Markham went to the window and stood, his hands behind him, looking down into the little paved rear yard. After several minutes he turned slowly.

“The situation, as I see it,” he said, “boils down to this:—The Odell girl has an engagement for dinner and the theatre with a man of some distinction. He calls for her a little after seven, and they go out together. At eleven o’clock they return. He goes with her into her apartment and remains half an hour. He leaves at half past eleven and asks the phone operator to call him a taxi. While he is waiting the girl screams and calls for help, and, in response to his inquiries, she tells him nothing is wrong and bids him go away. The taxi arrives, and he departs in it. Ten minutes later some one telephones her, and a man answers from her apartment. This morning she is found murdered, and the apartment ransacked.”

He took a long draw on his cigar.

“Now, it is obvious that when she and her escort returned last night, there was another man in this place somewhere; and it is also obvious that the girl was alive after her escort had departed. Therefore, we must conclude that the man who was already in the apartment was the person who murdered her. This conclusion is further corroborated by Doctor Doremus’s report that the crime occurred between eleven and twelve. But since her escort did not leave till half past eleven, and spoke with her after that time, we can put the actual hour of the murder as between half past eleven and midnight. … These are the inferable facts from the evidence thus far adduced.”

“There’s not much getting away from ’em,” agreed Heath.

“At any rate, they’re interestin’,” murmured Vance.

Markham, walking up and down earnestly, continued:

“The features of the situation revolving round these inferable facts are as follows:—There was no one hiding in the apartment at seven o’clock—the hour the maid went home. Therefore, the murderer entered the apartment later. First, then, let us consider the side door. At six o’clock—an hour before the maid’s departure—the janitor bolted it on the inside, and both operators disavow emphatically that they went near it. Moreover, you, Sergeant, found it bolted this morning. Hence, we may assume that the door was bolted on the inside all night, and that nobody could have entered that way. Consequently, we are driven to the inevitable alternative that the murderer entered by the front door. Now, let us consider this other means of entry. The phone operator who was on duty until ten o’clock last night asserts positively that the only person who entered the front door and passed down the main hall to this apartment was a man who rang the bell and, getting no answer, immediately walked out again. The other operator, who was on duty from ten o’clock until this morning, asserts with equal positiveness that no one entered the front door and passed the switchboard coming to this apartment. Add to all this the fact that every window on this floor is barred, and that no one from up-stairs can descend into the main hall without coming face to face with the operator, and we are, for the moment, confronted with an impasse.”

Heath scratched his head, and laughed mirthlessly.

“It don’t make sense, does it, sir?”

“What about the next apartment?” asked Vance, “the one with the door facing the rear passageway—No. 2, I think?”

Heath turned to him patronizingly. “I looked into that the first thing this morning. Apartment No. 2 is occupied by a single woman; and I woke her up at eight o’clock and searched the place. Nothing there. Anyway, you have to walk past the switchboard to reach her apartment the same as you do to reach this one; and nobody called on her or left her apartment last night. What’s more, Jessup, who’s a shrewd sound lad, told me this woman is a quiet, ladylike sort, and that she and Odell didn’t even know each other.”

“You’re so thorough, Sergeant!” murmured Vance.

“Of course,” put in Markham, “it would have been possible for some one from the other apartment to have slipped in here behind the operator’s back between seven and eleven, and then to have slipped back after the murder. But as Sergeant Heath’s search this morning failed to uncover any one, we can eliminate the possibility of our man having operated from that quarter.”

“I dare say you’re right,” Vance indifferently admitted. “But it strikes me, Markham old dear, that your own affectin’ recapitulation of the situation jolly well eliminates the possibility of your man’s having operated from any quarter. … And yet he came in, garroted the unfortunate damsel, and departed—eh, what? … It’s a charmin’ little problem. I wouldn’t have missed it for worlds.”

“It’s uncanny,” pronounced Markham gloomily.

“It’s positively spiritualistic,” amended Vance. “It has the caressin’ odor of a séance. Really, y’ know, I’m beginning to suspect that some medium was hovering in the vicinage last night doing some rather tip-top materializations. … I say, Markham, could you get an indictment against an ectoplasmic emanation?”

“It wasn’t no spook that made those finger-prints,” growled Heath, with surly truculence.

Markham halted his nervous pacing and regarded Vance irritably.

“Damn it! This is rank nonsense. The man got in some way, and he got out, too. There’s something wrong somewhere. Either the maid is mistaken about some one being here when she left, or else one of those phone operators went to sleep and won’t admit it.”

“Or else one of ’em’s lying,” supplemented Heath.

Vance shook his head. “The dusky fille de chambre[39], I’d say, is eminently trustworthy. And if there was any doubt about any one’s having come in the front door unnoticed, the lads on the switchboard would, in the present circumstances, be only too eager to admit it. … No, Markham, you’ll simply have to approach this affair from the astral plane, so to speak.”

Markham grunted his distaste of Vance’s jocularity.

“That line of investigation I leave to you with your metaphysical theories and esoteric hypotheses.”

“But, consider,” protested Vance banteringly. “You’ve proved conclusively—or, rather, you’ve demonstrated legally—that no one could have entered or departed from this apartment last night; and, as you’ve often told me, a court of law must decide all matters, not in accord with the known or suspected facts, but according to the evidence; and the evidence in this case would prove a sound alibi for every corporeal being extant. And yet, it’s not exactly tenable, d’ ye see, that the lady strangled herself. If only it had been poison, what an exquisite and satisfying suicide case you’d have! … Most inconsiderate of her homicidal visitor not to have used arsenic instead of his hands!”

“Well, he strangled her,” pronounced Heath. “Furthermore, I’ll lay my money on the fellow who called here last night at half past nine and couldn’t get in. He’s the bird I want to talk to.”

“Indeed?” Vance produced another cigarette. “I shouldn’t say, to judge from our description of him, that his conversation would prove particularly fascinatin’.”

An ugly light came into Heath’s eyes.

“We’ve got ways,” he said through his teeth, “of getting damn interesting conversation outa people who haven’t no great reputation for repartee.”

Vance sighed. “How the Four Hundred needs you, my Sergeant!”

Markham looked at his watch.

“I’ve got pressing work at the office,” he said, “and all this talk isn’t getting us anywhere.” He put his hand on Heath’s shoulder. “I leave you to go ahead. This afternoon I’ll have these people brought down to my office for another questioning—maybe I can jog their memories a bit. … You’ve got some line of investigation planned?”

“The usual routine,” replied Heath drearily. “I’ll go through Odell’s papers, and I’ll have three or four of my men check up on her.”