Vance and I went home after lunch, and at about four o'clock Markham telephoned to say that he had made the necessary arrangements for the evening with Spotswoode, Mannix, and Cleaver. Immediately following this confirmation Vance left the house and did not return until nearly eight o'clock. Though I was filled with curiosity at so unusual a proceeding, he refused to enlighten me. But when, at a quarter to nine, we went downstairs to the waiting car, there was a man I did not know in the tonneau; and I at once connected him with Vance's mysterious absence.

    "I've asked Mr. Allen to join us tonight," Vance vouchsafed, when he had introduced us. "You don't play poker, and we really need another hand to make the game interestin', y' know. Mr. Allen, by the bye, is an old antagonist of mine."

    The fact that Vance would, apparently without permission, bring an uninvited guest to Markham's apartment amazed me but little more than the appearance of the man himself. He was rather short, with sharp, shrewd features; and what I saw of his hair beneath his jauntily tipped hat was black and sleek, like the painted hair on Japanese dolls. I noted, too, that his evening tie was enlivened by a design of tiny white forget-me-nots, and that his shirtfront was adorned with diamond studs.

    The contrast between him and the immaculately stylish and meticulously correct Vance was aggressively evident. I wondered what could be the relationship between them. Obviously it was neither social nor intellectual.

    Cleaver and Mannix were already on hand when we were ushered into Markham's drawing room, and a few minutes later Spotswoode arrived. The amenities of introduction over, we were soon seated comfortably about the open log fire, smoking, and sipping very excellent Scotch highballs. Markham had, of course, accepted the unexpected Mr. Allen cordially, but his occasional glances in the latter's direction told me he was having some difficulty in reconciling the man's appearance with Vance's sponsorship.

    A tense atmosphere lay beneath the spurious and affected affability of the little gathering. Indeed, the situation was scarcely conducive to spontaneity. Here were three men each of whom was known to the others to have been interested in the same woman; and the reason for their having been brought together was the fact that this woman had been murdered. Markham, however, handled the situation with such tact that he largely succeeded in giving each one the feeling of being a disinterested spectator summoned to discuss an abstract problem. He explained at the outset that the "conference" had been actuated by his failure to find any approach to the problem of the murder. He hoped, he said, by a purely informal discussion, divested of all officialism and coercion, to turn up some suggestion that might lead to a fruitful line of inquiry. His manner was one of friendly appeal, and when he finished speaking, the general tension had been noticeably relaxed.

    During the discussion that followed I was interested in the various attitudes of the men concerned. Cleaver spoke bitterly of his part in the affair, and was more self-condemnatory than suggestive. Mannix was voluble and pretentiously candid, but beneath his comments ran a strain of apologetic wariness. Spotswoode, unlike Mannix, seemed loath to discuss the matter and maintained a consistently reticent attitude. He responded politely to Markham's questions, but he did not succeed entirely in hiding his resentment at thus being dragged into a general discussion. Vance had little to say, limiting himself to occasional remarks directed always to Markham. Allen did not speak but sat contemplating the others with a sort of canny amusement.

    The entire conversation struck me as utterly futile. Had Markham really hoped to garner information from it, he would have been woefully disappointed. I realized, though, that he was merely endeavoring to justify himself for having taken so unusual a step and to pave the way for the game of poker which Vance had requested. When the time came to broach the subject, however, there was no difficulty about it.

    It was exactly eleven o'clock when he made the suggestion. His tone was gracious and unassuming; but by couching his invitation in terms of a personal request, he practically precluded declination. But his verbal strategy, I felt, was unnecessary. Both Cleaver and Spotswoode seemed genuinely to welcome the opportunity of dropping a distasteful discussion in favor of playing cards; and Vance and Allen, of course, concurred instantly. Mannix alone declined. He explained that he knew the game only slightly and disliked it; though he expressed an enthusiastic desire to watch the others. Vance urged him to reconsider but without success; and Markham finally ordered his man to arrange the table for five.

    I noticed that Vance waited until Allen had taken his place, and then dropped into the chair at his right. Cleaver took the seat at Allen's left. Spotswoode sat at Vance's right; and then came Markham. Mannix drew up his chair midway behind Markham and Cleaver. Thus:

    Cleaver first named a rather moderate limit, but Spotswoode at once suggested much larger stakes. Then Vance went still higher, and as both Markham and Allen signified their agreement, his figure was accepted. The prices placed on the chips somewhat took my breath away, and even Mannix whistled softly.

    That all five men at the table were excellent players became obvious before the game had progressed ten minutes. For the first time that night Vance's friend Allen seemed to have found his milieu and to be wholly at ease.

    Allen won the first two hands, and Vance the third and fourth. Spotswoode then had a short run of good luck, and a little later Markham took a large jackpot, which put him slightly in the lead. Cleaver was the only loser thus far; but in another half-hour he had succeeded in recovering a large portion of his losses. After that Vance forged steadily ahead, only to relinquish his winning streak to Allen. Then for a while the fortunes of the game were rather evenly distributed. But later on both Cleaver and Spotswoode began to lose heavily. By half past twelve a grim atmosphere had settled over the party; for so high were the stakes, and so rapidly did the betting pyramid, that even for men of means—such as all these players undoubtedly were—the amounts which continually changed hands represented very considerable items.

    Just before one o'clock, when the fever of the game had reached a high point, I saw Vance glance quickly at Allen and pass his handkerchief across his forehead. To a stranger the gesture would have appeared perfectly natural; but, so familiar was I with Vance's mannerisms, I immediately recognized its artificiality. And simultaneously I noticed that it was Allen who was shuffling the cards preparatory to dealing. Some smoke from his cigar evidently went into his eye at that moment, for he blinked, and one of the cards fell to the floor. Quickly retrieving it, he reshuffled the deck and placed it before Vance to cut.

    The hand was a jackpot, and there was a small fortune in chips already on the table. Cleaver, Markham, and Spotswoode passed. The decision thus reached Vance, and he opened for an unusually large amount. Allen at once laid down his hand, but Cleaver stayed. Then Markham and Spotswoode both dropped out, leaving the entire play between Vance and Cleaver. Cleaver drew one card, and Vance, who had opened, drew two. Vance made a nominal wager, and Cleaver raised it substantially. Vance in turn raised Cleaver, but only for a small amount; and Cleaver again raised Vance—this time for an even larger sum than before. Vance hesitated, and called him. Cleaver exposed his hand triumphantly.

    "Straight flush—jack high," he announced. "Can you beat that?"

    "Not on a two-card draw," said Vance ruefully. He put his cards down to show his openers. He had four kings.

    About half an hour later Vance again took out his handkerchief and passed it across his forehead. As before, I noted that it was Allen's deal, and also that the hand was a jackpot which had been twice sweetened. Allen paused to take a drink of his highball and to light his cigar. Then, after Vance had cut the cards, he dealt them.

    Cleaver, Markham, and Spotswoode passed, and again Vance opened, for the full amount of the pot. No one stayed except Spotswoode; and this time it was a struggle solely between him and Vance. Spotswoode asked for one card; and Vance stood pat. Then there followed a moment of almost breathless silence. The atmosphere seemed to me to be electrically charged, and I think the others sensed it too, for they were watching the play with a curiously strained intentness. Vance and Spotswoode, however, appeared frozen in attitudes of superlative calm. I watched them closely, but neither revealed the slightest indication of any emotion.

    It was Vance's first bet. Without speaking he moved a stack of yellow chips to the center of the table—it was by far the largest wager that had been made during the game. But immediately Spotswoode measured another stack alongside of it. Then he coolly and deftly counted the remainder of his chips, and pushed them forward with the palm of his hand, saying quietly, "The limit."

    Vance shrugged almost imperceptibly.

    "The pot, sir, is yours." He smiled pleasantly at Spotswoode and put down his hand face up, to establish his openers. He had held four aces!

    "Gad! That's poker!" exclaimed Allen, chuckling.

    "Poker?" echoed Markham. "To lay down four aces with all that money at stake?"

    Cleaver also grunted his astonishment, and Mannix pursed his lips disgustedly.

    "I don't mean any offense, y' understand, Mr. Vance," he said. "But looking at that play from a strictly business standpoint, I'd say you quit too soon."

    Spotswoode glanced up.

    "You gentlemen wrong Mr. Vance," he said. "He played his hand perfectly. His withdrawal, even with four aces, was scientifically correct."

    "Sure it was," agreed Allen. "Oh, boy! What a battle that was!"

    Spotswoode nodded and, turning to Vance, said, "Since the exact situation is never likely to occur again, the least I can do, by way of showing my appreciation of your remarkable perception, is to gratify your curiosity. I held nothing."

    Spotswoode put down his hand and extended his fingers gracefully toward the upturned cards. There were revealed a five, six, seven, and eight of clubs, and a knave of hearts.

    "I can't say that I follow your reasoning, Mr. Spotswoode," Markham confessed. "Mr. Vance had you beaten—and he quit."

    "Consider the situation," Spotswoode replied, in a suave, even voice. "I most certainly would have opened so rich a pot, had I been able to, after Mr. Cleaver and you had passed. But since I nevertheless stayed after Mr. Vance had opened for so large an amount, it goes without saying that I must have had either a four-straight, a four-flush, or a four-straight-flush. I believe I may state without immodesty that I am too good a player to have stayed otherwise. . . ."

    "And I assure you, Markham," interrupted Vance, "that Mr. Spotswoode is too good a player to have stayed unless he had actually had a four-straight-flush. That is the only hand he would have been justified in backing at the betting odds of two to one. You see, I had opened for the amount in the pot, and Mr. Spotswoode had to put up half the amount of the money on the table in order to stay—making it a two-to-one bet. Now, these odds are not high, and any nonopening hand smaller than a four-straight-flush would not have warranted the risk. As it was, he had, with a one-card draw, two chances in forty-seven of making a straight-flush, nine chances in forty-seven of making a flush, and eight chances in forty-seven of making a straight; so that he had nineteen chances in forty-seven—or more than one chance in three—of strengthening his hand into either a straight-flush, a flush, or a straight."

    "Exactly," assented Spotswoode. "However, after I had drawn my one card, the only possible question in Mr. Vance's mind was whether or not I had made my straight-flush. If I had not made it—or had merely drawn a straight or a flush—Mr. Vance figured, and figured rightly, that I would not have seen his large bet and also have raised it the limit. To have done so, in those circumstances, would have been irrational poker. Not one player in a thousand would have taken such a risk on a mere bluff. Therefore, had Mr. Vance not laid down his four aces when I raised him, he would have been foolhardy in the extreme. It turned out, of course, that I was actually bluffing; but that does not alter the fact that the correct and logical thing was for Mr. Vance to quit."

    "Quite true," Vance agreed. "As Mr. Spotswoode says, not one player in a thousand would have wagered the limit without having filled his straight-flush, knowing I had a pat hand. Indeed, one might also say that Mr. Spotswoode, by doing so, has added another decimal point to the psychological subtleties of the game; for, as you see, he analyzed my reasoning, and carried his own reasoning a step further."

    Spotswoode acknowledged the compliment with a slight bow; and Cleaver reached for the cards and began to shuffle them. But the tension had been broken, and the game was not resumed.

    Something, however, seemed to have gone wrong with Vance. For a long while he sat frowning at his cigarette and sipping his highball in troubled abstraction. At last he rose and walked to the mantel, where he stood studying a Cézanne watercolor he had given Markham years before. His action was a typical indication of his inner puzzlement.

    Presently, when there came a lull in the conversation, he turned sharply and looked at Mannix.

    "I say, Mr. Mannix"—he spoke with only casual curiosity—"how does it happen you've never acquired a taste for poker? All good businessmen are gamblers at heart."

    "Sure they are," Mannix replied, with pensive deliberation. "But poker, now, isn't my idea of gambling—positively not. It's got too much science. And it ain't quick enough for me—it hasn't got the kick in it, if you know what I mean. Roulette's my speed. When I was in Monte Carlo last summer, I dropped more money in ten minutes than you gentlemen lost here this whole evening. But I got action for my money."

    "I take it, then, you don't care for cards at all."

    "Not to play games with." Mannix had become expansive. "I don't mind betting money on the draw of a card, for instance. But no two out of three, y' understand. I want my pleasures to come rapid." And he snapped his thick fingers several times in quick succession to demonstrate the rapidity with which he desired to have his pleasures come.

    Vance sauntered to the table and carelessly picked up a deck of cards. "What do you say to cutting once for a thousand dollars?"

    Mannix rose instantly. "You're on!"

    Vance handed the cards over, and Mannix shuffled them. Then he put them down and cut. He turned up a ten. Vance cut and showed a king.

    "A thousand I owe you," said Mannix, with no more concern than if it had been ten cents.

    Vance waited without speaking, and Mannix eyed him craftily.

    "I'll cut with you again—two thousand this time. Yes?"

    Vance raised his eyebrows. "Double? . . . By all means." He shuffled the cards and cut a seven.

    Mannix's hand swooped down and turned a five.

    "Well, that's three thousand I owe you," he said. His little eyes had now narrowed into slits, and he held his cigar clamped tightly between his teeth.

    "Like to double it again—eh, what?" Vance asked. "Four thousand this time?"

    Markham looked at Vance in amazement, and over Allen's face there came an expression of almost ludicrous consternation. Everyone present, I believe, was astonished at the offer, for obviously Vance knew that he was giving Mannix tremendous odds by permitting successive doubling. In the end he was sure to lose. I believe Markham would have protested if at that moment Mannix had not snatched the cards from the table and begun to shuffle them.

    "Four thousand it is!" he announced, putting down the deck and cutting. He turned up the queen of diamonds. "You can't beat the lady—positively not!" He was suddenly jovial.

    "I fancy you're right," murmured Vance; and he cut a trey.

    "Want some more?" asked Mannix, with good-natured aggressiveness.

    "That's enough." Vance seemed bored. "Far too excitin'. I haven't your rugged constitution, don't y' know."

    He went to the desk and made out a check to Mannix for a thousand dollars. Then he turned to Markham and held out his hand. "Had a jolly evening and all that sort of thing. . . . And, don't forget: we lunch together tomorrow. One o'clock at the club, what?"

    Markham hesitated. "If nothing interferes."

    "But really, y' know, it mustn't," insisted Vance. "You've no idea how eager you are to see me."

    He was unusually silent and thoughtful during the ride home. Not one explanatory word could I get out of him. But when he bade me good night he said, "There's a vital part of the puzzle still missing, and until it's found, none of it has any meaning."