes play among ourselves when the good abbess doth hap to be away. The Nun then produced the eighteen cards that are shown in the illustration. It is easy enough if you work backwards, but the reader should try to arrive at the required order without doing this, or using any actual cards. Of the Merchant the poet writes, "Forsooth he was a worthy man withal.
Sounding away the increase of his winning. He thereupon said, "Be it so? Here then is a riddle in numbers that I will set before this merry company when next we do make a halt. There be thirty of us in all riding over the common this morn.
Truly we [Pg 34] may ride one and one, in what they do call the single file, or two and two, or three and three, or five and five, or six and six, or ten and ten, or fifteen and fifteen, or all thirty in a row. In no other way may we ride so that there be no lack of equal numbers in the rows. Now, a party of pilgrims were able thus to ride in as many as sixty-four different ways.
Prithee tell me how many there must perforce have been in the company. The Sergeant of the Law was "full rich of excellence. Discreet he was, and of great reverence. These prisoners be numbered in order, 7, 5, 6, 8, 2, 1, 4, 3, and I desire to know how they can, in as few moves as possible, put themselves in the order 1, 2, 3, 4, 5, 6, 7, 8.
One prisoner may move at a time along the passage to the dungeon that doth happen to be empty, but never, on pain of death, may two men be in any dungeon at the same time.
How may it be done? As there is never more than one vacant dungeon at a time to be moved into, the moves may be recorded in this simple way: When the Weaver brought out a square piece of beautiful cloth, daintily embroidered with lions and castles, as depicted in the illustration, the pilgrims disputed among themselves as to the meaning of these ornaments.
The Knight, however, who was skilled in heraldry, explained that they were probably derived from the lions and castles borne in the arms of Ferdinand III. In this he was undoubtedly correct. The puzzle that the Weaver proposed was this. No cut may pass through any part of a lion or a castle. We find that there was a cook among the company; and his services were no doubt at times in great request, "For he could roast and seethe, and broil and fry, And make a mortress and well bake a pie.
There be eleven pilgrims seated at this board on which is set a warden pie and a venison pasty, each of which may truly be divided into four parts and no more.
Now, mark ye, five out of the eleven pilgrims can eat the pie, but will not touch the pasty, while four [Pg 37] will eat the pasty but turn away from the pie.
Moreover, the two that do remain be able and willing to eat of either. By my halidame, is there any that can tell me in how many different ways the good Franklin may choose whom he will serve? Strange to say, while the company perplexed their wits about this riddle the cook played upon them a merry jest. In the midst of their deep thinking and hot dispute what should the cunning knave do but stealthily take away both the pie and the pasty.
Then, when hunger made them desire to go on with the repast, finding there was nought upon the table, they called clamorously for the cook. There be excellent bread and cheese in the pantry. The Sompnour, or Summoner, who, according to Chaucer, joined the party of pilgrims, was an officer whose duty was to summon delinquents to appear in ecclesiastical courts.
In later times he became known as the apparitor. Our particular individual was a somewhat quaint though worthy man. One evening ten of the company stopped at a village inn and [Pg 39] requested to be put up for the night, but mine host could only accommodate five of them. The Sompnour suggested that they should draw lots, and as he had had experience in such matters in the summoning of juries and in other ways, he arranged the company in a circle and proposed a "count out.
He therefore gave the Wife of Bath a number and directed her to count round and round the circle, in a clockwise direction, and the person on whom that number fell was immediately to step out of the ring. The count then began afresh at the next person. But the lady misunderstood her instructions, and selected in mistake the number eleven and started the count at herself. As will be found, this resulted in all the women falling out in turn instead of the men, for every eleventh person withdrawn from the circle is a lady.
Can any tell me what number the good Wife should have used withal, and at which pilgrim she should have begun her count so that no other than the five men should have been counted out?
The Monk that went with the party was a great lover of sport. Of riding and of hunting for the hare Was all his love, for no cost would he spare. Nine kennels have I for the use of my dogs, and they be put in the form of a square; though the one in the middle I do never use, it not being of a useful nature. Now the riddle is to find in how many different ways I may place my dogs in all or any of the outside kennels so that the [Pg 40] number of dogs on every side of the square may be just ten.
Any kennels may be left empty. This puzzle was evidently a variation of the ancient one of the Abbess and her Nuns. Of this person we are told, "He knew well all the havens, as they were, From Gothland to the Cape of Finisterre, And every creek in Brittany and Spain: In each year my good ship doth sail over every one of the ten courses depicted thereon, but never may she pass along the same course twice in any year. Is there any among the company who can tell me in how many different ways I may direct the Magdalen's ten yearly voyages, always setting out from the same island?
But the good Abbot of Chertsey did once tell me that the cross may be so cunningly cut into four pieces that they will join and make a perfect square; though on my faith I know not the manner of doing it.
It is recorded that "the pilgrims did find no answer to the riddle, [Pg 42] and the Clerk of Oxenford thought that the Prioress had been deceived in the matter thereof; whereupon the lady was sore vexed, though the gentle knight did flout and gibe at the poor clerk because of his lack of understanding over other of the riddles, which did fill him with shame and make merry the company.
This Doctor, learned though he was, for "In all this world to him there was none like To speak of physic and of surgery," and "He knew the cause of every malady," yet was he not indifferent to the more material side of life. He produced two spherical phials, as shown in our illustration, and pointed out that one phial was exactly a foot in circumference, and the other two feet in circumference.
Of course the thickness of the glass, and the neck and base, are to be ignored. The Ploughman—of whom Chaucer remarked, "A worker true and very good was he, Living in perfect peace and charity"—protested that riddles were not for simple minds like his, but he [Pg 44] would show the good pilgrims, if they willed it, one that he had frequently heard certain clever folk in his own neighbourhood discuss.
Once on a time a man of deep learning, who happened to be travelling in those parts, did say that the sixteen trees might have been so planted that they would make so many as fifteen straight rows, with four trees in every row thereof. Can ye show me how this might be? Many have doubted that 'twere possible to be done.
How can we make fifteen? Of fish and flesh, and that so plenteous, It snowed in his house of meat and drink, Of every dainty that men could bethink. One day, at an inn just outside Canterbury, the company called on him to produce the puzzle required of him; whereupon he placed on the table sixteen bottles numbered 1, 2, 3, up to 15, with the last one marked 0. Of a truth will I set before ye another that may seem to be somewhat of a like kind, albeit there be little in common betwixt them.
Here be set out sixteen bottles in form of a square, and I pray you so place them afresh that they shall form a magic square, adding up to thirty in all the ten straight ways. But mark well that ye may not remove more than ten of the bottles from their present places, for therein layeth the subtlety of the riddle. The young Squire, twenty years of age, was the son of the Knight that accompanied him on the historic pilgrimage.
Singing he was or fluting all the day, He was as fresh as is the month of May. The Knight turned to him after a while and said, "My son, what is it over which thou dost take so great pains withal? To him who first shall show it unto me will I give the portraiture. I am able to present a facsimile of the original drawing, which was won by the Man of Law. It may be here remarked that the pilgrimage set out from Southwark on 17th April , and Edward the Third died in The Friar was a merry fellow, with a sweet tongue and twinkling eyes.
There was a man nowhere so virtuous. One day he produced four money bags and spoke as follows: The Parson was a really devout and good man.
He produced a plan of part of his parish, through which a small river ran that joined the sea some hundreds of miles to the south. I give a facsimile of the plan. Upon this island doth stand my own poor parsonage, and ye may all see the whereabouts of the village church. Mark ye, also, that there be eight bridges and no more over the river in my parish.
On my way to church it is my wont to visit sundry of my flock, and in the doing thereof I do pass over every one of the eight bridges once and no more. Can any of ye find the path, after this manner, from the house to the church, without going out of the parish? Nay, nay, my friends, I do never cross the river in any boat, neither by swimming nor wading, nor do I go underground like unto the mole, nor fly in the air as doth the eagle; but only pass over by the [Pg 49] bridges.
Can the reader discover it? At first it seems impossible, but the conditions offer a loophole. Many attempts were made to induce the Haberdasher, who was of the party, to propound a puzzle of some kind, but for a long time without success. At last, at one of the Pilgrims' stopping-places, he said that he would show them something that would "put their brains into a twist like unto a bell-rope.
He produced a piece of cloth in the shape of a perfect equilateral triangle, as shown in the illustration, and said, "Be there any among ye full wise in the true cutting of cloth? Every man to his trade, and the scholar may learn from the varlet and the wise man from the fool. Show me, then, if ye can, in what manner this piece of cloth may be cut into four several pieces that may be put together to make a perfect square.
Now some of the more learned of the company found a way of doing it in five pieces, but not in four. But when they pressed the Haberdasher for the correct answer he was forced to admit, after much beating about the bush, that he knew no way of doing it in any number of pieces.
But the curious point of the puzzle is that I have found that the feat may really be performed in so few as four pieces, and without turning over any piece when placing them together.
The method of doing this is subtle, but I think the reader will find the problem a most interesting one. One of the pilgrims was a Dyer, but Chaucer tells us nothing about him, the Tales being incomplete. Time after time the company had pressed this individual to produce a puzzle of some kind, but without effect.
The poor fellow tried his best to follow the examples of his friends the Tapiser, the Weaver, and the Haberdasher; but the necessary idea would not come, rack his brains as he would. All things, however, come to those who wait—and persevere—and one morning he announced, in a state of considerable excitement, that he had a poser to set before them. He brought out a square piece of silk on which were embroidered a number of fleurs-de-lys in rows, as shown in our illustration.
Since I was awakened at dawn by the crowing of cocks—for which [Pg 51] din may our host never thrive—I have sought an answer thereto, but by St. Bernard I have found it not. There be sixty-and-four flowers-de-luce, and the riddle is to show how I may remove six of these so that there may yet be an even number of the flowers in every row and every column. The Dyer was abashed when every one of the company showed without any difficulty whatever, and each in a different way, how this might be done.
But the good Clerk of Oxenford was seen to whisper something to the Dyer, who added, "Hold, my masters! What I have said is not all. Ye must find in how many different ways it may be done! And only a few of the company got the right answer. Chaucer records the painful fact that the harmony of the pilgrimage was broken on occasions by the quarrels between the Friar and the Sompnour. At one stage the latter threatened that ere they reached Sittingbourne he would make the Friar's "heart for to mourn;" but the worthy Host intervened and patched up a [Pg 52] temporary peace.
Unfortunately trouble broke out again over a very curious dispute in this way. At one point of the journey the road lay along two sides of a square field, and some of the pilgrims persisted, in spite of trespass, in cutting across from corner to corner, as they are seen to be doing in the illustration.
Now, the Friar startled the company by stating that there was no need for the trespass, since one way was exactly the same distance as the other! If the reader will refer to the diagrams that we have given, he will be able to follow the Friar's argument. If we suppose the [Pg 53] side of the field to be yards, then the distance along the two sides, A to B, and B to C, is yards. He undertook to prove that the diagonal distance direct from A to C is also yards. Now, if we take the diagonal path shown in Fig.
No matter how many steps we make in our zigzag path, the result is most certainly always the same. In this way, the Friar argued, we may go on straightening out that zigzag path until we ultimately reach a perfectly straight line, and it therefore follows that the diagonal of a square is of exactly the same length as two of the sides.
Now, in the face of it, this must be wrong; and it is in fact absurdly so, as we can at once prove by actual measurement if we [Pg 54] have any doubt.
Yet the Sompnour could not for the life of him point out the fallacy, and so upset the Friar's reasoning.
It was this that so exasperated him, and consequently, like many of us to-day when we get entangled in an argument, he utterly lost his temper and resorted to abuse. In fact, if some of the other pilgrims had not interposed the two would have undoubtedly come to blows.
The reader will perhaps at once see the flaw in the Friar's argument. Chaucer himself accompanied the pilgrims. Being a mathematician and a man of a thoughtful habit, the Host made fun of him, he tells us, saying, "Thou lookest as thou wouldst find a hare, For ever on the ground I see thee stare. It is an interesting fact that in the "Parson's Prologue" Chaucer actually [Pg 55] introduces a little astronomical problem.
In modern English this reads somewhat as follows: I calculate that it was four o'clock, for, assuming my height to be six feet, my shadow was eleven feet, a little more or less. At the same moment the moon's altitude she being in mid-Libra was steadily increasing as we entered at the west end of the village. This speaks well for Chaucer's accuracy, for the first line of the Tales tells us that the pilgrimage was in April—they are supposed to have set out on 17th April , as stated in No.
Though Chaucer made this little puzzle and recorded it for the interest of his readers, he did not venture to propound it to his fellow-pilgrims. The puzzle that he gave them was of a simpler kind altogether: Of a truth, as he did show me, a mug will hold less liquor at the top of this mountain than in the valley beneath.
Prythee tell me what mountain this may be that has so strange a property withal. This person joined the party on the road.
Fast have I ridden,' saith he, 'for your sake, Because I would I might you overtake, To ride among this merry company. He showed them the diamond-shaped arrangement [Pg 56] of letters presented in the accompanying illustration, and said, "I do call it the rat-catcher's riddle.
In how many different ways canst thou read the words, 'Was it a rat I saw? The Manciple was an officer who had the care of buying victuals for an Inn of Court—like the Temple. The particular individual who accompanied the party was a wily man who had more than thirty masters, and made fools of them all. Yet he was a man "whom purchasers might take as an example How to be wise in buying of their victual.
It happened that at a certain stage of the journey the Miller and the Weaver sat down to a light repast. The Miller produced five loaves and the Weaver three.
The Manciple coming upon the scene asked permission to eat with them, to which they agreed. When the Manciple had fed he laid down eight pieces of money and said with a sly smile, "Settle betwixt yourselves how the money shall be fairly divided. A discussion followed, and many of the pilgrims joined in it. The Reve and the Sompnour held that the Miller should receive five pieces and the Weaver three, the simple Ploughman was ridiculed for suggesting that the Miller should receive seven and the Weaver only one, while the Carpenter, the Monk, and the Cook insisted that the money should be divided equally between the two men.
Various other opinions were urged with considerable vigour, until it was finally decided that the Manciple, as an expert in such matters, should himself settle the point. His decision was quite correct.
Of course, all three are supposed to have eaten equal shares of the bread. Everybody that has heard of Solvamhall Castle, and of the quaint customs and ceremonies that obtained there in the olden times, is familiar with the fact that Sir Hugh de Fortibus was a lover of all kinds of puzzles and enigmas. Certes, I wot not the riddle that he may not rede withal. The selection has been made to suit all tastes, and while the majority will be found sufficiently easy to interest those who like a puzzle that is a puzzle, but well within the scope of all, two that I have included may perhaps be found worthy of engaging the attention of the more advanced student of these things.
Bandy-ball, cambuc, or goff the game so well known to-day by the name of golf , is of great antiquity, and was a special favourite [Pg 59] at Solvamhall Castle.
Sir Hugh de Fortibus was himself a master of the game, and he once proposed this question. They had nine holes, , , , , , , , , and yards apart. If a man could always strike the ball in a perfectly straight line and send it exactly one of two distances, so that it would either go towards the hole, pass over it, or drop into it, what would the two distances be that would carry him in the least number of strokes round the whole course?
Two very good distances are and 75, which carry you round in 28 strokes, but this is not the correct answer. Can the reader get round in fewer strokes with two other distances?
Another favourite sport at the castle was tilting at the ring. A horizontal bar was fixed in a post, and at the end of a hanging supporter was placed a circular ring, as shown in the above illustrated title. By raising or lowering the bar the ring could be adjusted to the proper height—generally about the level of the left eyebrow of the horseman.
The object was to ride swiftly some eighty paces and run the lance through the ring, which was easily detached, and remained on the lance as the property of the skilful winner. It was a very difficult feat, and men were not unnaturally proud of the rings they had succeeded in capturing. At one tournament at the castle Henry de Gournay beat Stephen Malet by six rings. Each had his rings made into a chain—De Gournay's chain being exactly sixteen inches in length, and Malet's six inches.
Now, as the rings were all of the same size and made of metal half an inch thick, the little puzzle proposed by Sir Hugh was to discover just how many rings each man had won. Seated one night in the hall of the castle, Sir Hugh desired the company to fill their cups and listen while he told the tale of his [Pg 60] adventure as a youth in rescuing from captivity a noble demoiselle who was languishing in the dungeon of the castle belonging to his father's greatest enemy.
The story was a thrilling one, and when he related the final escape from all the dangers and horrors of the great Death's-head Dungeon with the fair but unconscious maiden in his arms, all exclaimed, "'Twas marvellous valiant! Sir Hugh then produced a plan of the thirty-five cells in the dungeon and asked his companions to discover the particular cell that the demoiselle occupied. He said that if you started at one of the outside cells and passed through every doorway once, and once only, you were bound to end at the cell that was sought.
Can you find the cell? Unless you start at the correct outside cell it is impossible to pass through all the doorways once and once only. Try tracing out the route with your pencil. The butt or target used in archery at Solvamhall was not marked out in concentric rings as at the present day, but was prepared in [Pg 61] fanciful designs.
In the illustration is shown a numbered target prepared by Sir Hugh himself. It is something of a curiosity, because it will be found that he has so cleverly arranged the numbers that every one of the twelve lines of three adds up to exactly twenty-two. One day, when the archers were a little tired of their sport, Sir Hugh de Fortibus said, "What ho, merry archers!
Of a truth it is said that a fool's bolt is soon shot, but, by my faith, I know not any man among you who shall do that which I will now put forth. Let these numbers that are upon the butt be set down afresh, so that the twelve lines thereof shall make twenty and three instead of twenty and two. To rearrange the numbers one to nineteen so that all the twelve lines shall add up to twenty-three will be found a fascinating puzzle.
Half the lines are, of course, on the sides, and the others radiate from the centre. On one occasion Sir Hugh greatly perplexed his chief builder. He took this worthy man to the walls of the donjon keep and pointed to a window there.
I trow thou art but a sorry craftsman if thou canst not, forsooth, set such a window in a keep wall. It will be noticed that Sir Hugh ignores the thickness of the bars. When Sir Hugh's kinsman, Sir John de Collingham, came back from the Holy Land, he brought with him a flag bearing the sign of a crescent, as shown in the illustration.
It was noticed that De Fortibus spent much time in examining this crescent and comparing it with the cross borne by the Crusaders on their own banners. One day, in the presence of a goodly company, he made the following striking announcement: Truly it was shown to me in a dream that this crescent of the enemy may be exactly converted into the cross of our own banner.
Herein is a sign that bodes good for our wars in the Holy Land. Sir Hugh de Fortibus then explained that the crescent in one banner might be cut into pieces that would exactly form the perfect cross in the other. It is certainly rather curious; and I show how the conversion from crescent to cross may be made in ten [Pg 64] pieces, using every part of the crescent.
The flag was alike on both sides, so pieces may be turned over where required. A strange man was one day found loitering in the courtyard of the castle, and the retainers, noticing that his speech had a foreign accent, suspected him of being a spy.
So the fellow was brought before Sir Hugh, who could make nothing of him. He ordered the varlet to be removed and examined, in order to discover whether any secret letters were concealed about him. All they found was a piece of parchment securely suspended from the neck, bearing this mysterious inscription: To-day we know that Abracadabra was the supreme deity of the Assyrians, and this curious arrangement of the letters of the word was commonly worn in Europe as an amulet or charm against diseases.
But Sir Hugh had never heard of it, and, regarding the document rather seriously, he sent for a learned priest. Place your pencil on the A at the top and count in how many different ways you can trace out the word downwards, always passing from a letter to an adjoining one. It would often be interesting if we could trace back to their origin many of the best known puzzles. Some of them would be found to have been first propounded in very ancient times, and there can be very little doubt that while a certain number may have improved with age, others will have deteriorated and even lost their original point and bearing.
It is curious to find in the Solvamhall records our familiar friend the climbing snail puzzle, and it will be seen that in its modern form it has lost its original subtlety.
On the occasion of some great rejoicings at the Castle, Sir Hugh [Pg 66] was superintending the flying of flags and banners, when somebody pointed out that a wandering snail was climbing up the flagstaff. One wise old fellow said: Can the reader give the answer to this version of a puzzle that we all know so well?
It was inlaid with pieces of wood, and a strip of gold ten inches long by a quarter of an inch wide. When young men sued for the hand of Lady Isabel, Sir Hugh promised his consent to the one who would tell him the dimensions of the top of the box from these facts alone: Many young men failed, but one at length succeeded. The puzzle is not an easy one, but the dimensions of that strip of gold, combined with those other conditions, absolutely determine the size of the top of the casket.
Thus passed away the life of the jovial and greatly beloved Abbot of the old monastery of Riddlewell. The monks of Riddlewell Abbey were noted in their day for the quaint enigmas and puzzles that they were in the habit of propounding.
The Abbey was built in the fourteenth century, near a sacred spring known as the Red-hill Well. This became in the vernacular Reddlewell and Riddlewell, and under the Lord Abbot David the monks evidently tried to justify the latter form by the riddles they propounded so well.
The solving of puzzles became the favourite recreation, no matter whether they happened to be of a metaphysical, philosophical, mathematical, or mechanical kind. It grew into an absorbing passion with them, and as I have shown above, in the case of the Abbot this passion was strong even in death. It would seem that the words "puzzle," "problem," "enigma," etc.
They were accustomed to call every poser a "riddle," no matter whether it took the form of "Where was Moses when the light went out? On one of the walls in the refectory were inscribed [Pg 69] the words of Samson, "I will now put forth a riddle to you," to remind the brethren of what was expected of them, and the rule was that each monk in turn should propose some riddle weekly to the community, the others being always free to cap it with another if disposed to do so.
Abbot David was, undoubtedly, the puzzle genius of the monastery, and everybody naturally bowed to his decision. Only a few of the Abbey riddles have been preserved, and I propose to select those that seem most interesting. I shall try to make the conditions of the puzzles perfectly clear, so that the modern reader may fully understand them, and be amused in trying to find some of the solutions.
At the bottom of the Abbey meads was a small fish-pond where the monks used to spend many a contemplative hour with rod and line.
One day, when they had had very bad luck and only caught twelve fishes amongst them, Brother Jonathan suddenly declared [Pg 70] that as there was no sport that day he would put forth a riddle for their entertainment.
He thereupon took twelve fish baskets and placed them at equal distances round the pond, as shown in our illustration, with one fish in each basket. Start at any basket you like, and, always going in one direction round the pond, take up one fish, pass it over two other fishes, and place it in the next basket. Go on again; take up another single fish, and, having passed that also over two fishes, place it in a basket; and so continue your journey.
Six fishes only are to be removed, and when these have been placed, there should be two fishes in each of six baskets, and six baskets empty. Which of you merry wights will do this in such a manner that you shall go round the pond as few times as possible? I will explain to the reader that it does not matter whether the two fishes that are passed over are in one or two baskets, nor how many empty baskets you pass. And, as Brother Jonathan said, you must always go in one direction round the pond without any doubling back and end at the spot from which you set out.
One day, when the monks were seated at their repast, the Abbot announced that a messenger had that morning brought news that a number of pilgrims were on the road and would require their hospitality. There must be eleven persons sleeping on each side of the building, and twice as many on the upper floor as on the lower floor. Of course every room must be occupied, and you know my rule that not more than three persons may occupy the same room.
I give a plan of the two floors, from which it will be seen that the sixteen rooms are approached by a well staircase in the centre. After the monks had solved this little problem and arranged for [Pg 71] the accommodation, the pilgrims arrived, when it was found that they were three more in number than was at first stated. This necessitated a reconsideration of the question, but the wily monks succeeded in getting over the new difficulty without breaking the Abbot's rules.
The curious point of this puzzle is to discover the total number of pilgrims. It seems that it was Friar Andrew who first managed to "rede the riddle of the Tiled Hearth. The square hearth, where they burnt their Yule logs and round which they had such merry carousings, was floored with sixteen large ornamental tiles. When these became cracked and burnt with the heat of the great fire, it was decided to put down new tiles, which had to be selected from four different patterns the Cross, the Fleur-de-lys, the Lion, and the Star ; but plain tiles were also available.
The Abbot proposed that they should be laid as shown in our sketch, without any plain tiles at all; but Brother Richard broke in,—. Listen, then, to that which I shall put forth. Let these [Pg 72] sixteen tiles be so placed that no tile shall be in line with another of the same design"— he meant, of course, not in line horizontally, vertically, or diagonally —"and in such manner that as few plain tiles as possible be required.
All had used too many plain tiles. One evening, when seated at table, Brother Benjamin was called upon by the Abbot to give the riddle that was that day demanded of him. Mark me take a glass of sack from this bottle that contains a pint of wine and pour it into that jug which contains a pint of water. Now, I fill the glass with the mixture from the jug and pour it back into the bottle holding [Pg 73] the sack.
Pray tell me, have I taken more wine from the bottle than water from the jug? Or have I taken more water from the jug than wine from the bottle? I gather that the monks got nearer to a great quarrel over this little poser than had ever happened before. One brother so far forgot himself as to tell his neighbour that "more wine had got into his pate than wit came out of it," while another noisily insisted that it all depended on the shape of the glass and the age of the wine. But the Lord Abbot intervened, showed them what a simple question it really was, and restored good feeling all round.
Then Abbot David looked grave, and said that this incident brought to his mind the painful fact that John the Cellarer had been caught robbing the cask of best Malvoisie that was reserved for special occasions.
He ordered him to be brought in. What hast thou to say for thyself? There were a hundred pints in the cask at the start, and I have taken me a pint every day this month of June—it being to-day the thirtieth thereof—and if my Lord Abbot can tell me to a nicety how much good wine I have taken in all, let him punish me as he will. It is a curious fact that this is the only riddle in the old record that is not accompanied by its solution.
Is it possible that it proved too hard a nut for the monks? There is merely the note, "John suffered no punishment for his sad fault. On another occasion a certain knight, Sir Ralph de Bohun, was a guest of the monks at Riddlewell Abbey. Towards the close of a sumptuous repast he spoke as follows: A body of Crusaders went forth to fight the good cause, and such was their number that they were able to form themselves into a square.
But on the way a stranger took up arms and joined them, and they were then able to form exactly thirteen smaller squares. Pray tell me, merry monks, how many men went forth to battle?
In the first place there were men, who would make a square 18 by 18, and afterwards men would make 13 squares of 25 Crusaders each. But which of you can tell me how many men there would have been if, instead of 13, they had been able to form squares under exactly the like conditions?
Edmondsbury," said Father Peter on one occasion, "that many years ago they were so overrun with mice that the good abbot gave orders that all the cats from the country round should be obtained to exterminate the vermin. A record was kept, and at the end of the year it was found that every cat had killed an equal number of mice, and the total was exactly 1,, mice. How many cats do you suppose there were? They told me it was merely a question of the division of numbers, but I know not the answer to the riddle.
One Christmas the Abbot offered a prize of a large black jack mounted in silver, to be engraved with the name of the monk who should put forth the best new riddle. This tournament of wit was won by Brother Benedict, who, curiously enough, never before or [Pg 77] after gave out anything that did not excite the ridicule of his brethren. It was called the "Frogs' Ring. 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: 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.
But since it is thy wish, thou shalt have thy choice whether I kill thee or the executioner. 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 [Pg 79] 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.
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? 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 [Pg 80] 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?
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 [Pg 81] 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? I was now face to face with the castle moat, which was, indeed, very wide and very deep. 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 [Pg 82] 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?
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, [Pg 83] 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.
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