Labyrinths of Reason (15 page)

I found Holmes lying on the couch in the drawing room in a blue haze of tobacco smoke. He was still wearing last night’s clothes. “All but one were quite trivial,” he announced.

“Really?” I sat down at the table. A thousand mad diagrams dissected a thousand squares into crazy quilts. At the top of one sheet of paper was the desperate neologism
UNDACHSHUND
, heavily crossed out. With difficulty I refrained from commenting. Below it was the correct answer.

“Which did you get first?”

“I got the
UND
puzzle
last.”

“That was supposed to be the easiest.”

“So I thought as well,” Holmes conceded. “The riddle is hard because there is no systematic way of solving it. If inspiration fails, the best you can do is to run through all the possible combinations of letters beginning and ending with
UND
.

“Look here,” Holmes said, proffering a sheet of paper covered with letters. “Since
UND
is not a word, nor
UNDUND
, we have
UNDAUND, UNDBUND, UNDCUND
,
and so on through
UNDZUND
.
If none of these 26 seven-letter combinations are common English words (they aren’t, I quickly found), you must run through the eight-letter groups:
UNDAAUND, UNDABUND … UNDZZUND
.
This time, however, there are 26 times 26 combinations. That’s 676 all told.”

“And you still won’t have the answer,” I added.

“No. As you check longer words, the number of combinations goes up in a geometric progression. The correct answer, you will observe, is long enough so that one would have to check
millions
of letter combinations to stumble on it. That is why I find this problem unfair. No one could ever solve this problem logically; it’s too taxing.”

“How did you get the answer?”

“A lucky guess. The so-called subconscious mind. Either of
which is unsatisfying. I had hoped to deduce it logically. One moment all was darkness; the next,
UNDERGROUND
popped into my head.”
²

“There were perhaps other instances of a lucky guess triumphing over deduction?” I suggested.

“Tying the string? To a degree. By now I recognize a red herring when I see one, Watson. You must not be too upset when I tell you I suspected right off that the more fanciful of your set of objects were quite likely irrelevant. The clever thing on your part was that the solution depends not on a specific object but on any of them.

“I used the Swiss Army knife. The bottle of ether would work as well, and the firecracker or ice might serve the purpose. The cat would wriggle—I suppose the ether could remedy that. I tied the knife to a string and set it to swinging. Then I grabbed the other string, caught the knife, and tied the strings in a graceful parabola. It is simplicity itself—in retrospect.”

“A graceful catenary,” I corrected.

“I commend you on bringing the gas, water, and electricity puzzle to my renewed attention,” Holmes said. “I gather this is what you had in mind? Your topological solution?” He produced a neat pencil diagram of my solution.

Holmes explained: “The riddle is given as a problem on a plane. That the earth is actually a sphere makes no difference. Any network of points and lines on a sphere is equivalent to a network on a plane, since the sphere may be ‘punctured’ at the antipodes and deformed to a plane. The paths of the pipes need not cross on certain other topological surfaces. The problem is soluble on a Möbius strip or on a torus—a doughnut shape, with a hole in it. Any natural tunnel renders the earth a torus. ‘Natural bridge’ or ‘window rock’ formations, caverns and sea grottoes with two openings, blowholes, prairie-dog burrows—any will do. The tunnel is, in effect, a free crossing. If you have two pipes that must otherwise cross, you may route one pipe through the tunnel, and the other over the mountain, so to speak.

“The torus hole must be within the network of pipes. It is presumptuous to assume a priori that any tunnel or burrow exists in the vicinity of the houses and the utility companies. This threw me for a while. Then I realized that if the mountain won’t come to
Mohammed, Mohammed may go to the mountain. It is proper to assume that the puzzle takes place on this earth, and there are many natural tunnels in the earth. The utility companies could snake three of their pipes out to the nearest such tunnel or burrow and then back to the houses.”

“I trust you got the one about the company rumor mill,” I said. “That was pure deduction, I should say.”

“It was most singular. The answer is pure deduction, and yet I am not at all sure that I deduced. I fear it was another fortunate guess.”

“A guess?”

“The Medicis were said to have had a slow poison which proved fatal only after as many days as it had been cured in the sun. If one wanted a collateral heir or indiscreet mistress to die in fifteen days, one administered a preparation that had baked in the Florentine sun for fifteen days. The formula has been lost to us—”

“As a medical man, I say that’s a fairy tale. What about the riddle?”

“I mentioned the Medicis’ alleged poison only because thinking of it (quite by chance) led me to the solution. Everyone in the company scheduled to be fired will deduce it and resign on the same day. It will take as many days to arrive at this deduction as there are people to be fired. If the company is letting 79 people go, then on the 79th day after the announcement, all 79 people will resign.”

“And how did you come to that conclusion?” I asked.

“It is a wonderful—a monstrous—bit of logic. I simplified the puzzle by supposing that just one person is to be fired. The rumor mill learns his identity, and everyone in the company except that
person knows the situation. That night, the doomed man is tossing and turning in bed. He knows that someone is going to be fired. Isn’t it peculiar that he hasn’t heard who? The company grapevine is so efficient … The only possible conclusion is that he
and he alone
is to be fired. Were he one of a group of persons to be fired, he would have learned the names of the others. Consequently, this single unlucky man must resign the next morning. It is the only logical possibility.
If
just one person is to be fired, that is precisely what does happen.

“Allow instead that there are two persons to be fired. Thanks to the grapevine, everyone learns the name of at least one person to be fired and sleeps soundly the first night. Each employee can suppose that the one-person scenario I have just outlined is taking place. The
second
night after the announcement, the two persons to be fired are stricken with insomnia. Each thinks, ‘It’s too bad about so-and-so getting the sack. What I can’t figure is, why didn’t he resign this morning?’ All employees being perfectly logical and having ample time to consider the implications of actions, so-and-so could have failed to resign
only
if he knew the name of another employee to be terminated. Each of the two employees must conclude that that other employee can only be himself. Both employees must resign on the second morning after the announcement.

“Then it all fell into place. If three persons are to be fired, each can deduce his fate from the facts that no one resigned on the first and second mornings and that each knows only two persons who are scheduled to be fired. It makes no difference how many persons are involved, just as long as all can trust in the unflinching logic of their fellows. After 999 days with no resignations, all 1000 employees would spend a sleepless night concluding that the entire work force is to be released.”

“And the man in the graveyard?”

“I told you, Watson, that was the easy one.”

“To you perhaps. I’m not sure there is an objective way of saying what’s easy and what’s difficult.”

“I suppose you’re right. In any event, the answer is of course no, the man didn’t vote for Roosevelt. The solution depends on realizing that the aforesaid crescent moon cannot be visible in the middle of the night. Not from most parts of the world, anyway. It is shocking how many of the so-called educated class are unaware of this elemental fact known to every goatherd. The exception is in the polar regions, where the sun—and nearby crescent moon—can be
visible all through a twenty-four hour cycle. Therefore, if the man lives in the United States at all, he must live in Alaska, near or above the Arctic Circle. The citizens of the territory of Alaska are not permitted to vote for President. Whatever his politics, the man did not vote for Roosevelt.”
³

“Congratulations, Holmes,” I said. “Well then, it must be the land-division puzzle that baffles you.”

Holmes nodded. “It is that which is responsible for keeping me up all night. I feel this puzzle is qualitatively different from the others. With the others, the number of conceivable solutions is in some sense limited. But just as there are an infinity of lines in a plane, so are the possible dissections of a plane figure beyond number. Not only did I fail, I did not even see how to
start.”

“Are you ready to concede defeat?”

“Yes. Show me how to do it.”

“I have given it some thought, and think it best that I inform you by letter, once I am safely back in London.”

“Why?”

“You will not be happy.”

“Watson, tell me at once!”

Only upon arriving home the next day did I post this diagram to Holmes:

P
UZZLES AND PARADOXES are subtly related. In a puzzle just one of many conceivable hypotheses avoids creating a contradiction. That single hypothesis is the puzzle’s solution. In a paradox, no hypothesis at all is tenable.

Like raw oysters, logic puzzles are an idiosyncratic taste. Some find them challenging or amusing; others, annoying. An important question is whether there is any general method for solving logic problems. Is there a cut-and-dried procedure, a trick, a recipe that anyone can learn and use to tackle any logic problem? If there is, it would be invaluable in the scientific realm and elsewhere.

In practice, logic is a blend of step-by-step deduction and exhaustive search of possible hypotheses. The first approach is illustrated by a set of classical paradoxes.

When Theseus returned to Athens after slaying the Minotaur, his ship “was preserved by the Athenians down even to the time of Demetrius Phalereus, for they took away the old planks as they decayed, putting in new and stronger timber in their place,” wrote Plutarch. “This ship became a standing example among philosophers, for the logical question as to things that grow; one side holding that the ship remained the same, and the other contending that it was not the same.”