Part 3

‘The sacrifice that I made to luck on the occasion of our last competition,’ said Major Byles, ‘has brought me success at last. I claim to be a winner, and await your decision with confidence. It happened that two of my friends both wanted a furnished house at Brightgate for the winter, and did not want the bother of going down to make their selection. I saw my chance at once. I might never have thought of it, but I didn’t miss it when it was shoved at me. I said at once that I was thinking of running down to Brightgate for a day or two, and that it always interested me to look over houses. They told me their requirements and let me take on the job for them.

‘The house-agent at Brightgate had only six houses on his books that were at all suitable. He gave me orders to view, and I started business at eight one morning. I started badly.

‘At the first house a proud but pretty parlour-maid told me that it was not usual to show furnished houses at that hour, but I could call again at eleven. At the second house there was only a caretaker. That left me with, so to speak, four cartridges and three birds to kill. I hurried on to the third house, which was half a mile away. By a bit of luck I met the owner on the doorstep, and told him my alleged business.

‘“You’re very early,” he said. “Why, we haven’t had breakfast yet.”

‘“No more have I,” I said. “But last year I lost a good house through being too late, and I thought I wouldn’t make the same mistake again.”

‘He was a genial old chap. He said the best thing I could do was to come in and breakfast with him, and by the time I had finished the servants would have got the bedrooms tidied up. I did my best to accept with decent hesitation.

‘At lunch-time I tried the fourth house on my list and struck another caretaker. I couldn’t afford another miss. I got lunch at the fifth house, but I had to be no end complimentary before I could get them up to the point. In fact, it wasn’t till I told the woman that her pimply-faced son was a fine upstanding young fellow that she decided to order the extra chop.

‘But at the sixth house I had no trouble about dinner. The owner turned out to be a friend of a friend of mine. He fetched up a bottle of the ’87 in my honour and insisted on my stopping the night.

‘They were all _pukka_ meals, and all the conditions were observed. Am I a winner, Mr Chairman?’

‘Certainly. Does anybody else claim to be a winner?’

‘I do,’ said Dr Alden. ‘The day before yesterday a doctor rang me up and asked me to see a patient of his—a woman with a wealthy, devoted, and very nervous husband. That was at eight in the morning. My car happened to be at the door, and it suited me to go right away. I saw the patient, was able to reassure the husband, and had breakfast with him. Later in the morning a man was introduced to me who was interested in old glass and had heard of me as a collector. He was very keen that I should lunch with him and see what he had got. He was a pleasant chap and I accepted. When I got back, a doctor, quite an old pal of mine, said that he was going to take me to dine that night with a man I had never seen before. It seemed that the stranger had staying with him for one night a French specialist in my own line. The Frenchman was anxious to meet me, and his host was anxious to please him. So he had tried to arrange it through a mutual friend. I was myself keen to meet that Frenchman, and so he succeeded.

‘Of course, I didn’t arrange all this—couldn’t have arranged it. As a matter of fact, I had never intended to compete this time. But destiny decided to take a hand in this competition. I claim to be a winner.’

‘An interesting point,’ said the chairman. ‘Can a man be said to win who has never competed? I shall decide in Dr Alden’s favour. Leonard says nothing of intention. He only demands certain facts. And these facts the doctor by an amazing stroke of luck has been able to provide. The prize of one hundred and ten pounds will be divided equally between him and Major Byles, unless there is any further claim.’

No further claim was forthcoming. The chairman then announced that Mr Matthews would preside at the next meeting, and read out the problem set for the following month, called ‘The Win-and-Lose Problem,’ and there was a general feeling that it would take some doing.

No. IV.

The Win-and-Lose Problem

At the forty-sixth meeting of the Problem Club, the waiters having left the room, Mr Matthews, smiling and rubicund, took his place as chairman. He finished his glass of an old and veritable cognac, lit with care and a cedar-wood spill a cigar that can only be obtained by the favour of the planter, and read out the terms of the Win-and-Lose Problem.

‘It is required to win an even bet of one pound, resulting in a net loss of one pound to the winner; and to lose an even bet of one pound resulting in a net gain of one pound to the loser. No competitor is to make more than two bets.’

‘Well, gentlemen,’ said Mr Matthews, ‘I’m supposed to make one or two preliminary observations. Now here’s a thing that strikes me. You may remember that when we tackled the Kiss Problem, our reverend friend Mr Cunliffe said that it revealed the artful Leonard as an apostle of morality. Of course, the padre took the jack-pot on that occasion, and so he may have been prejudiced, but it looks to me now as if he may have been right. See for yourselves. You’ve got to win a bet and lose money by it, and then you’ve got to lose a bet and make money by it, and at the end of it you’re left just where you were when you started. There’s not much deadly fascination and excitement about that—why, it’s enough to make you lose your taste for gambling.

‘Yes, and there’s one more point. I noticed a good deal of preoccupation at dinner to-night. Very few of you seemed to be putting your heart into the work, and I believe I was the only man who had the _vol-au-vent_ brought back to him for further reference. Great mistake that of yours. Some of you tried to work out sums on the back of your menus. I detected Major Byles, with corrugated brows, in the act of making pencil calculations on the tablecloth. Yes, there’s not a doubt that Leonard has given you a worrying time, and some of you were wrestling with it right up to the last moment. It won’t surprise me if there’s not a winner among the whole lot of you. However, we’ll begin with a likely chance. You, Sir Charles, have got a reputation as a learned man; can I ask the secretary to draw a cheque in your favour?’

‘I’d be sorry to stop you,’ said Sir Charles, ‘but I’m afraid I can’t claim it. Archæology don’t help with arithmetic. As an eminent classical scholar once observed, I’ve not got the low cunning that makes a mathematician. The only thing I could think of was to insure the chances of each bet appropriately, but it seemed to me that you would regard such insurance as being in itself a bet.’

‘I certainly should. You don’t change a thing by changing its name. You are limited to the two bets, and I shall not allow four even if you call two of them insurance. Come now, Jimmy, have you profited sufficiently by your racing experiences to have won the prize to-night?’

‘Profited by my racing experiences?’ said the Hon. James Feldane wearily. ‘If you’d go and talk to the bank that has charge of my overdraft you wouldn’t use words like those. But backing horses, though it’s a mug’s game, is, at any rate, easy. There are too many complications in Leonard’s fancy work for a simple child of Nature like myself. I can’t engineer a two-cylinder gamble with a double back-jump actuated by the cam-shaft. The only man I know who could face it without mental overstrain is my bookmaker. He’s a wonder. He’d give you fifteen different ways of perforating this problem inside a minute. No juggle with figures can beat him. I don’t know if you’d call it a talent or a disease, but I’ve not got it. As a competitor, I’ve failed, but I don’t mind admitting that I’ve made a little actual money out of the competition.’ Jimmy smiled reminiscently.

‘May we have the details?’ asked the chairman.

‘I’d sooner you got them from Hesseltine.’

The chairman called upon Mr Hesseltine.

‘I don’t wonder,’ said that young man, ‘that Jimmy don’t like to tell you. If I’d stolen money from a crossing-sweeper in St James’s Street I shouldn’t be proud of it myself. The silly ass thinks he’s scored off me, but as I was out to lose a quid anyhow——’

‘May we have the actual facts?’ suggested the chairman.

‘Certainly. I was thinking about this problem and I got a sudden brain-wave. I saw how to do the first half—to win a bet of a pound that would leave me one pound down when I’d won it. Well, I happened to be going up St James’s Street with Jimmy later that morning, and by way of leading up to it I asked him what he generally gave to a street-beggar. “Nix,” he said. “What do you?” So I told him that I generally gave a sovereign. He told me in his coarse sort of way that he didn’t believe it. That was what I had expected. “All right,” I said, “I’ll bet you a pound that I give two golden sovereigns to the next beggar or crossing-sweeper I come across.” He thought about it and then said: “I’ll take that, and to guard against accidents I’ll be the next beggar. Give me a little assistance, kind sir?”

‘Of course, in that way he put himself on velvet. Whether I decided to win my bet or to lose it, Jimmy had to make one sovereign out of me. Didn’t affect me at all, for according to Leonard I’d got to win my bet and lose a pound by it, which I did. The only person hit was the crossing-sweeper up the street, who would otherwise have made two quid. Of course, what I ought to have done was to have handed Jimmy over to the police for begging—wish I’d thought of it.

‘Well, I negotiated the first half of the problem, but the second half beat me. I’m inclined to think the sting of the beast is in its tail. It takes two people to make a bet. I’m not a poet or any sort of imaginative chap, but I could think of a bet which for a dead certainty it would pay me to lose. I couldn’t think of anybody, even including that rotter Jimmy, who would be fool enough to take it. You must try somebody else, Mr Chairman.’

‘Major Byles?’ the chairman suggested.

‘As a head-waiter,’ said the Major, ‘I’ve got nothing against Leonard. As a setter of problems he’s given general satisfaction, but this time I should like to back my bill to the effect that he has mixed up too much arithmetic with the sport. I’ve spent a month on this win-and-lose business, all the time with the feeling that a boy fresh from school would work out the whole thing on the back of an envelope in ten minutes, and I’ve done nothing. I spent the first fortnight at home, and at the end of it I had contracted insomnia, headache, and what you might call pardonable irritability. At the end of that time my wife said that of course she had noticed the change, and that I seemed to be doing sums all day, and that if we were ruined I had better say so and she would face it bravely. I reassured her and came to town on important business. I used tons of the club notepaper for my calculations, put an undue strain on the club wastepaper-baskets, quarrelled with two of my best friends, was sarcastic in addressing club servants, and am expecting a letter from the committee to ask for my resignation. The amazing thing is that all the time I have been on the very point of getting the solution. In my opinion it’s the most horribly worrying thing that Leonard has ever given us.’

‘Well,’ said the chairman, ‘artists are not generally supposed to be particularly strong at arithmetic, but I’ll ask Mr Wildersley what he’s done about it.’

‘Can’t say I agree with the Major,’ said Wildersley. ‘I call it a jolly easy problem, and I claim to be a winner. It didn’t take me any time to think of it, either. I got a man into my studio, to see alleged works of art, and I said to him that I would bet him a pound I would give him two pounds. He took me. “You’ve lost,” I said. “Pay up, and then I’ll pay up.” He handed me a sovereign and I handed him two pounds of potatoes in a paper bag. So I’d won a pound in money and lost two pounds in potatoes. If you win one pound and lose two, that makes a net loss of one pound on the transaction, and so I’d done the first half of the problem.

‘The chap seemed to be grumbling rather. “What’s the matter with you?” I said. “The green-grocer told me that they were the kind he eats himself, and that he could guarantee them.”

‘“I don’t want the beastly potatoes,” he said. “The whole thing’s a dirty swindle.” I thought he’d say that. So I told him that it was no swindle and I would be quite willing to take the same bet myself. He jumped at it, but to make sure he said he would bet me a sovereign he would give me two pounds. I took him, lost, paid the sovereign, and got back my two pounds of potatoes. That finished the second half of the problem. I’d lost a bet of one pound, and had made two pounds, giving a net gain of one pound. Naturally he wanted to know what I had done it for, and I said it was to stop him from trying to talk about art—the chap’s a critic.’

Mr Matthews took two minutes and a brandy-and-soda before giving his decision as follows:—

‘Ingenious, but it won’t do. Mr Wildersley professes to have subtracted money to the value of a sovereign from two pounds by weight of potatoes, and to have got a result of one pound. Of what did that pound consist? Even after dinner we can’t have mental confusion of this kind. The claim is disallowed.’

Mr Harding Pope, M.P., made an uninteresting confession of failure, and the chairman then called upon Mr Quillian, K.C., who was acting as secretary for the evening.

Mr Quillian removed his pince-nez and glanced round the room with that look of amiable superiority that some people found irritating.

‘I claim to have won this fairly simple competition,’ he said. ‘Of course, it has a psychological as well as an arithmetical side; the bets have to be actually made and not merely worked out on paper. I made my plan one afternoon, and then went over to my club to see if I could find my friend Blenkinsop. He is generally at the club at that hour, and I felt sure that he would accept the two bets that I had to propose.

‘Well, as it happened, Blenkinsop was not at the club, but I found Mr Pusely-Smythe alone in the smaller reading-room. I’ve had to submit to a good deal of chaff—not particularly amusing—from Pusely-Smythe, and by way of return it seemed appropriate that he should help me to win our one hundred and ten pound prize. Also, if he will forgive me for saying so, he has just the commonplace shrewdness that I required in my victim.

‘After a little preliminary conversation, I produced my sovereign-case. I told him that there was a certain sum of money in gold in that case, and that I was willing to bet him a sovereign I would make him a present of it. He said, as I knew he would, that this meant that the sum of money in the case was half a sovereign, and that in consequence he would lose ten shillings on the transaction if he took the bet.

‘“Yes,” I said, “there is that possibility, but I am willing to protect you against it by a second bet. We will agree that the loser by the first transaction shall have the option to give the other man double what he has lost for double the sum now in my sovereign-case. And I will bet you a sovereign that he will not exercise that option. You see how it works out. If the sum in my sovereign-case is half a sovereign, as you suppose, you will lose ten shillings on the first transaction, but you will win a sovereign on the second transaction by exercising an option to exchange twenty shillings for twenty shillings.”

‘Without taking the time to think, he accepted both bets. I then opened my sovereign-case and showed him that it contained two pounds. I gave them to him, and as by so doing I had won my bet he gave me one of them back again. Kindly observe that I had now solved the first part of Leonard’s problem. I had won an even bet of one pound the net result of which was that I had lost a pound. Having made myself the loser on the first transaction, I now had the option to exchange twice my loss against twice the sum that had been in the sovereign-case—that is, to exchange two pounds for four pounds. I had bet that the loser would not exercise this option. I lost the bet and exercised the option. Thus, I lost an even bet of one pound with the net result that I made one pound. This settles the second half of the problem. I await, sir, with confidence, your decision in my favour.’

Mr Matthews referred once more to the terms of the problem. ‘Yes,’ he said, ‘it seems to me that you have met all Leonard’s requirements. Very smart bit of work, in my opinion. You take the club cheque for one hundred and ten pounds, unless, of course, some claim to share it with you is substantiated. Is there any such claim?’

‘Naturally, there’s mine,’ said Pusely-Smythe, with his deceptive air of melancholy.

‘Yours? How did you do it?’

‘My learned friend has just been telling you. I was going away for a brief and well-earned holiday, and I had decided to give the competition a miss this time. As I was sitting in the club, studying a guide-book, in came Quillian looking like a thimble-rigger who has just set up his little plush-covered table. He offered me his first bet. I put it aside. He offered the second, and he says I didn’t take time to think. Thought with me does not take the prolonged period of gestation that it does in the case of the nobler animals, such as K.C.’s. I thought two thoughts. The first was that Quillian was out after this competition. The second was that when two men gamble together what one wins the other loses and vice versa. That was enough. I took him. He won the first bet but lost a pound by it. It follows that I lost the first bet but won a pound by it. Similarly, when he lost the bet but won a pound I won the bet and lost a pound. It’s all very simple and elementary. I hope he’s going to make a victim of me again soon. This time without any effort on my part he has shoved fifty-five pounds at me. I’ve only had to take it. And I don’t care whether it was benevolence or mental short-sightedness—I’m going to thank him just the same.’

‘The claim’s allowed, of course,’ said the chairman. ‘The thing that makes me mad is that I didn’t see it myself until you pointed it out. It’s obvious. It simply shrieks at you. My mind must be going.’

‘The menu that you devised for our dinner to-night, sir,’ said Pusely-Smythe, ‘was sufficient proof of the contrary. Those that study the recondite must sometimes find the obvious out of their focus.’

‘Thank you,’ said Mr Matthews. ‘I’ll learn the last sentence by heart—it’ll make a ripping excuse next time I do a dam’ silly thing.’

Cheques were drawn for Quillian and Pusely-Smythe, and the chairman then opened the envelope containing the problem that Leonard had set for the following month. It was entitled ‘The Handkerchief Problem,’ and on the face of it scarcely supported the theory that the ingenious Leonard was a Great Moral Teacher. The Hon. James Feldane was reminded that it would be his duty to preside on the next occasion and to adjudicate on this problem, which was as follows: ‘It is required to steal as many handkerchiefs as possible from a member or members of the Problem Club. Violence may not be used and thefts detected in the act will not score. Restitution will be made of the stolen handkerchiefs within twenty-four hours of the adjudication, but felonious intent is to be presumed in every case.’

‘Rotten luck,’ said Feldane, to his friend, Hesseltine. ‘I should have enjoyed working on this problem. It appeals to my natural instincts. I should probably have won it, and in that case might have given one or two of them something on account. And so this has to be the occasion when I’m shut out of the competition and have to act as chairman.’

‘Yes,’ said Hesseltine. ‘Nobody’s so sure of himself as the non-starter.’

No. V.

The Handkerchief Problem

At the forty-seventh meeting of the Problem Club, the chair was taken by the youngest member, the Hon. James Feldane. That weary young gentleman having provided himself with a double portion of green Chartreuse, for the purpose, as he said, of supporting the dignity of the position, opened his adjudication a little informally.

‘Let’s get started,’ he said. ‘The first job is to read out the particular teaser with which the wily Leonard has been worrying you poor old things during the past month. Here goes.’

The terms of the Handkerchief Problem were then read out. They were as follows: ‘It is required to steal as many handkerchiefs as possible from a member or members of the Problem Club. Violence may not be used and thefts detected in the act will not score. Restitution will be made of the stolen handkerchiefs within twenty-four hours of the adjudication, but felonious intent is to be presumed in every case.’

‘I wish I could have been a competitor this time,’ the chairman continued, ‘instead of being stuck up here to give the momentous decision. I should have had some sport, and handkerchief-sneaking falls nicely within my line of intellect; I might have scooped the prize. But as I’m debarred from scoring off you, I’ve taken jolly good care that none of you should score off me. For the past month every handkerchief I’ve used has been attached to the interior of the pocket by a steel chain and swivel, and those not in use have been locked away in a safe. My valet thinks I’ve gone off my head, of course, but then he’d have been bound to have thought that sooner or later, anyhow. The great point is that not one of you low pickpockets has been able to get a handkerchief out of me. We’ll now pursue the inquiry. Hesseltine, are you guilty or not guilty?’

‘Guilty, m’lord,’ said young Hesseltine cheerfully. ‘I may not be winner, but I think it would be safe to back me for a place. I struck early. At our last meeting, as soon as this problem was announced, I slipped stealthily and unobserved from the room. I had rightly concluded that there would be no attendant in the cloak-room at that hour. If there had been I should have sent him away to get me a box of matches. From the overcoats of members I secured a nice little haul of nine handkerchiefs. One of them, a silk bandanna, the property of Major Byles, was big enough for two, and ought to count as two.’

‘Might count two on a division, as they say at the elections,’ said the chairman. ‘But in the undivided state it counts one. Anything further to say?’

‘That was my only _coup_. The only thing to add is that one of the nine belonged to a gentleman who did not start keeping them in the safe quite soon enough.’

‘All right,’ said Jimmy. ‘Speaking entirely _sotto voce_ and _ex officio_, I’ll be even with you for that one of these days. Meanwhile, Mr Matthews, it will be your painful duty as secretary to give that thief a score of nine.’

The chairman then called upon Mr Quillian, K.C., whose story was connected with the story of Mr Pusely-Smythe. In both cases it was a story of failure. Both men had hit on precisely the same idea.

Quillian called on Pusely-Smythe at a time when he knew he would be out, but would be expected back shortly. He, as he anticipated, was recognised by the servant and asked if he would wait. During the period of waiting Quillian made a swift and silent excursion to Pusely-Smythe’s bedroom with a view to abstracting his available store of handkerchiefs. But the chairman was not the only member who had taken the precaution of keeping his handkerchiefs in an unlikely place. Not one solitary handkerchief could Quillian find. And while he was thus engaged Pusely-Smythe had been calling on Quillian with similar intentions, similar practice, and a similar result.