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A Fancy Riddle and Three Kinds of Reasoning

Riddle Recap

Last month, the riddle below was posted in Nerds' Corner for your consideration and subsequent titillation. In this article, I will reveal the solution to the riddle and show you what Andy Hunter would look like holding an enormous corndog. Then I will use the solution to the riddle to introduce and explain the form of reasoning around which most scientific controversy centers.

Awww, yeah.

If you've already read the riddle in part one of this article, here's the answer if you're ready for it. If you haven't read the riddle, or you want to give it some more thought, continue on.

Four Dudes in the Sand

Four teammates in a Survivor-esque reality TV show are competing for instant celebrity and $1,000,000,000,000,000,000,000,000,000,000. At the moment, they are buried in the sand, up to their necks.

One teammate is buried on one side of a brick wall, and the other three are buried on the other side of the wall. All of them are facing the wall, and none of them can turn his head. The host of the show has painted X's on the backs of their heads — white X's on teammates 1 and 3, and black X's on teammates 2 and 4.

Because teammate 4 is buried behind teammates 3 and 2, teammate 4 can see the white X on teammate 3, and he can see the black X on teammate 2. Because he's buried behind teammate 2, teammate 3 can see the black X on teammate 2. Because he's buried in front of teammates 3 and 4 and facing the wall, teammate 2 cannot see any X's on anybody. Teammate 1 obviously cannot see any X's on anybody. None of the teammates has any way of seeing his own X or gaining any information about the color of his X from shadows, reflections in the TV cameras, paint spilled on the ground, etc.

The host tells the teammates that they will be eliminated in an hour unless one of them can figure out the color of his own X. Each teammate knows that there are four teammates total, and each teammate is told that two of them are wearing a white X and two of them are wearing a black X. Each teammate, therefore, knows that his own X is either black or white, but none of them knows which it is.

So, here is what each teammate knows:

Teammate 4: I have either a black or a white X on my head, as do the other teammates. Teammate 2 has a black X on his head, and teammate 3 has a white X on his head.

Teammate 3: I have either a black or a white X on my head, as do the other teammates. Teammate 2 has a black X on his head.

Teammate 2: I have either a black or a white X on my head, as do the other teammates.

Teammate 1: I have either a black or a white X on my head, as do the other teammates.

The team will be given exactly one chance to speak. The first teammate to make a noise must immediately state the color of his own X and explain exactly how he knows what color his X is. If he is wrong about the color of his X, the team will be eliminated. If he says anything that doesn't explain how he knows what color it is, the team will be eliminated. If his explanation gives the host the impression that he was guessing about the color of his X and got lucky, the team will be eliminated. If all of the teammates remain silent for the full hour, the team will be eliminated.

One of the teammates does figure out the color of the X painted on the back of his head and he is able to give the host a satisfactory explanation for how he knows.

Which teammate figures it out, and what is his explanation?

Know the answer? If you need time, don't read any further — go back and take another look at the facts. Of course, if you want the answer, then continued reading is OK.

And the Winner is …

In spite of the fact that teammate 4 has the most information, teammate 3 is the only one in a position to figure out the color of his own X. Here's how he does it:

The moment the clock starts, teammate 3 brings to mind everything he knows that's relevant to the situation and quickly discovers that he doesn't know very much. He knows there are four X's total — two black and two white. He knows that, if he knew the colors of the other teammates' X's, then he could deduce the color of his own X. For example, if he knew that teammate 1 was wearing a white X and knew that teammates 2 and 4 were wearing black X's, then he would be able to deduce that he was wearing a white X.

But, of course, he doesn't know what color the X's of teammates 1 and 4 are. Teammate 3 can only see the X on teammate 2, which is black.

After a while, teammate 3 realizes that if he knew that teammate 4 was also wearing a black X, then he really wouldn't need to know what color teammate 1 was wearing on the other side of the wall. That's because in this case teammates 2 and 4 would both be wearing black X's, from which it would follow that he would be one of the two teammates wearing white X's. And the same would be the case if teammate 3 knew that teammate 1 was wearing a black X. But, of course, he neither knows the color of teammate 4's X nor the color of teammate 1's X, and all that follows from the fact that teammate 2 is wearing a black X is that, out of himself and teammates 1 and 4, one of them is wearing a black X and the other two are wearing white X's.

Teammate 3 realizes that he has no way to deduce the color of his own X, so he sits there, waiting, hoping that one of the other teammates will figure it out and thereby win $1,000,000,000,000,000,000,000,000,000,000 for the whole group.

After a few more minutes, however, teammate 3 starts to suspect that there's something significant about the fact that he's still sitting there, buried in the sand. After all, he thinks, the riddle shouldn't be that hard for teammate 4, since he can see my X and the X on teammate 2. … Teammate 2 is wearing a black X. If I'm also wearing a black X, then teammate 4 can see that teammate 2 and I both have black X's, and from this he can easily deduce the color of his own X — namely, white. Why hasn't teammate 4 deduced the color of his own X, then?

No sooner does teammate 3 ask this question than the answer seems obvious to him: If he — teammate 3 — was wearing a black X, then teammate 4 would surely have figured out the color of his own X, and all of them would have $1,000,000,000,000,000,000,000,000,000,000. The only explanation for the fact that we're all still sitting here in the sand is that I'm wearing a white X.

Teammate 3 raises his eyebrows at the game show host, trying to grab his attention in order to tell him exactly how he has solved the riddle.

But then he realizes that this isn't the only explanation for the fact that they're all still sitting there, buried in the sand. In fact, he realizes that there are hundreds, perhaps even thousands of explanations for this fact.

Perhaps I am wearing a black X and teammate 4 is just being a bit slow to realize that, if teammate 2 and I are both wearing black X's, then he must have a white X on his own head. Perhaps teammate 4 has lost his vision due to dehydration and can't see the color of our X's. Perhaps teammate 4 is a narcoleptic and he's fallen fast asleep. Perhaps teammate 4 knows the answer but can't get the attention of the host. Perhaps teammate 4 knows the answer but he's intent on playing mind games with us and isn't going to speak up until one minute before the hour is up. Perhaps teammate 4 has suddenly lost his interest in money and intends to lose it on purpose. Perhaps I'm really just a brain in a vat and this whole game is an illusion; teammate 4 doesn't really even exist; the computer that's hooked up to my brain is fabricating his appearance and running a program in which he remains silent; Alpha Centurion super-scientists are studying my every response. …

Teammate 3 decides not to answer and resolves instead to wait, hoping that he's wearing a black X, and hoping that teammate 4 will give the host the straightforward deductive explanation he has been considering in his head.

But nothing happens.

Twenty minutes pass, then 30, then 40, 50 and 55. Finally, teammate 3 reasons that, since teammate 4 has remained silent for the first 55 minutes, he's going to let the entire hour pass without saying anything. Teammate 3 decides he should probably try to solve the riddle. He realizes he can't be 100 percent certain that he's wearing a white X — after all, he's got little to go on beyond the fact that teammate 4 hasn't solved the riddle, and there are many possible explanations for this fact.

Nevertheless, teammate 3 is confident that his X is white, and he's confident that the host will approve of his explanation. After all, he reasons, even though other explanations are possible, my wearing a white X is clearly the best explanation available. And besides, only five minutes remain. If I don't say something, we're going to be eliminated anyway, so I really don't have anything to lose.

Teammate 3 can't be absolutely positive that he's wearing a white X, and he can't, via the explanation he'll give to the host, prove that he's wearing a white X, but he figures his reasons are strong enough.

Teammate 3 speaks up and explains to the host that, if he had been wearing a black X, then teammate 4 would probably have solved the puzzle; therefore he — teammate 3 — is probably wearing a white X. The host accepts this answer as both true and rational, and the teammates win instant celebrity and $1,000,000,000,000,000,000,000,000,000,000.

Look. Andy Hunter with an enormous corndog.

Three Kinds of Reasoning

So, there's actually a point to this little story, and it has to do with a form of reasoning that's integral to human judgment yet rarely mentioned by name or given any explanation. It's an important form of reasoning. Among other things, it's the form of reasoning by which juries conclude that defendants are guilty, and it's the form of reasoning around which the most heated scientific controversies center.

It's also the form of reasoning that allowed teammate 3 to make good use of teammate 4's silence.

Let me explain.

In order to figure out the color of his own X, teammate 3 relied heavily on three kinds of reasoning: induction, deduction and abduction.¹ But he was only able to engage fruitfully in abductive reasoning after first engaging in inductive and deductive reasoning. What are inductive, deductive and abductive reasoning, then?

One reasons inductively when one reasons from parts to wholes, or from particular cases to general rules. The following is an example of inductive reasoning:

Micah writes for TrueU, and he wears tube socks; Denise writes for TrueU, and she wears tube socks; Blake writes for TrueU, and he wears tube socks; therefore, all TrueU writers wear tube socks.

In contrast to inductive reasoning, one reasons deductively when one reasons from wholes to parts, or from general rules to particular cases. For example:

Everyone who writes for TrueU wears tube socks; Lauren Winner writes for TrueU; therefore, Lauren Winner wears tube socks.

Abductive reasoning bears obvious formal similarities to inductive and deductive reasoning, but one engages in abductive reasoning for wholly different reasons. Abductive reasoning consists in reasoning from a particular case to an explanation (hopefully, the best explanation) for that case. The following is an example of abductive reasoning:

Michael Moore wears tube socks; everyone who writes for TrueU wears tube socks; therefore, Michael Moore writes for TrueU.

In the story above, teammate 3 engaged in inductive reasoning when he reasoned from the fact that teammate 4 had not said anything in the first 55 minutes to the conclusion that teammate 4 wasn't going to say anything throughout the whole hour.

Teammate 3 engaged in deductive reasoning when he reasoned from the fact that two of the teammates were wearing white X's and two of the teammates were wearing black X's to the conclusion that, if he and teammate 2 were both wearing black X's, then teammates 1 and 4 were both wearing white X's.

Finally, teammate 3 engaged in abductive reasoning when he reasoned from the fact that teammate 4 had not answered the riddle to the best explanation for that fact that he — teammate 3 — was wearing a white X.

Abductive reasoning served teammate 3 well: It led him to the correct conclusion, and in such a way that we're inclined to think he arrived at it rationally. But as the Michael Moore example demonstrates, abduction can also lead us wildly astray.

As a matter of fact, many TrueU authors don't wear tube socks. And so far as I know, neither does Michael Moore. But even if both Michael Moore and all TrueU authors did wear tube socks, we still couldn't rationally reason from Michael Moore wearing tube socks to his writing for TrueU. Tube-sock-wearing obviously isn't evidence for TrueU writing, even if writing for TrueU would explain why one wears tube socks.

C O F F E E  S H O P

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What do you think about Blake's answer to the riddle and the limits of abductive reasoning?

Join the discussion!

In any case, understanding the nature and limits of abductive reasoning is essential to understanding scientific controversy in general. I'll address abductive reasoning further in my next article. And because I'm pretty sure that not reading my next article is actually a carcinogen, I strongly suggest that you read it.