Bill burst in eagerly on the top of the sentence.  “Yes, now what was all that about?  You were so damn Sherlocky yesterday all of a sudden.  We’d been doing the thing together all the time, and you’d been telling me everything, and then suddenly you become very mysterious and private and talk enigmatically—­is that the word?—­about dentists and swimming and the ‘Plough and Horses,’ and—­well, what was it all about?  You simply vanished out of sight; I didn’t know what on earth we were talking about.”

Antony laughed and apologized.

“Sorry, Bill.  I felt like that suddenly.  Just for the last half-hour; just to end up with.  I’ll tell you everything now.  Not that there’s anything to tell, really.  It seems so easy when you know it—­so obvious.  About Mr. Cartwright of Wimpole Street.  Of course he was just to identify the body.”

“But whatever made you think of a dentist for that?”

“Who could do it better?  Could you have done it?  How could you?  You’d never gone bathing with Mark; you’d never seen him stripped.  He didn’t swim.  Could his doctor do it?  Not unless he’d had some particular operation, and perhaps not then.  But his dentists could—­at any time, always—­if he had been to his dentist fairly often.  Hence Mr. Cartwright of Wimpole Street.”

Bill nodded thoughtfully and went back again to the letter.

“I see.  And you told Cayley that you were telegraphing to Cartwright to identify the body?”

“Yes.  And then of course it was all up for him.  Once we knew that Robert was Mark we knew everything.”

“How did you know?”

Antony got up from the breakfast table and began to fill his pipe.

“I’m not sure that I can say, Bill.  You know those problems in Algebra where you say, ‘Let x be the answer,’ and then you work it out and find what x is.  Well, that’s one way; and another way, which they never give you any marks for at school, is to guess the answer.  Pretend the answer is 4—­well, will that satisfy the conditions of the problem?  No.  Then try 6; and if 6 doesn’t either, then what about 5?—­and so on.  Well, the Inspector and the Coroner and all that lot had guessed their answer, and it seemed to fit, but you and I knew it didn’t really fit; there were several conditions in the problem which it didn’t fit at all.  So we knew that their answer was wrong, and we had to think of another—­an answer which explained all the things which were puzzling us.  Well, I happened to guess the right one.  Got a match?”

Bill handed him a box, and he lit his pipe.

“Yes, but that doesn’t quite do, old boy.  Something must have put you on to it suddenly.  By the way, I’ll have my matches back, if you don’t mind.”

Antony laughed and took them out of his pocket.

“Sorry ....  Well then, let’s see if I can go through my own mind again, and tell you how I guessed it.  First of all, the clothes.”