D A C 123 567 9 10 II 12 13 14 15 F FIG. 13. B paragraphs. Three of the four blocks meant for squares 1, 2, 3, 4, must be either in the rectangle] C B or in the rectangle D B. In either case we can bring them (directly in one case, by rotation around D B in the other) to the squares 4, 8, 12. If the remaining block is in the oblong C F, we E get the four into right order, down the righthand column of the oblong C B by the last case, and rotate to the required squares 1 2, 3, 4. If the fourth block is in one of the squares 1, 5, 9, rotate the blocks in 11, 15 (bringing the one in 15 to corner B), and then the four blocks lie in the oblong A E, and can be brought to the squares 1, 2, 3, 4, as in last case. Lastly, if the fourth block is at 13, push down the blocks in 4, 8, 12, rotate those in 7, 3, bringing the one in 3 to corner square 4, and then the four blocks are in the oblong D B, and can be brought into the lowest row in the required order, as in the last case, and thence rotated to the squares 1, 2, 3, 4. After this, the rest of the square, namely, the oblong D B, can be arranged, as shown in the last case, so that all the blocks, except those in the squares 14, 15, are in assigned positions. I might here go on to show that in any square or oblong whatever, no matter how great the number of blocks in the length and breadth, all except the two can be brought into any assigned order. To do this, all that would be necessary would be to show that, if in an oblong or square of given numbers of blocks in length and breadth the blocks can so be arranged, they can also be arranged in an oblong or square having one more row added either to its length or breadth. For then, having already shown that we can so arrange an oblong of two by three, an oblong of two by four, a square of three by three, an oblong of three by four, and a square of four by four, it follows that we can similarly arrange an oblong of three by five and of four by five, a square of five by five, and so on, without limit. But I leave this as an exercise for the reader, noting only that the method is precisely similar to that by which the last case above dealt with was obtained from the last but one, that from the preceding, and so forth. In a paper which appeared in the Australasian for August 21, 1880, I have proved the above relations, and also the general case, in another way, not quite so simple but more concise; showing that from any given position a certain number of positions must always be obtainable, and that number being (with the given position) exactly one-half of the total number of possible arrangements, must include all the cases of its own kind, that is, either winning or losing, as the case may be. I have there also established the following rules for distinguishing winning from losing positions in an oblong or rectangle of any number of squares in the length and breadth. First, if the number both of horizontal and vertical rows be even (as in the Fifteen Puzzle), the won position, in which the blocks succeed each other in numerical sequence, following the lines as in reading, and leaving the last square vacant, can be obtained from any position in which the "total discrepancy" and the number of the partly vacant square are either both even or both odd; but if the "total discrepancy" is even and the number of the partly vacant line odd, or vice versa, the won position cannot be obtained. Secondly, if the number of horizontal rows be odd, and the num ber of vertical rows even, then the won position can be obtained if the "total discrepancy " is even and the number of the partly vacant line odd, or vice versa. But if the "total discrepancy" and the number of the incomplete line are either both odd or both even, the won position cannot be obtained. Thirdly and Fourthly. If the number of vertical rows be odd, then, whether the number of horizontal lines be (iii) even or (iv) odd, the won position can be obtained if the "total discrepancy" is even, and cannot be obtained if the "total discrepancy" is odd. These four laws include all possible cases. Let me add, in conclusion, that the total number of possible arrangements in a square of ten blocks in the side is so great, that if we imagine each case represented by a tiny globe one millionth of an inch in diameter, and these globes gathered in the form of a great sphere, the extent of that sphere would be greater than that of the entire region of space over which the mightiest telescope yet made by man extends his survey, though, from the remotest star reached by such a telescope, light, with its stupendous velocity of 187,000 miles a second, takes thousands of years in reaching this earth. It may be noticed, in conclusion, that the above study of the ways of solving the puzzle for six-block and eight-block rectangles will be found to indicate the proper way of dealing with the only cases of difficulty which ever arise in dealing with the Fifteen Puzzle. I wrote the whole of this paper, for instance, without having before me any actual set of blocks, simply drawing mental pictures of the various cases before writing the paragraphs respectively relating to them. Yet, on the first trial with the actual puzzle, I found that four or five minutes sufficed to resolve any position into the final (won or lost) position of its own kind; and after half-an-hour's practice (based on the principles above explained) I found the solutions averaged only two minutes. RICHARD A. PROCTOR. THE DISCROWNED FINGO. INGOISM is not dead, and it would not be prudent to assume even that it is sleeping. It is a passion too deeply implanted in the human breast to afford reasonable hope of its final eradication. It is one of the touches of nature that make the whole world kin. We call it Jingoism in England, in France it is called Chauvinism, and in the United States Bunkum. Seen across the Atlantic, or even over the narrower seas that divide us from France, we laugh at it, and thank heaven we are not as other men are-as this poor Yankee with his bird o' freedom, or as these French colonels with their baggy red trousers and their blustering demand to be led in flat-bottomed boats to the shores of England. But when the time comes we succumb to infection ourselves, and are as foolish as any of our kin across the sea. The seeds of disease are sown in our constitution, and in due course we break forth into a sort of delirious idiocy in which we see visions of England maintaining her "ascendency in the councils of Europe," and dream dreams of what noble fellows we are personally, and what abject cravens are those who will not shriek with us or march to and fro defiant at beat of circus drum. It is an essential part of Jingoism that its warlike spirit should never carry it further than this theatrical marching and countermarching, and that, since the drum is the proper accompaniment of its martial ebullition, the particular instrument selected should be the gaily painted cylinder of the circus. The attitude of the Jingo in view of actual hostilities is set forth with charming frankness in the famous couplet of his battle-song : We don't want to fight, but by Jingo if we do, We've got the ships, we've got the men, we've got the money too. Observe, "we don't want to fight." On the contrary, we prefer to stop at home and bray. If it comes to the worst, and there follows, in consequence of our shrieking, what Mr. Mantalini (who, if he had lived in the year 1877, would certainly have been a Jingo) calls "demnition hard knocks," we don't mind paying a small share of the cost of war. We have got the men, fellows of no particular account out of their home circles, who will go and be shot for a shilling a day. We have got the ships, which would make short work of Moscow and effectively blockade Siberia. Finally, we have got the money-chiefly other people's. What the country did not have was Jingo himself, body and bones, to be placed in the forefront of the battle, and there finally and exceptionally justify the reason of his existence. Jingo, in those still recent days, had a very distinct idea of the subdivision of labour. He would stop at home and shout, rattling his beer-glass in Music Hall or his wine-glass at Guildhall. Somebody else should go and fight, and so the ascendency of England would be maintained in the councils of Europe. The Jingo is the aggregation of the bully. An individual may be a bully, but in order to create Jingoism there must be a crowd. To this extent a bully is a more respectable individual than a Jingo. Equally with the Jingo, a bully need not want to fight. But his profession made him constantly prone to accidents that sometimes led to his having to strip and give battle. In Mr. Pepys' diary there is a narrative of an encounter between two bullies, which shows how bullydom can sometimes rise to the heights of heroism. It happened one sultry night in July 1667. "Two young bloods, Sir Henry Bellasses and Tom Porter, having dined, were conversing; Sir Henry Bellasses talking in a loud voice. Some of the company standing by said, 'What, are they quarrelling that they talk so high?' Sir Henry Bellasses, hearing it, said, 'No, I would have you know I never quarrel but I strike; that is a rule of mine.' 'How,' says Tom Porter, 'strike! I would I could see the man in England who durst give me a blow.' With that Sir Henry Bellasses did give him a box on the ear, and so they were going to fight there, but were hindered. However, they fought later the same night, and Sir Henry Bellasses was wounded so much that it is feared he will die, and finding himself severely wounded he called to Tom Porter and kissed him, and bade him shift for himself: for, says he, Tom, thou hast hurt me; but I will make shift to stand upon my legs till thou mayest withdraw, and the world not take notice of you: for I would not have thee troubled for what thou hast done.'" This is a speech which glorifies much swashbucklery, and makes us think tenderly of this roysterer of the Stuart time. Sir Henry Bellasses was a bully, and would doubtless on occasion prove himself a Jingo. But he would not have sung "We don't want to fight," or indulged in cheap heroics blustering about his ships, and shaking his money-bags at the head of the surprised enemy. Sir Henry Bellasses was a member of Parliament in the reign of Charles II., and doubtless from his place below the gangway blustered a good deal against the Dutch. If he had happened to have been born two hundred years later, he would certainly have obtained a seat in the Parliament the return of which marked the establishment of those great principles of which Mr. Disraeli was the embodiment. Whether he would have felt quite at home in the place, and whether he would have thoroughly enjoyed the association with his fellow Jingoes, may be questioned. The natural reluctance which the Jingo has for fighting when he meets on equal terms with the adversary does not influence him when he finds the opportunity of doing a little safe blustering. In the last Parliament the Jingoes had it all their own way, and noisily lorded it over the minority. It is doubtful whether a man, the chivalry of whose nature impelled him, whilst his life-blood was ebbing by a wound that three days later proved fatal, to make shift to stand in order that his adversary might escape, would have approved the general conduct of the Jingoes at this period. Certainly he would not have taken part in the famous, or perhaps infamous, scene which happened on the night when Mr. Gladstone and some others, having exercised their right of private judgment and voted against the views of the majority, were waylaid in the corridor and hooted by hon. gentlemen, much after the fashion that the dog which inevitably turns up on the Derby Day is chevied by the crowd. I am inclined to think that Sir H. Bellasses, whatever might have been his opinion of the political views of Mr. Gladstone, would have turned upon the well-dressed mob and beaten them off. Jingo to-day is discrowned and set in the dust, comforted only by the sure and certain hope of resurrection. One result of the general election of last spring was considerably to reduce his numerical proportions. He found himself not only dispirited but decimated. The dunghill on which he used to crow, one of a famous company, is now a lonely waste, and the sound of his own voice frightens him. In his best days he was not highly gifted with eloquence. But the theme at his command was one which easily lent itself to speech-making. The talk was tall and filled much space. The colours were bright green and scarlet, which caught the eye from remote distances. A man was not under the necessity of being logical or sensible or even truthful. He had to talk blatant nonsense, the more blatant the better, the more nonsensical the more successful. Ignorance and evil passions were at the bottom of the whole business, and whilst one could not be alarmed by discrepan |