“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.”
“Yes?”
“To Cayley the clothes seemed an enormously important clue. I didn’t quite see why, but I did realize that to a man in Cayley’s position the smallest clue would have an entirely disproportionate value. For some reason, then, Cayley attached this exaggerated importance to the clothes which Mark was wearing on that Tuesday morning; all the clothes, the inside ones as well as the outside ones. I didn’t know why, but I did feel certain that, in that case, the absence of the collar was unintentional. In collecting the clothes he had overlooked the collar. Why?”
“It was the one in the linen-basket?”
“Yes. It seemed probable. Why had Cayley put it there? The obvious answer was that he hadn’t. Mark had put it there. I remembered what you told me about Mark being finicky, and having lots of clothes and so on, and I felt that he was just the sort of man who would never wear the same collar twice.” He paused, and then asked, “Is that right, do you think?”
“Absolutely,” said Bill with conviction.