Vance slept late the following morning and spent the hour or so before lunch checking a catalog of ceramics which were to be auctioned next day at the Anderson Galleries. At one o'clock we entered the Stuyvesant Club and joined Markham in the grill.

    "The lunch is on you, old thing," said Vance. "But I'll make it easy. All I want is a rasher of English bacon, a cup of coffee, and a croissant."

    Markham gave him a mocking smile.

    "I don't wonder you're economizing after your bad luck of last night."

    Vance's eyebrows went up. "I rather fancied my luck was most extr'ordin'ry."

    "You held four of a kind twice and lost both hands."

    "But, y' see," blandly confessed Vance, "I happened to know both times exactly what cards my opponents held."

    Markham stared at him in amazement.

    "Quite so," Vance assured him. "I had arranged before the game, d' ye see, to have those particular hands dealt." He smiled benignly. "I can't tell you, old chap, how I admire your delicacy in not referring to my rather unique guest, Mr. Allen, whom I had the bad taste to introduce so unceremoniously into your party. I owe you an explanation and an apology. Mr. Allen is not what one would call a charming companion. He is deficient in the patrician elegancies, and his display of jewelry was a bit vulgar—though I infinitely preferred his diamond studs to his piebald tie. But Mr. Allen has his points—decidedly he has his points. He ranks with Andy Blakely, Canfield, and Honest John Kelly as an indoor soldier of fortune. In fact, our Mr. Allen is none other than Doc Wiley Allen, of fragrant memory."

    "Doc Allen! Not the notorious old crook who ran the Eldorado Club?"

    "The same. And, incidentally, one of the cleverest card manipulators in a once lucrative but shady profession."

    "You mean this fellow Allen stacked the cards last night?" Markham was indignant.

    "Only for the two hands you mentioned. Allen, if you happen to remember, was the dealer both times. I, who purposely sat on his right, was careful to cut the cards in accordance with his instructions. And you really must admit that no stricture can possibly attach to my deception, inasmuch as the only beneficiaries of Allen's manipulations were Cleaver and Spotswoode. Although Allen did deal me four of a kind on each occasion, I lost heavily both times."

    Markham regarded Vance for a moment in puzzled silence and then laughed good-naturedly. "You appear to have been in a philanthropic mood last night. You practically gave Mannix a thousand dollars by permitting him to double the stakes on each draw. A rather quixotic procedure, I should say."

    "It all depends on one's point of view, don't y' know. Despite my financial losses—which, by the bye, I have every intention of charging up to your office budget—the game was most successful. . . . Y' see, I attained the main object of my evening's entertainment."

    "Oh, I remember!" said Markham vaguely, as if the matter, being of slight importance, had for the moment eluded his memory. "I believe you were going to ascertain who murdered the Odell girl."

    "Amazin' memory! . . . Yes, I let fall the hint that I might be able to clarify the situation today."

    "And whom am I to arrest?"

    Vance took a drink of coffee and slowly lit a cigarette.

    "I'm quite convinced, y' know, that you won't believe me," he returned in an even, matter-of-fact voice. "But it was Spotswoode who killed the girl."

    "You don't tell me!" Markham spoke with undisguised irony. "So it was Spotswoode! My dear Vance, you positively bowl me over. I would telephone Heath at once to polish up his handcuffs, but, unfortunately, miracles—such as strangling persons from across town—are not recognized possibilities in this day and age. . . . Do let me order you another croissant."

    Vance extended his hands in a theatrical gesture of exasperated despair. "For an educated, civilized man, Markham, there's something downright primitive about the way you cling to optical illusions. I say, y' know, you're exactly like an infant who really believes that the magician generates a rabbit in a silk hat, simply because he sees it done."

    "Now you're becoming insulting."

    "Rather!" Vance pleasantly agreed. "But something drastic must be done to disentangle you from the Lorelei of legal facts. You're so deficient in imagination, old thing."

    "I take it that you would have me close my eyes and picture Spotswoode sitting upstairs here in the Stuyvesant Club and extending his arms to 71st Street. But I simply couldn't do it. I'm a commonplace chap. Such a vision would strike me as ludicrous; it would smack of a hasheesh dream. . . . You yourself don't use Cannabis indica, do you?"

    "Put that way, the idea does sound a bit supernatural. And yet: Certum est quia impossibile est. I rather like that maxim, don't y' know; for, in the present case, the impossible is true. Oh, Spotswoode's guilty—no doubt about it. And I'm going to cling tenaciously to that apparent hallucination. Moreover, I'm going to try to lure you into its toils; for your own—as we absurdly say—good name is at stake. As it happens, Markham, you are at this moment shielding the real murderer from publicity."

    Vance had spoken with the easy assurance that precludes argument; and from the altered expression on Markham's face I could see he was moved.

    "Tell me," he said, "how you arrived at your fantastic belief in Spotswoode's guilt."

    Vance crushed out his cigarette and folded his arms on the table.

    "We begin with my quartet of possibilities—Mannix, Cleaver, Lindquist, and Spotswoode. Realizing, as I did, that the crime was carefully planned with the sole object of murder, I knew that only someone hopelessly ensnared in the lady's net could have done it. And no suitor outside of my quartet could have been thus enmeshed, or we would have learned of him. Therefore, one of the four was guilty. Now, Lindquist was eliminated when we found out that he was bedridden in a hospital at the time of Skeel's murder; for obviously the same person committed both crimes—"

    "But," interrupted Markham, "Spotswoode had an equally good alibi for the night of the Canary's murder. Why eliminate one and not the other?"

    "Sorry, but I can't agree with you. Being prostrated at a known place surrounded by incorruptible and disinterested witnesses, both preceding and during an event, is one thing; but being actually on the ground, as Spotswoode was that fatal evening, within a few minutes of the time the lady was murdered, and then being alone in a taxicab for fifteen minutes or so following the event—that is another thing. No one, as far as we know, actually saw the lady alive after Spotswoode took his departure."

    "But the proof of her having been alive and spoken to him is incontestable."

    "Granted. I admit that a dead woman doesn't scream and call for help and then converse with her murderer."

    "I see." Markham spoke with sarcasm. "You think it was Skeel, disguising his voice."

    "Lord no! What a priceless notion! Skeel didn't want anyone to know he was there. Why should he have staged such a masterpiece of idiocy? That certainly isn't the explanation. When we find the answer it will be reasonable and simple."

    "That's encouraging," smiled Markham. "But proceed with your reason for Spotswoode's guilt."

    "Three of my quartet, then, were potential murderers," Vance resumed. "Accordingly, I requested an evening of social relaxation, that I might put them under the psychological microscope, as it were. Although Spotswoode's ancestry was wholly consistent with his having been the guilty one, nevertheless I confess I thought that Cleaver or Mannix had committed the crime; for, by their own statements, either of them could have done it without contradicting any of the known circumstances of the situation. Therefore, when Mannix declined your invitation to play poker last night, I put Cleaver to the first test. I wig-wagged to Mr. Allen, and he straightway proceeded to perform his first feat of prestidigitation."

    Vance paused and looked up.

    "You perhaps recall the circumstances? It was a jackpot. Allen dealt Cleaver a four-straight-flush and gave me three kings. The other hands were so poor that everyone else was compelled to drop out. I opened, and Cleaver stayed. On the draw, Allen gave me another king, and gave Cleaver the card he needed to complete his straight-flush. Twice I bet a small amount, and each time Cleaver raised me. Finally I called him, and, of course, he won. He couldn't help but win d' ye see. He was betting on a sure thing. Since I opened the pot and drew two cards, the highest hand I could possibly have held would have been four of a kind. Cleaver knew this, and having a straight-flush, he also knew, before he raised my bet, that he had me beaten. At once I realized that he was not the man I was after."

    "By what reasoning?"

    "A poker player, Markham, who would bet on a sure thing is one who lacks the egotistical self-confidence of the highly subtle and supremely capable gambler. He is not a man who will take hazardous chances and tremendous risks, for he possesses, to some degree, what the psychoanalysts call an inferiority complex, and instinctively he grasps at every possible opportunity of protecting and bettering himself. In short, he is not the ultimate, unadulterated gambler. And the man who killed the Odell girl was a supreme gambler who would stake everything on a single turn of the wheel, for, in killing her, that is exactly what he did. And only a gambler whose paramount self-confidence would make him scorn, through sheer egotism, to bet on a sure thing, could have committed such a crime. Therefore, Cleaver was eliminated as a suspect."

    Markham was now listening intently.

    "The test to which I put Spotswoode a little later," Vance went on, "had originally been intended for Mannix, but he was out of the game. That didn't matter, however, for, had I been able to eliminate both Cleaver and Spotswoode, then Mannix would undoubtedly have been the guilty man. Of course I would have planned something else to substantiate the fact; but, as it was, that wasn't necessary. . . . The test I applied to Spotswoode was pretty well explained by the gentleman himself. As he said, not one player in a thousand would have wagered the limit against a pat hand, when he himself held nothing. It was tremendous—superb! It was probably the most remarkable bluff ever made in a game of poker. I couldn't help admiring him when he calmly shoved forward all his chips, knowing, as I did, that he held nothing. He staked everything, d' ye see, wholly on his conviction that he could follow my reasoning step by step and, in the last analysis, outwit me. It took courage and daring to do that. And it also took a degree of self-confidence which would never have permitted him to bet on a sure thing. The psychological principles involved in that hand were identical with those of the Odell crime. I threatened Spotswoode with a powerful hand—a pat hand—just as the girl, no doubt, threatened him; and instead of compromising—instead of calling me or laying down—he outreached me; he resorted to one supreme coup, though it meant risking everything. . . . My word, Markham! Can't you see how the man's character, as revealed in that amazing gesture, dovetails with the psychology of the crime?"

    Markham was silent for a while; he appeared to be pondering the matter. "But you yourself, Vance, were not satisfied at the time," he submitted at length. "In fact, you looked doubtful and worried."

    "True, old dear. I was no end worried. The psychological proof of Spotswoode's guilt came so dashed unexpectedly—I wasn't looking for it, don't y' know. After eliminating Cleaver I had a parti pris, so to speak, in regard to Mannix; for all the material evidence in favor of Spotswoode's innocence—that is, the seeming physical impossibility of his having strangled the lady—had, I admit, impressed me. I'm not perfect, don't y' know. Being unfortunately human, I'm still susceptible to the malicious animal magnetism about facts and appearances, which you lawyer chaps are continuously exuding over the earth like some vast asphyxiating effluvium. And even when I found that Spotswoode's psychological nature fitted perfectly with all the factors of the crime, I still harbored a doubt in regard to Mannix. It was barely possible that he would have played the hand just as Spotswoode played it. That is why, after the game was over, I tackled him on the subject of gambling. I wanted to check his psychological reactions."

    "Still, he staked everything on one turn of the wheel, as you put it."

    "Ah! But not in the same sense that Spotswoode did. Mannix is a cautious and timid gambler as compared with Spotswoode. To begin with, he had an equal chance and an even bet, whereas Spotswoode had no chance at all—his hand was worthless. And yet Spotswoode wagered the limit on a pure bit of mental calculation. That was gambling in the higher ether. On the other hand, Mannix was merely tossing a coin, with an even chance of winning. Furthermore, no calculation of any kind entered into it; there was no planning, no figuring, no daring. And, as I have told you from the start, the Odell murder was premeditated and carefully worked out with shrewd calculation and supreme daring. . . . And what true gambler would ask an adversary to double a bet on the second flip of the coin, and then accept an offer to redouble on the third flip? I purposely tested Mannix in that way, so as to preclude any possibility of error. Thus I not only eliminated him, I expunged him, eradicated him, wiped him out utterly. It cost me a thousand dollars, but it purged my mind of any lingering doubt. I then knew, despite all the contr'ry material indications, that Spotswoode had done away with the lady."

    "You make your case theoretically plausible. But, practically, I'm afraid I can't accept it." Markham was more impressed, I felt, than he cared to admit. "Damn it, man!" he exploded after a moment. "Your conclusion demolishes all the established landmarks of rationality and sane credibility.—Just consider the facts." He had now reached the argumentative stage of his doubt. "You say Spotswoode is guilty. Yet we know, on irrefutable evidence, that five minutes after he came out of the apartment, the girl screamed and called for help. He was standing by the switchboard, and, accompanied by Jessup, he went to the door and carried on a brief conversation with her. She was certainly alive then. Then he went out the front door, entered a taxicab, and drove away. Fifteen minutes later he was joined by Judge Redfern as he alighted from the taxicab in front of the club here—nearly forty blocks away from the apartment house! It would have been impossible for him to have made the trip in less time; and, moreover, we have the chauffeur's record. Spotswoode simply did not have either the opportunity or the time to commit the murder between half past eleven and ten minutes of twelve, when Judge Redfern met him. And, remember, he played poker in the club here until three in the morning—hours after the murder took place."

    Markham shook his head with emphasis.

    "Vance, there's no human way to get round those facts. They're firmly established; and they preclude Spotswoode's guilt as effectively and finally as though he had been at the North Pole that night."

    Vance was unmoved.

    "I admit everything you say," he rejoined. "But as I have stated before, when material facts and psychological facts conflict, the material facts are wrong. In this case, they may not actually be wrong, but they're deceptive."

    "Very well, magnus Apollo!" The situation was too much for Markham's exacerbated nerves. "Show me how Spotswoode could have strangled the girl and ransacked the apartment, and I'll order Heath to arrest him."

    "'Pon my word, I can't do it," expostulated Vance. "Omniscience was denied me. But—deuce take it!—I think I've done rather well in pointing out the culprit. I never agreed to expound his technic, don't y' know."

    "So! Your vaunted penetration amounts only to that, does it? Well, well! Here and now I become a professor of the higher mental sciences, and I pronounce solemnly that Doctor Crippen murdered the Odell girl. To be sure, Crippen's dead; but that fact doesn't interfere with my newly adopted psychological means of deduction. Crippen's nature, you see, fits perfectly with all the esoteric and recondite indications of the crime. Tomorrow I'll apply for an order of exhumation."

    Vance looked at him with waggish reproachfulness and sighed. "Recognition of my transcendent genius, I see, is destined to be posthumous. Omnia post obitum fingit majora vetustas. In the meantime I bear the taunts and jeers of the multitude with a stout heart. 'My head is bloody, but unbowed.'"

    He looked at his watch and then seemed to become absorbed with some line of thought.

    "Markham," he said, after several minutes, "I've a concert at three o'clock, but there's an hour to spare. I want to take another look at that apartment and its various approaches. Spotswoode's trick—and I'm convinced it was nothing more than a trick—was enacted there; and if we are ever to find the explanation, we shall have to look for it on the scene."

    I had got the impression that Markham, despite his emphatic denial of the possibility of Spotswoode's guilt, was not entirely unconvinced. Therefore, I was not surprised when, with only a halfhearted protest, he assented to Vance's proposal to revisit the Odell apartment.