Labyrinths of Reason (14 page)

As we shared a late supper that night, I remarked that I had been analyzing our old adventures at my club. “Holmes, I think you are one of the most misunderstood men in Britain. All think your renown is due to the difficulty of your cases. I believe that what made the tales popular is that the solutions were so
simple.”

Holmes put his fingertips together, his face now betraying amusement. “You believe the British public wants to hear of simple-minded detection?”

Feeling pleasantly articulate from the port, I continued: “The public likes a case where the solution, once stated, is pellucidly obvious and self-evident … however difficult it may have been to come up with that solution. Only when the solution is so obviously correct can the reader kick himself for not having thought of it himself.”

“All
problems are easy in retrospect,” Sherlock Holmes countered. “It is like solving a labyrinth by going backward from the goal.”

“No,” I objected. “I must differ.
Some
labyrinths are just as difficult when you start from the goal. There are many problems whose solutions are as abstruse as the problems themselves. If you had gone around hanging men on dreary ballistics and fingerprints, the way Scotland Yard does most of the time, my accounts of your exploits would not have found one-tenth the audience they did. The public wants a readily understandable solution.”

“An interesting point,” Holmes said dreamily. “I subscribe to several esoteric journals to ease the tedium of the apiarist’s life. I was reading in one of them that William Shanks, a mathematician of our fair island, has recently computed pi to 707 decimal places. It took him twenty years. His result filled a whole page with quite senseless, random numbers. Should anyone doubt Mr. Shanks’s result, he would have to budget an equal amount of time and duplicate his work. In that case also, verifying the answer would be precisely as difficult as coming up with the answer in the first place—the very antithesis of an ‘obvious’ solution.”
¹

“Precisely. I was telling this to a school chum in London, and he said it’s like this riddle: What common English word starts and ends with the letters
UND
? It’s difficult to think of the word, but once you think of it, you can’t doubt you’ve got the right answer. It’s no fair checking in the dictionary, of course.”

Holmes wrinkled his forehead at this but said nothing.

“I told several acquaintances at the club that I was visiting you and wanted to pass the time with a few riddles. They gave me some good, tough puzzles. They’re all the sort you seem to specialize in, where the answer is obvious once you see it. That way, you can mull
them over for weeks if need be, and I needn’t be here to tell you you’re correct.”

“Weeks? I hardly think so.”

After the meal, I directed Holmes into the guest room next to mine. It had known little use under Holmes’s tenancy and was sparsely furnished. That afternoon, I had removed the bed and chair, leaving it quite bare.

Hanging from the ceiling were two lengths of string, each six feet long. The strings were ten feet apart. As the room was also ten feet high, the lower ends of the strings hung four feet above the floor.

The only other feature of the room was a eclectic assortment of objects laid out on the floor. There was a Swiss Army knife, a firecracker, a small vial of ether, a twenty-five-pound block of ice, and a tortoiseshell kitten. The ice was in a pan to avoid harm to the Indian rug.

“I concede, Watson. What are you up to?” Holmes asked.

“The problem,” I said, “is to tie the two ends of the string together. You will notice that the distance between the hanging strings is about four feet greater than your arm span. While holding one string, you cannot touch any part of the other string. All you are allowed to use in your solution are the Swiss Army knife, the firecracker, the ether, the ice, and—
or
the kitten. You may hot use the curtain rods, wallpaper, the carpeting, or anything else in this room. That includes clothing and objects on your person.”

Holmes’s eyes inspected the floor and ceiling minutely. “The ladder you used to hang the string had a loose third rung.”

Ignoring this, I continued: “I have tied the strings to the ceiling fixtures with simple slip knots. They will not support your weight. As a hint, I can only remind you that it’s one of your infuriating puzzles where the solution, difficult though it may be to deduce, is absurdly simple after you see it.”

Holmes spent a few moments in silent introspection. Then he asked, “The second puzzle?”

“Your next problem,” I said, “is one that was recently written up in the papers by Henry Ernest Dudeney. I spent some time mulling
this over, only to hear that it has no solution at all. Then I thought it over some more, and concluded that there is a solution.”

I showed Holmes the diagram that I had clipped from the paper. “There are three houses and three utility companies—gas, water, and electricity. Each company desires to lay pipes or cable to each house without any pipe or cable intersecting the path of any other. The paths of the pipes can bend and may be wastefully indirect; they cannot cross.”

Holmes scarcely glanced at the clipping. “I am quite familiar with that puzzle, Watson. It is older than electric lighting or even gas lighting. Less modernistic versions spoke of paths to a dovecote, a well, and a haystack. I can assure you you are mistaken if you think you have a solution. It can’t be done.”

“I think you will agree with me that there
is a
solution nonetheless.”

Holmes sighed tolerantly. “I am aware of the
unfair
solutions that have been offered. One pipe could go through one house. The water pipe could be concentrically embedded in the gas pipe. I confess a certain admiration for the cleverness of these schemes, but I trust you appreciate that they are cheats. They violate the essentially topological spirit of the original puzzle. The houses and utilities should be regarded as dimensionless points, the pipes as curves of zero breadth.”

“I could not agree with you more. There is a solution that is true to this topological spirit of which you speak.”

My third puzzle went like this: “A certain large business firm has 1000 employees, and a peculiar method of terminating them,” I began. “Never is anyone told that he is fired. Each employee scheduled
for termination is allowed to deduce his impending fate and resign rather than be fired.

“All the company’s employees live in constant fear of losing their jobs. Rumors of impending firings spread instantaneously through the company. This rumor mill is completely accurate. There is such a steady stream of firings that no one invents falsehoods out of malice or boredom. Whenever someone is scheduled to be let go, everyone in the company knows about it except for the unfortunate person himself. He is literally the last to know. No one has the grit to tell anyone that he is going to be fired, and everyone has learned through constant practice to act precisely as if nothing is wrong when in the company of a doomed fellow worker.

“This environment of gossip and duplicity has made the logical powers of the employees all the keener. Each man lies awake in bed each night thinking over what he heard and what he didn’t hear, hashing over all possible hypotheses concerning his position in the company. No nuance, no slight goes unrecognized or unpondered. All the employees being quite bright (and quite paranoid), no one fails to see all logical implications of any action. If an employee deduces that he is to be fired, he hands in his resignation the first thing the next morning.

“One day the company was acquired by a larger firm. The manager of the larger company called a meeting of all the employees and said, ‘It’s time somebody trimmed the fat over here. Heads will roll!’ The manager did not say who was to be fired. He did not even say how many people were to go. As always, there were no secrets to the company grapevine. Immediately after the meeting, the grapevine learned who was to go. What happened next?”

“What do you mean, what happened?” Holmes asked.

“A rather beautiful deduction about the firings is possible. The puzzle is to see what that deduction is.”

“There isn’t enough information!”

“The appeal of this little conundrum, Holmes, is how much can be deduced from a minimum of information.”

Holmes appeared to toy with several ideas and reject them all. “I suppose some of the wretches could figure out what was up from the way other people acted.”

“No, no, you miss the point. They are consummate actors all, and so spineless that they would not tell their best friend of his fate.”

“I have noted that the pupils of the eye frequently betray the most practiced liar—”

“I said nothing about pupils, so it can’t be relevant.”

“The employees cannot get together and pool their knowledge?”

“Not other than in the company rumor mill as I have described it. No one, under any circumstances, tells anyone that he is to be fired, or allows him to be told by a third party outside the company.”

“Anonymous letters?”

“Not permitted.”

“Speaking of anonymous letters: A man gets an unsigned letter telling him to go to the local graveyard at midnight. He does not generally pay attention to such things, but complies out of curiosity. It is a deathly still night, lighted by a thin crescent moon. The man stations himself in front of his family’s ancestral crypt. The man is about to leave when he hears scraping footsteps. He yells out, but no one answers. The next morning, the caretaker finds the man dead in front of the crypt, a hideous grin on his face.

“Did the man vote for Teddy Roosevelt in the 1904 U.S. presidential election?”

“Well!” Holmes said with greater enthusiasm. “At last a problem open to
logical
solution!”

I next produced a set of three cardboard figures from my doctor’s bag: a triangle, a square with one quarter missing, and an entire square. “In a certain part of the American desert, there were three landowners I will call Smith, Jones, and Robinson. Smith had three sons, Jones had four, and Robinson had five. Americans being very democratic, they divide their estates equally among all their heirs.

“Smith’s property was in the form of a regular triangle. He did not want to favor any of his three sons, so he asked the county surveyor to divide the land into three tracts of exactly the same size and shape. That the surveyor was able to do.” With a pen I sketched the division on the cardboard triangle.

“Jones, with four sons, had this L-shaped property, three-quarters of a square. After much deliberation, the surveyor divided it into four parcels, each the same size and shape.

“Finally, Robinson, with five sons, had a perfectly square property. He asked the surveyor to divide it into five identical pieces.
The surveyor found the problem quite intractable. He could not put it aside, and he neglected his other work. He ended up tearing most of his hair out, and they had to feed him with a spoon. Your final puzzle is to divide a square into five pieces, all exactly the same size and shape. It
is
possible, but I warn you that there is only one way of doing it.

“I hope these diversions occupy you. I plan to retire for the evening. Please don’t stay up all night. If you do, I would appreciate not being woken when you hit on the answers. You shan’t need my verification anyway. The correct solutions will be as plain as the nose on your face.”

As I left Holmes, he was seated at the deal-topped table, scribbling notes and ignoring me.

I had bad dreams, occasioned, I think, by the weird strains of a violin playing through the night. I arose at eight the next morning. I first peeked into the empty room next door. The two strings attached to the ceiling now hung knotted just above my head.

I was relieved to find the kitten fit and unharmed. It was thoughtless of me to employ it as misdirection! Holmes’s compassion for animals was not excessive, I recalled.