“Then how do you know?”

“You’re the perfect Watson, Bill.  You take to it quite naturally.  Properly speaking, I oughtn’t to explain till the last chapter, but I always think that that’s so unfair.  So here goes.  Of course, I don’t really know that he’s got it, but I do know that he had it.  I know that when I came on him this afternoon, he had just locked the door and put the key in his pocket.”

“You mean you saw him at the time, but that you’ve only just remembered it—­reconstructed it in the way you were explaining just now?”

“No.  I didn’t see him.  But I did see something.  I saw the key of the billiard-room.”

“Where?

“Outside the billiard-room door.”

“Outside?  But it was inside when we looked just now.”

“Exactly.”

“Who put it there?”

“Obviously Cayley.”

“But—­”

“Let’s go back to this afternoon.  I don’t remember noticing the billiard-room key at the time; I must have done so without knowing.  Probably when I saw Cayley banging at the door I may have wondered subconsciously whether the key of the room next to it would fit.  Something like that, I daresay.  Well, when I was sitting out by myself on that seat just before you came along, I went over the whole scene in my mind, and I suddenly saw the billiard-room key there outside.  And I began to wonder if the office-key had been outside too.  When Cayley came up, I told you my idea and you were both interested.  But Cayley was just a shade too interested.  I daresay you didn’t notice it, but he was.”

“By Jove!”

“Well, of course that proved nothing; and the key business didn’t really prove anything, because whatever side of the door the other keys were, Mark might have locked his own private room from the inside sometimes.  But I piled it on, and pretended that it was enormously important, and quite altered the case altogether, and having got Cayley thoroughly anxious about it, I told him that we should be well out of the way for the next hour or so, and that he would be alone in the house to do what he liked about it.  And, as I expected, he couldn’t resist it.  He altered the keys and gave himself away entirely.”

“But the library key was still outside.  Why didn’t he alter that?”

“Because he’s a clever devil.  For one thing, the Inspector had been in the library, and might possibly have noticed it already.  And for another—­” Antony hesitated.

“What?” said Bill, after waiting for him to go on.

“It’s only guesswork.  But I fancy that Cayley was thoroughly upset about the key business.  He suddenly realized that he had been careless, and he hadn’t got time to think it all over.  So he didn’t want to commit himself definitely to the statement that the key was either outside or inside.  He wanted to leave it vague.  It was safest that way.”

“I see,” said Bill slowly.