"Offhand, what do you make of these new developments?" asked Vance, after a short pause.
"I'm somewhat foggy as to the details. The professor was not exactly phosphorescent. Sketchy, in fact. Drukker fell from the wall, like Humpty Dumpty, round ten o'clock, and was found this morning--that's all plain. But under what conditions did Lady Mae succumb to shock? Who, or what, shocked her? And how?"
"The murderer took Drukker's key and came here immediately after the crime. Mrs. Drukker caught him in her son's room. There was a scene, according to the cook, who listened from the head of the stairs; and during it Mrs. Drukker died from dilatation of the heart."
"Thereby relieving the gentleman of the bother of killing her."
"That seems clear enough," agreed Vance. "But the reason for the murderer's visit here is not so lucid. Can you suggest an explanation?"
Arnesson puffed thoughtfully on his pipe.
"Incomprehensible," he muttered at length. "Drukker had no valuables, or no compromising documents. Straightforward sort of cuss--not the kind to mix in any dirty business. . . . No possible reason for any one prowling about his room."
Vance lay back and appeared to relax.
"What was this quantum theory Drukker was working on?"
"Ha! Big thing!" Arnesson became animated. "He was on the path of reconciling the Einstein-Bohr theory of radiation with the facts of interference, and of overcoming the inconsistencies inherent in Einstein's hypothesis. His research had already led him to an abandonment of causal space-time coordination of atomic phenomena, and to its replacement by a statistical description.* . . . Would have revolutionized physics--made him famous. Shame he was told off before he'd put his data in shape."
* An important step toward the solution of these complex problems was taken a few years later by the de Broglie-Schrödinger theory as laid down in de Broglie's "Ondes et Mouvements" and Schrödinger's "Abhandlungen zur Wellenmechanik."
"Do you happen to know where Drukker kept the records of these computations?"
"In a loose-leaf note-book--all tabulated and indexed. Methodical and neat about everything. Even his chirography was like copperplate."
"You know, then, what the note-book looked like?"
"I ought to. He showed it to me often enough. Red limp-leather cover--thin yellow pages--two or three clips on every sheet holding notations--his name gold-stamped in large letters on the binding. . . . Poor devil! Sic transit. . . ."
"Where would this note-book be now?"
"One of two places--either in the drawer of his desk in the study or else in the escritoire in his bedroom. In the daytime, of course, he worked in the study; but he fussed day and night when wrapped up in a problem. Kept an escritoire in his bedroom, where he put his current records when he retired, in case he got an inspiration to monkey with 'em during the night. Then, in the morning, back they'd go to the study. Regular machine for system."
Vance had been gazing lazily out of the window as Arnesson rambled on. The impression he gave was that he had scarcely heard the description of Drukker's habits; but presently he turned and fixed Arnesson with a languid look.
"I say," he drawled; "would you mind toddling up-stairs and fetching Drukker's note-book? Look in both the study and the bedroom."
I thought I noticed an almost imperceptible hesitation on Arnesson's part; but straightway he rose.
"Good idea. Too valuable a document to be left lying round." And he strode from the room.
Markham began pacing the floor, and Heath revealed his uneasiness by puffing more energetically on his cigar. There was a tense atmosphere in the little drawing-room as we waited for Arnesson's return. Each of us was in a state of expectancy, though just what we hoped for or feared would have been difficult to define.
In less than ten minutes Arnesson reappeared at the door. He shrugged his shoulders and held out empty hands.
"Gone!" he announced. "Looked in every likely place--couldn't find it." He threw himself into a chair and relighted his pipe. "Can't understand it. . . . Perhaps he hid it."
"Perhaps," murmured Vance.
CHAPTER XX
THE NEMESIS
(Saturday, April 16; 1 p.m.)
It was past one o'clock, and Markham, Vance and I rode to the Stuyvesant Club. Heath remained at the Drukker house to carry on the routine work, to draw up his report, and to deal with the reporters who would be swarming there shortly.
Markham was booked for a conference with the Police Commissioner at three o'clock; and after lunch Vance and I walked to Stieglitz's Intimate Gallery and spent an hour at an exhibition of Georgia O'Keeffe's floral abstractions. Later we dropped in at Aeolian Hall and sat through Debussy's G-minor quartette. There were some Cézanne water-colors at the Montross Galleries; but by the time we had pushed our way through the late-afternoon traffic of Fifth Avenue the light had begun to fail, and Vance ordered the chauffeur to the Stuyvesant Club, where we joined Markham for tea.
"I feel so youthful, so simple, so innocent," Vance complained lugubriously. "So many things are happenin', and they're bein' manipulated so ingeniously that I can't grasp 'em. It's very disconcertin', very confusin'. I don't like it--I don't at all like it. Most wearin'." He sighed drearily and sipped his tea.
"Your sorrows leave me cold," retorted Markham.
"You've probably spent the afternoon inspecting arquebuses and petronels at the Metropolitan Museum. If you'd had to go through what I've suffered--"
"Now, don't be cross," Vance rebuked him. "There's far too much emotion in the world. Passion is not going to solve this case. Cerebration is our only hope. Let us be calm and thoughtful." His mood became serious. "Markham, this comes very near being the perfect crime. Like one of Morphy's great chess combinations, it has been calculated a score of moves ahead. There are no clews; and even if there were, they'd probably point in the wrong direction. And yet . . . and yet there's something that's trying to break through. I feel it: sheer intuition--that is to say, nerves. There's an inarticulate voice that wants to speak, and can't. A dozen times I've sensed the presence of some struggling force, like an invisible ghost trying to make contact without revealing its identity."
Markham gave an exasperated sigh.
"Very helpful. Do you advise calling in a medium?"
"There's something we've overlooked," Vance went on, disregarding the sarcasm. "The case is a cipher, and the key-word is somewhere before us, but we don't recognize it. 'Pon my soul, it's dashed annoyin'. . . . Let's be orderly. Neatness--that's our desideratum. First, Robin is killed. Next, Sprigg is shot. Then Mrs. Drukker is frightened with a black bishop. After that, Drukker is shoved over a wall. Makin' four distinct episodes in the murderer's extravaganza. Three of 'em were carefully planned. One--the leaving of the bishop at Mrs. Drukker's door--was forced on the murderer, and was therefore decided on without preparation. . . ."
"Clarify your reasoning on that point."
"Oh, my dear fellow! The conveyor of the black bishop was obviously acting in self-defence. An unexpected danger developed along his line of campaign, and he took this means of averting it. Just before Robin's death Drukker departed from the archery-room and installed himself in the arbor of the yard, where he could look into the archery-room through the rear window. A little later he saw some one in the room talking to Robin. He returned to his house, and at that moment Robin's body was thrown on the range. Mrs. Drukker saw it, and at the same time she probably saw Drukker. She screamed--very natural, what? Drukker heard the scream, and told us of it later in an effort to establish an alibi for himself after we'd informed him that Robin had been killed. Thus the murderer learned that Mrs. Drukker had seen something--how much, he didn't know. But he wasn't taking any chances. He went to her room at midnight to silence her, and took the bishop to leave beside her body as a signature. But he found the door locked, and left the bishop outside, by way of warning her to say nothing on pain of death. He didn't know that the poor woman suspected her own son."
; "But why didn't Drukker tell us whom he saw in the archery-room with Robin?"
"We can only assume that the person was some one whom he couldn't conceive of as being guilty. And I'm inclined to believe he mentioned the fact to this person and thus sealed his own doom."
"Assuming the correctness of your theory, where does it lead us?"
"To the one episode that wasn't elaborately prepared in advance. And when there has been no preparation for a covert act there is pretty sure to be a weakness in one or more of the details.--Now, please note that at the time of each of the three murders any one of the various persons in the drama could have been present. No one had an alibi. That, of course, was cleverly calculated: the murderer chose an hour when all of the actors were, so to speak, waiting in the wings. But that midnight visit! Ah! That was a different matter. There was no time to work out a perfect set of circumstances,--the menace was too immediate. And what was the result? Drukker and Professor Dillard were, apparently, the only persons on hand at midnight. Arnesson and Belle Dillard were supping at the Plaza and didn't return home until half past twelve. Pardee was hornlocked with Rubinstein over a chess-board from eleven to one. Drukker is now of course eliminated. . . . What's the answer?"
"I could remind you," returned Markham irritably, "that the alibis of the others have not been thoroughly checked."
"Well, well, so you could." Vance lay back indolently and sent a long regular series of smoke-rings toward the ceiling. Suddenly his body tensed, and with meticulous care he leaned over and put out his cigarette. Then he glanced at his watch and got to his feet. He fixed Markham with a quizzical look.
"Allons, mon vieux. It's not yet six. Here's where Arnesson makes himself useful."
"What now?" expostulated Markham.
"Your own suggestion," Vance replied, taking him by the arm and leading him toward the door. "We're going to check Pardee's alibi."
Half an hour later we were seated with the professor and Arnesson in the Dillard Library.
"We've come on a somewhat unusual errand," explained Vance; "but it may have a vital bearing on our investigation." He took out his wallet, and unfolded a sheet of paper. "Here's a document, Mr. Arnesson, I wish you'd glance over. It's a copy of the official scoresheet of the chess game between Pardee and Rubinstein. Very interestin'. I've toyed with it a bit, but I'd like your expert analysis of it. The first part of the game is usual enough, but the play after the adjournment rather appeals to me."
Arnesson took the paper and studied it with cynical amusement.
"Aha! The inglorious record of Pardee's Waterloo, eh?"
"What's the meaning of this, Markham?" asked professor Dillard contemptuously. "Do you hope to run a murderer to earth by dilly-dallying over a chess game?"
"Mr. Vance hoped something could be learned from it."
"Fiddlesticks!" The professor poured himself another glass of port and, opening a book, ignored us completely.
Arnesson was absorbed in the notations of the chess score.
"Something a bit queer here," he muttered. "The time's askew. Let's see. . . . The scoresheet shows that, up to the time of adjournment, White--that is, Pardee--had played one hour and forty-five minutes, and Black, or Rubinstein, one hour and fifty-eight minutes. So far, so good. Thirty moves. Quite in order. But the time at the end of the game, when Pardee resigned, totals two hours and thirty minutes for White, and three hours and thirty-two minutes for Black--which means that, during the second session of the game, White consumed only forty-five minutes whereas Black used up one hour and thirty-four minutes."
Vance nodded.
"Exactly. There were two hours and nineteen minutes of play beginning at 11 p. m., which carried the game to 1.19 a. m. And Rubinstein' s moves during that time took forty-nine minutes longer than Pardee's.--Can you make out what happened?"
Arnesson pursed his lips and squinted at the notations.
"It's not clear. I'd need time."
"Suppose," Vance suggested, "we set up the game in the adjourned position and play it through. I'd like your opinion on the tactics."
Arnesson rose jerkily and went to the little chess table in the corner.
"Good idea." He emptied the men from the box. "Let's see now. . . . Oho! A black bishop is missing. When do I get it back, by the way?" He gave Vance a plaintive leer. "Never mind. We don't need it here. One black bishop was swapped." And he proceeded to arrange the men to accord with the position of the game at the time of adjournment. Then he sat down and studied the set-up.
"It doesn't strike me as a particularly unfavorable position for Pardee," ventured Vance.
"Me either. Can't see why he lost the game. Looks drawish to me." After a moment Arnesson referred to the scoresheet. "We'll run through the play and find out where the trouble lay." He made half a dozen moves; then, after several minutes' study, gave a grunt. "Ha! This is rather deep stuff of Rubinstein's. Amazing combination he began working up here. Subtle, by Gad! As I know Rubinstein, it took him a long time to figure it out. Slow, plodding chap."
"It's possible, isn't it," suggested Vance, "that the working out of that combination explains the discrepancy in time between Black and White?"
"Oh, undoubtedly. Rubinstein must have been in good form not to have made the discrepancy greater. Planning the combination took him all of forty-five minutes--or I'm a duffer."
"At what hour, would you say," asked Vance carelessly, "did Rubinstein use up that forty-five minutes?"
"Well, let's see. The play began at eleven: six moves before the combination started. . . . Oh, say, somewhere between half past eleven and half past twelve. . . . Yes, just about. Thirty moves before the adjournment: six moves beginning at eleven--that makes thirty-six: then on the forty-fourth move Rubinstein moved his pawn to Bishop-7-check, and Pardee resigned. . . . Yes--the working out of the combination was between eleven-thirty and twelve-thirty."
Vance regarded the men on the board, which were now in the position they had occupied at the time of Pardee's resignation.*
* For the benefit of the expert chess-player who may be academically interested I append the exact position of the game when Pardee resigned:--WHiTE: King at QKtsq; Rook at QB8; Pawns at QR2 and Q2. BLACK: King at Q5; Knight at QKt5; Bishop at QR6; Pawns at QKt7 and QB7.
"Out of curiosity," he said quietly, "I played the game through to the checkmate the other night.--I say, Mr. Arnesson; would you mind doin' the same. I could bear to hear your comment on it."
Arnesson studied the position closely for a few minutes. Then he turned his head slowly and lifted his eyes to Vance. A sardonic grin overspread his face.
"I grasp the point. Gad! What a situation! Five moves for Black to win through. And an almost unheard-of finale in chess. Can't recall a similar instance. The last move would be Bishop to Knight-7, mating. In other words, Pardee was beaten by the black bishop! Incredible!"*
* The final five unplayed moves for Black to mate, as I later obtained them from Vance, were:--45. RxP; KtxR. 46. KxKt; P--Kt8 (Queen). 47. KxQ; K--Q6. 48. K--Rsq; K--B7. 49. P--Q3; B--Kt7 mate.
Professor Dillard put down his book.
"What's this?" he exclaimed, joining us at the chess table. "Pardee was defeated by the bishop?" He gave Vance a shrewd, admiring look. "You evidently had good reason, sir, for investigating that chess game. Pray overlook an old man's temper." He stood gazing down at the board with a sad, puzzled expression.
Markham was frowning with deep perplexity.
"You say it's unusual for a bishop alone to mate?" he asked Arnesson.
"Never happens--almost unique situation. And that it should happen to Pardee of all people! Incomprehensible!" He gave a short ironic laugh. "Inclines one to believe in a nemesis. You know, the bishop has been Pardee's bête noir for twenty years--it's ruined his life. Poor beggar! The black bishop is the symbol of his sorrow. Fate, by Gad! It's the one chessman that defeated the Pardee gambit. Bishop-to-Knight-5 always broke up his calculations--disqualified his pet theory--made
a hissing and a mocking of his life's work. And now, with a chance to break even with the great Rubinstein, the bishop crops up again and drives him back into obscurity."
A few minutes later we took our departure and walked to West End Avenue, where we hailed a taxicab.
"It's no wonder, Vance," commented Markham, as we rode down-town, "that Pardee went white the other afternoon when you mentioned the black bishop's being at large at midnight. He probably thought you were deliberately insulting him--throwing his life's failure in his face."
"Perhaps. . . ." Vance gazed dreamily out into the gathering shadows. "Dashed queer about the bishop being his incubus all these years. Such recurring discouragements affect the strongest minds sometimes; create a desire for revenge on the world, with the cause of one's failure exalted to an Astraean symbol."
"It's difficult to picture Pardee in a vindictive rôle," objected Markham. Then, after a moment: "What was your point about the discrepancy in time between Pardee's and Rubinstein's playing? Suppose Rubinstein did take forty-five minutes or so to work out his combination. The game wasn't over until after one. I don't see that your visit to Arnesson put us ahead in any way."
"That's because you're unacquainted with the habits of chess players. In a clock game of that kind no player sits at the table all the time his opponent is figuring out moves. He walks about, stretches his muscles, takes the air, ogles the ladies, imbibes ice-water, and even indulges in food. At the Manhattan Square Masters Tournament last year there were four tables, and it was a common sight to see as many as three empty chairs at one time. Pardee's a nervous type. He wouldn't sit through Rubinstein's protracted mental speculations."
Vance lighted a cigarette slowly.
"Markham, Arnesson's analysis of that game reveals the fact that Pardee had three-quarters of an hour to himself around midnight."
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