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But at one shop they had; and that was odder still. The obliging salesman looked up the exact date, and found that it was just a month ago. Roger described his friend, and the salesman at once agreed that Roger's friend and his own customer were one and the same.

"Good gracious, and now I come to think of it," Roger cried, " I actually believe I've got my friend's photograph on me at this very minute. Let me see!" He rummaged in his pockets, and to his great astonishment produced the photograph in question.

The salesman most obligingly proceeded to identify his customer without hesitation. He then went on, just as obligingly, to sell Roger the second - hand Hamilton No. 4 which that enthusiastic detective felt he had not the face to refuse to buy. Detecting, Roger was discovering, is for the person without official authority to back him, a singularly expensive business. But like Mrs. Fielder - Flemming, he did not grudge money spent in a good cause.

He went back to his rooms to tea. There was nothing more to be done except await the call from Moresby.

It came sooner than he expected. "Is that you, Mr. Sheringham? There are fourteen taxi - drivers here, littering up my office," said Moresby offensively. "They all took fares from Piccadilly Circus to the Strand, or vice versa, at your time. What do you want me to do with 'em?"

"Kindly keep them till I come, Chief Inspector," returned Roger with dignity, and grabbed his hat. He had not expected more than three at the most, but he was not going to let Moresby know that.

The interview with the fourteen was brief enough however. To each grinning man in turn (Roger deduced a little heavy humour on the part of Moresby before he arrived) Roger showed the photograph, taking some pains to hold it so that Moresby could not see it, and asked if he could recognise his fare.

Not a single one could. Moresby dismissed the men with a broad grin.

"That's a pity, Mr. Sheringham. Puts a bit of a spoke in the case you're trying to work up, no doubt?"

Roger smiled at him in a superior manner. "On the contrary, my dear Moresby, it just about clinches it."

"It what did you say?" asked Moresby, startled out of his grammar. "What are you up to, Mr. Sheringham, eh?"

"I thought you knew all that. Aren't we being sleuthed?"

"Well!" Moresby actually looked a shade out of countenance. "To tell you the truth, Mr. Sheringham, all your people seemed to be going so far off the lines that I called my men off; it didn't seem worth while keeping 'em on."

"Dear, dear," said Roger gently. "Fancy that. Well, it's a small world, isn't it?"

"So what have you been doing, Mr. Sheringham? You've no objection to telling me that, I suppose? "

"None in the least, Moresby. Your work for you. Does it interest you to know that I've found out who sent those chocolates to Sir Eustace?"

Moresby eyed him for a moment. "It certainly does, Mr. Sheringham. If you really have."

"Oh, I have, yes," said Roger very nonchalantly; even Mr. Bradley himself could not have spoken more so. "I'll give you a report on it as soon as I've got my evidence in order. - It was an interesting case," he added. And suppressed a yawn.

"Was it now, Mr. Sheringham?" said Moresby, in a choked voice.

"Oh, yes; in its way. But absurdly simple once one had grasped the really essential factor. Quite ridiculously so. I'll let you have that report some time. So long, then." And he strolled out.

One cannot conceal the fact that Roger had his annoying moments.

CHAPTER XIII

ROGER called on himself.

"Ladies and gentlemen, as the one responsible for this experiment, I think I can congratulate myself. The three members who have spoken so far have shown an ingenuity of observation and argument which I think could have been called forth by no other agency. Each was convinced before beginning to speak that he or she had solved the problem and could produce positive proof in support of such solution, and each, I think, is still entitled to say that his or her reading of the puzzle has not yet been definitely disproved.

"Even Sir Charles's choice of Lady Pennefather is perfectly arguable, in spite of the positive alibi that Miss Dammers is able to give to Lady Pennefather herself; Sir Charles is quite entitled to say that Lady Pennefather has an accomplice, and to adduce in support of that the rather dubious circumstances attending her stay in Paris.

"And in this connection I should like to take the opportunity of retracting what I said to Bradley last night. I said that I knew definitely that the woman he had in mind could not have committed the murder. That was a mis-statement. I didn't know definitely at all. I found the idea, from what I personally know of her, to be quite incredible.

"Moreover," said Roger bravely, "I have some reason to suspect the origin of her interest in criminology, and I'm pretty sure it's quite a different one from that postulated by Bradley. What I should have said was, that her guilt of this crime was a psychological impossibility. But so far as facts go, one can't prove psychological impossibilities. Bradley is still perfectly entitled to believe her the criminal. And in any case she must certainly remain on the list of suspects."

"I agree with you, Sheringham, you know, about the psychological impossibility," remarked Mr. Bradley. "I said as much. The trouble is that I consider I proved the case against her."

"But you proved the case against yourself too," pointed out Mrs. Fielder - Flemming sweetly.

"Oh, yes; but that doesn't worry me with its inconsistency. That involves no psychological impossibility, you see."

"No," said Mrs. Fielder - Flemming. "Perhaps not."

"Psychological impossibility!" contributed Sir Charles robustly. "Oh, you novelists. You're all so tied up with Freud nowadays that you've lost sight of human nature altogether. When I was a young man nobody talked about psychological impossibilities. And why? Because we knew very well that there's no such thing."

"In other words, the most improbable person may, in certain circumstances, do the most unlikely things," amplified Mrs. Fielder - Flemming. "Well, I may be old - fashioned, but I'm inclined to agree with that."

"Constance Kent," led Sir Charles.

"Lizzie Borden," Mrs. Fielder - Flemming covered.

"The entire Adelaide Bartlett case," Sir Charles brought out the ace of trumps.

Mrs. Fielder - Flemming gathered the cards up into a neat pack. "In my opinion, people who talk of psychological impossibilities are treating their subjects as characters in one of their own novels - they're infusing a certain percentage of their own mental make - up into them and consequently never see clearly that what they think may be the impossible for themselves may quite well be the possible (however improbable) in somebody else."

"Then there is something to be said after all for the detective story merchant's axiom of the most unlikely person," murmured Mr. Bradley. "Good!"

"Shall we hear what Mr. Sheringham's got to say about the case now? " suggested Miss Dammers.

Roger took the hint. "I was going on to say how interestingly the experiment had turned out, too, in that the three people who have already spoken happen each to have hit on a different person for the criminal. I, by the way, am going to suggest another, or even if Miss Dammers and Mr. Chitterwick each agree with one of us, that gives us four entirely different possibilities. I don't mind confessing that I'd hoped something like that would happen, though I hardly looked for such an excellent result.

"Still, as Bradley has pointed out in his remarks about closed and open murders, the possibilities in this case really are almost infinite. That, of course, makes it so much more interesting from our point of view. For instance, I began my own investigations from the point of view of Sir Eustace's private life. It was there, I felt convinced, that the clue to the murder was to be found. Just as Bradley did. And like him, I thought that this clue would be in the form of a discarded mistress; jealousy or revenge, I was sure, would turn out to be the mainspring of the crime. Lastly, like him I was convinced from the very first glance at the business that the crime was the work of a woman.