In The Matrix, Neo sees a black cat walking by. A second later, an eerie feeling creeps over him as the same cat walks by again, making the exact same movements. He finds out that a déjà vu is usually a “glitch in the Matrix,” meaning that their digital reality has been reprogrammed and is now misfiring.

They give a better example of a glitch in the Matrix in Beyond, the first short film in the Animatrix series. Some kids have found a “haunted house” where glass bottles shatter and reassemble, rain falls from a clear sky, broken light bulbs flicker, shadows aren’t attached to the objects that cast them, and they can jump from a height and stop before impact. (A team of “rodent exterminators” clears everyone out and repairs the glitch.)

Neo himself is also a glitch. The Architect tells him: “Your life is the sum of a remainder of an unbalanced equation inherent to the programming of the Matrix. You are the eventuality of an anomaly, which despite my sincerest efforts I have been unable to eliminate from what is otherwise a harmony of mathematical precision.”

When I talk about a glitch in the matrix, I’m not necessarily talking about Neo’s Matrix, but any kind of system that gives a certain perception of reality. The glitch is what shatters that perception, making you realize that the whole thing was an illusion.

During a lucid dream, you’re conscious, but at first you don’t know that you’re dreaming because your brain makes everything so real. However, it doesn’t get everything exactly right. If you just get the idea to test the dream world, you can easily find some glitches: lights that stay on when you flip the switch off, books and clocks that change when you look away, people who say things that don’t make sense, etc. Discovering one glitch tells you it’s all a dream.

Isaac Newton worked out a theory of gravity that held up well to the observations people could make at the time. However, his theory had a rather large glitch that he just swept under the rug. He was forced to assume that a gravitational field propagated at infinite speed. He hated this, but without a theory of relativity, he had no choice.

It was more than 100 years before before observations of Mercury’s orbit showed a glitch, paving the way for a new theory of gravity. Similar things are happening today, as theories of quantum mechanics are being developed to address glitches in classical mechanics at the subatomic level.

Life, the universe, and everything

What is the universe? Essentially, it’s just a very sophisticated program.

This program is made up a number of rules. Things such as “for every action, there is an equal and opposite reaction,” “F = ma,” “E = mc^2,” and so on. But the rules alone don’t do anything. They need some objects to act on. Objects such as you.

Human DNA consists of 3.2 billion pairs of nucleotide bases, with four possibilities for each. This caps the maximum number of genetically distinct people at 4^3,200,000,000 (and in fact far less, since most combinations won’t work).

At conception, your DNA was determined from one of these possibilities. Perhaps you were model #62,085,423,678,990,876,543,357,896,534,567,897,634,524,790,043,446,854,568,987,434,543.

But your DNA didn’t fully describe you, even at such a young age. There were many other factors, such as where and when you were born, what your family was like, etc. But regardless, you were just an object described by a handful of variables.

The objects plus the rules make the system, and now the program is running. But like all programs, it has glitches.

A simple glitch in Doom

I recently stumbled across this page about a bug with the “picked up a medikit” message in Doom. In their rush to revolutionize the first-person shooter genre, id Software apparently didn’t have time to test everything (or retest everything after last-minute changes).

When you pick up a “medikit,” you get an extra 25 health points, and you see a message saying “Picked up a medikit.” If your health was below 25 when you picked it up, the message was supposed to say “Picked up a medikit that you REALLY need!” However, since the code does the < 25 check after adding the 25 health points, the “REALLY need” message will never be displayed:

case SPR_MEDI:
  if (!P_GiveBody (player, 25))
    return;

  if (player->health < 25)
    player->message = GOTMEDINEED;
  else
    player->message = GOTMEDIKIT;
break;

The authors of that web page helpfully posted a corrected version of the code. Ironically, the corrected version is much worse than the original:

case SPR_MEDI:
  if (player->health < 25)
    player->message = GOTMEDINEED;
  else
    player->message = GOTMEDIKIT;

  if (!P_GiveBody (player, 25))
    return;
break;

If this had been done, the “Picked up a medikit” message would have been displayed even if it wasn’t actually picked up (like if you already had 100% health). The correct fix would have been to simply change 25 to 50 in the original code.

This is just meant to show how easy it is to introduce a glitch. In this case, it’s an easy fix, and it could be made without any complications. But it’s not always that easy.

A more complicated glitch in Pac-Man

Pac-Man theoretically has an infinite number of levels, with no ending. But because of a bug in level 256, it’s impossible to go any further.

The current level is stored as a single byte (8 bits), and therefore can’t get any higher than 255. When it tries to increment to 256, it rolls over to 0. But this actually doesn’t cause any problems, except for one big one with the fruit-drawing routine.

Normally, 0 to 7 fruits are shown at the bottom right, depending on what level you’re on. But when the level counter goes back to 0, the game attempts to draw 256 fruits, corrupting the right half on the screen and leaving an insufficient number of dots to finish the level.

How is this glitch different from the Doom glitch? First, it was much harder to catch. Who would think they needed to test 256 levels of Pac-Man? What player would spend enough quarters to even get close to that point? Why stop at 256? Why not 1,000 or 1,000,000? What about testing what happens when other high numbers get high, like lives, points, or time?

Second, there’s the question of how to fix it. When the level counter goes back to 0, you know it’s really level 256. But there’s no way to know the difference between levels 1 and 257. So how do you know whether to draw 0 fruits or 7? Or should they draw more than 7 fruits at the higher levels? Should they use two bytes to store the level?  Then they’d have the same issue at level 65,536. Should they use another bit to indicate the level is 256+, and just leave it at 7 fruits? Should they end the game after level 255? Whatever change they make, they have to retest it.

You can’t fix every bug

But the Pac-Man glitch is still a relatively small issue. Come on, it’s just about drawing a few pieces of fruit. But when programs grow in complexity, they rapidly become more difficult to fix.

In a software engineering class I took, we learned a surprising fact about fixing bugs in a sufficiently complicated program. The number of bugs starts high, and when you start fixing them, the number of course comes down. But the number of bugs can only get so low. Past a certain point, continuing to fix bugs causes the total number of bugs to increase.

I’m not exactly sure what explains this counter-intuitive result. It’s partly because of workarounds that people put in place to accommodate known bugs, which suddenly become bugs themselves when the original bugs are fixed. And it’s partly because people who use the program will come up with new requirements that aren’t properly implemented.

Anyway, since you can’t fix all the bugs, you get to a point where you either have to decide to keep putting out fires, live with the bugs you have, or start over. Starting over isn’t as bad as it sounds: Microsoft wrote Windows NT from scratch to greatly improve a buggy Windows 3.1. And the Architect wanted Neo to start over by rebooting the Matrix and repopulating the Earth from 23 people.

A self-referential trap

In 1998, two companies called Google and Amazon.com were all the rage. Google was a new search engine that we all used because we heard it was the best, but didn’t really know why. And Amazon.com was an online bookstore that claimed to have a book about everything.

When you did a search for something in Google, along with the search results, you’d get a list of books that Amazon.com had on that topic. Search for dogs, and Google said “Amazon.com has these books about dogs…” Search for magnesium phosphate tribasic, and Google said “Amazon.com has these books about magnesium phosphate tribasic…”

It was a little hard to believe. There was no way that Amazon had books about everything. But how could we catch them in a lie?

I figured that a self-referential statement would likely do the trick. I did a Google search for “topics that Amazon.com has no books about.” And Google then said “Amazon.com has these books about topics that Amazon.com has no books about.” Whether such books existed, I didn’t know, but I could be sure that Amazon didn’t have them. This was a glitch in the matrix.

The system needs rules

Google’s problem in that case was that they put no restrictions on what you could type in. But every system needs rules, or it will crash.

A simple example is the liar’s paradox. Consider this sentence: “This sentence is false.” That sentence gives a contradiction, but that’s not really a problem. A consistent system just needs to consist of rules that don’t allow such a sentence to be constructed.

The so-called “naive set theory” in math has a similar flaw, as discovered by British philosopher Bertrand Russell in 1901. Russell’s paradox says this: Let S be the set of all sets that are not elements of themselves. Now, is S an element of S?

If you think about it, you’ll see that the answer is simultaneously yes and no. The paradox can be handled by using a set theory based on axioms that prevent us from forming sets like S. But this safety comes at a price.

Cartoon by xkcd

Gödel’s incompleteness theorems

I’m about to get into a mathematical concept that’s easier to understand in a non-mathematical context. So to warm up, consider this:

- A government big enough to give you everything you want is big enough to take everything you have.
- If our brains were simple enough to understand them, we would be so simple we couldn’t.
- Every justice system either puts some guilty people back on the street, or some innocent people behind bars.

Makes sense, right? The fact that difficulties arise in social systems and our own brain is not surprising. What is surprising is that something similar happens in every mathematical system, where we theoretically have complete control.

In 1931, Austrian logician Kurt Gödel proved his two incompleteness theorems, which I’ve always seen as the best example of a glitch in the matrix. Unfortunately, it gets ridiculously complicated, so I’ll have to do my best to simplify.

In math, we construct things called formal systems. A formal system consists of a language and rules. Examples of formal systems include particular types of arithmetic, geometry, and set theory.

Formal systems can express statements in their language, such as “2 + 2 = 4″ or “Every integer is even.” Some statements are true, and some are false. Also, some statements can be proven, and some cannot.

Ideally, you’d like every statement to be provable if and only if it’s true. That is, you’d like your system to be both consistent (all provable statements are true) and complete (all true statements are provable).

What Gödel proved is that every formal system of sufficient complexity is either inconsistent or incomplete (or both). That is, it’s either too weak to prove everything it should, or it’s strong enough to prove something it shouldn’t. In other words, there’s a glitch in every matrix.

He did this by showing that you can always construct a statement G that essentially says “G is not provable,” but without explicitly referencing itself, and being constructed within the rules of the system. However, self-referential statements aren’t the only ones that can blow up.

Here’s the simplest example I have. Consider the statement “There is no set whose cardinality [size] is between that of the natural numbers and that of the real numbers.” We don’t know whether this statement is true. But we know that in ZFC set theory (the current standard), the statement can’t be proven either true or false.

We’ll eventually figure out whether it’s true or false by jacking out of the matrix and using a more powerful system, but either way, there’s a problem with ZFC set theory. We’ll have a true statement (either the one above, or its negation) that can’t be proven, and therefore ZFC is incomplete (and maybe inconsistent, too).

If there’s a glitch in every matrix, then what is real? How do you define real? Do you think that’s air you’re breathing now?

Is it best to make rational decisions with your brain, or just go with your gut? Are people who insist on logic making the best use of all available information, or are they missing out on something far more powerful?

We always hear that sometimes you just have to listen to your gut. What exactly is the gut, anyway? Dictionary.com offers this definition:

“the alimentary canal, esp. between the pylorus and the anus, or some portion of it”

If I had to pick a body part other than the brain to listen to, I’m not sure this would have been my first choice. Why not the skin, heart, or solar plexus, or even the appendix? But anyway, I’m willing to consider that maybe we do underestimate the decision-making power of our intestines.

I asked about it on Twitter, and got a couple of responses. @Armen said:

“My gut has told me some very smart things that I have ignored and paid for, but I hear your point there [that the brain is more likely to be right]…It sure is overrated. On the other hand, it is underrated by folks who ignore it until problems show up…Some that come to mind here are gut telling to see dentist, or to come clean on lie, or to try a biz opportunity”

But even if the gut works in these cases, is it the best source of advice?

Regarding the dentist, you can listen to the calendar that says to go every six months, or to your nerves that say you have a toothache. Regarding the lie, you could listen to your conscience and not lie in the first place.

As for the business opportunity, this is where I can see the gut being helpful. Many business ideas that looked crazy on paper have become huge successes. In these cases, only a gut feeling could convince someone to follow through without a logical reason.

But gut feelings can also lead people astray, such as the gambler who “just knows” that his luck is about to change (only it doesn’t). How do you sort out the accurate gut feelings from all the rest?

Maybe the best idea is to use the gut not as a replacement for the brain, but as an idea generator to brainstorm (intestinestorm?) potential options before handing them over to the brain for evaluation.

Back to Twitter, @MiscBytes said:

“Gut” is just our brain using shortcuts it’s already figured out! http://www.miscbytes.com/gut-feelings/

The linked post mentions a book that talks about the brain quickly using rules of thumb to make its best guess without analyzing all the data. This best guess is known as a “gut feeling.”

It’s not always right, of course. Gut feelings would tell you that a bowling ball falls faster than a grape, that Saddam Hussein had weapons of mass destruction, that there are no irrational numbers in the Cantor set, and that it’s better to upgrade a Prius than a Suburban (see When Logic And Intuition Fail).

But while poring over all the data might be better in theory, a gut feeling often works well when facing a shortage of time. An excess of data can also overwhelm you, blinding you to the answer that your intuition can clearly see.

Right now, think of some either-or decision you have to make, something you haven’t thought out yet. Going to work tomorrow vs. taking a day off, having a healthy meal vs. junk food, buying this house vs. the other one, something like that.

I’m going to flip a coin to help you decide. Heads, you take the first option. Tails, you take the second. Ready?

The coin is in the air…I’ve caught it, and it’s…

But I don’t need to say what it is. You already know what you want it to be. This is your gut talking. Does it conflict with your brain? And which organ will win?

Photo by mikebaird

I recently had the opportunity to interview Milton Bradley. No, not that Milton Bradley, though there is a board game connection. I happened to stumble across Milt’s site, where he talks about the benefits of the ancient board game of Go, why he finds it far superior to chess, and how he taught it to hundreds of kids in an experimental after school program.

Perhaps most interesting to me was his claim that many of the world’s biggest problems are caused by poor decisions resulting from undeveloped reasoning skills, and that we can actually make progress towards solving them by learning this game. I couldn’t pass up the chance to ask him a few questions.

Hunter: First things first. What is reasoning, and why is it important?

Milt: Wikipedia says: “Reasoning is the cognitive process of looking for reasons, beliefs, conclusions, actions or feelings.” I prefer to think of it more simply, as “the logical mental process through which one arrives at answers to real world problems.”

My own definition appears in the Preface to my autobiography. [Bottom of the post]

Hunter: Don’t schools already teach reasoning? Maybe not explicitly as a subject in its own right, but don’t we pick it up along the way? And if not, then what is the purpose of formal education?

Milt: No, absolutely not! The emphasis in the schools is primarily on facts and the application of formulas to the solution of problems with exact solutions (e.g. math and the sciences). The vital subject of decision making in messy real world situations is really never addressed.

Hunter: You say that a full solution to the problem would require that schools teach reasoning, starting in pre-school and continuing throughout one’s entire academic career. Noting that this isn’t going to happen anytime soon, you suggest an alternative partial solution – teaching the strategy board game of Go, which can start right now.

Most people in the West have never heard of Go. I had barely heard of it until recently, and while I’m beginning to gain an appreciation for it, I know I’m far from really getting it. So I’m sure you’ve encountered a lot of people who are skeptical of your ambitious claim. Tell us, how can a board game play an important role in solving the world’s problems?

Milt: In terms of learning the process of situational appraisal and then deciding upon an appropriate strategy and the specific tactics with which to implement it, the process in Go is much like that involved in solving real world problems. So mastering the former process theoretically should help learning it in the latter, but there remains the difficult problem of skill transference from the neat, clearly defined realm of the Go board to the messy, immensely complex real world (especially considering its emotional implications). So whether or not my inference in this regard will prove correct is currently unknown.

Hunter: In the U.S., we play board games with a large element of luck, such as Monopoly, Life, and Sorry (to say nothing of roulette, the lottery, and Super Bowl squares). Eurogames, such as The Settlers of Catan, Puerto Rico, and Imperial, require much more thought and planning. And in Asia, they play Go and games resembling chess.

Does this tell us something about the different cultures, or am I reading too much into it?

Milt: Hard to be sure, although that’s a reasonable inference. But that’s a possibly unknowable question about historical origins which is almost entirely irrelevant to the key issue of what we do with and about the games we now play.

Hunter: I’m very interested in brain plasticity, the ability of our neurons to adapt to new experiences, an ability that decreases with age. I don’t think it’s true that you can’t teach an old dog new tricks, but let’s face it, some tricks are much easier for young dogs. Perfect pitch is a good example; anyone is probably inherently capable of it, but if you don’t speak a tonal language or receive musical training very early, the window closes forever.

It seems that all the great Go players started learning from a very early age. Go Seigen was an exception, becoming possibly the greatest player of all time, despite not starting until the ripe old age of 9.

For people who are far older than 9, is it too late to learn Go, or reasoning?

Milt: A good question. It’s certainly harder to attain the highest levels of proficiency, and the main reason for that is that Go is a game of pattern recognition, and it seems that the brain’s ability to absorb and internalize patterns declines quickly as a child ages. And a key here that should not be overlooked is that these top players who began as small children only were able to do so because the patterns they were exposed to as young children were on a very high level, so what they learned “by osmosis” was correct. If their exposure had been to error full low level play the result would have been quite different!

Hunter: Forgive my playing devil’s advocate, but they already play Go in Japan, and that’s not exactly a utopia. Like any other country, they have their pros and cons (see my discussion with Akemi Gaines where we compared the U.S. and Japan, or my tongue-in-cheek 10 Reasons America is Better Than Japan).

Is this because Go has been insufficient to fully develop reasoning in the Japanese, or because good reasoning isn’t enough?

Milt: Both! And because of the skill transference problem I alluded to earlier. It’s relatively easy to be objective in playing Go compared to the real world where all kinds of emotional issues intrude on the decision making process. As earlier noted, this is a key issue that must be addressed.

Hunter: Being smart isn’t considered cool. Any ideas on how to change that?

Milt: What you’re talking about is a quite temporary (and manifestly counterproductive) artifact of our current prevalent “pop” culture! Our society has enough problems that we aren’t currently coming near to solving for this sort of mass stupidity to continue indefinitely without sounding its own death knell! So this will either change sometime in the fairly near future or our survival prospects will be even dimmer than they already are.

Milt’s Go Page is filled with insightful articles about the benefits of Go and his experience with teaching it to kids after school for eight years. When I asked him to write a bio for this post, he provided the entire preface to his autobiography, so read on for more content!

In January 1943, age 15 years and 10 months, I graduated from the newly created Bronx High School of Science, then arguably the best high school in the entire United States. At the same time I was also an overweight, friendless, indifferent student who was seriously contemplating committing suicide! But on my 16th birthday only two months later, thru sheer force of will I overcame that negative thinking, and began to transform both mind and body to turn my life around.

As a result, I can look back today with pride at the creative output and epiphany of my “retirement” years, which have resulted in the writing of 8 books, 3 of which are in print and one that’s published FREE on the internet, all brought to fruition after the age of 75, with the last just this year at age 82! But I’m now also unquestionably in the twilight of my life, suffering from incurable, invariably fatal Acute Myeloid Leukemia, while my 88 year old wife has fallen victim to both Parkinson’s and Alzheimer’s diseases. But I soldier on despite those burdens, still writing, and still hoping to see my most significant work published. This autobiography is high in that category as is my novel The Vigilante Murders, but first and foremost is what I consider to be my most important work, Reasoning And Problem Solving, which is the result of some unique insights which I believe have made my life worth living and writing about.

The primary theme of this autobiography can be viewed as the story of my triumph of wit and will over adversity. But that’s an oft told tale, not infrequently by others who’ve had far more serious challenges to overcome than mine. So a reasonable reader might question why they should be interested in a rather detailed exposition of my life and its accompanying problems, trials and triumphs. The most straightforward answer is that I believe my story is intrinsically interesting! But even more important is that it lays out the intellectual substrate upon which my sometimes unique insights were generated.

This memoir begins with a brief look at my familial pre-history, then continues by relating the events and circumstances of my childhood that brought me to the devastating state of mind noted above, in which I was prepared to prematurely end my then still very young life. It then proceeds through my WWII Navy service and subsequent “GI Bill” education, meeting my wife and beginning my now 62 year long marriage, and the many, often traumatic professional and personal triumphs and tragedies that followed during my working career.

Among the most unusual and noteworthy of my many personal interfaces detailed herein were intimate daily contact with four individuals who made local and national front page headlines! The first of these occurred at Bronx Science, when for 3 years I sat at the next desk to Harold Brown, who later went on to become US Secretary Of The Air Force in The Johnson Defense and Secretary of Defense in the Carter Administration. Slightly more than a decade after that, I spent 4 years as a Quality Control Engineer at the RCA Receiving Tube Plant in Harrison, New Jersey, working daily with John Butenko, who soon after was unmasked, tried and convicted as the second most important Soviet spy ever in the US! Fast forward another decade or so to when I was supervising the redesign of the NYC Parking Violations Bureau’s computer system, when I spent year long daily working interface with PVB’s Deputy Director Geoffrey Lindenauer, who was soon thereafter exposed as a leading conspirator in their infamous scandal! And not too long after that, Eric Klein, who had been one of my personal programmer/analyst staff at the NYC DOT, was arrested and convicted as the largest counterfeiter of subway tokens in City history!

In attempting to establish a context in which an objective assessment of this autobiography is possible, I believe that it’s essential for the reader to consider some interesting facts. Even for those few in human history who have achieved great renown, the details of their daily lives all too often don’t offer much insight into the origins of their monumental achievements.

As a result, it’s reasonable to conclude that perhaps value in an autobiography shouldn’t be sought in the facts of the author’s life, but rather in the quality of the insights that his story generates in the reader. Or at least that’s what I try to convince myself of in attempting to justify this effort.

To really appreciate the uniqueness of my insights, it would help greatly if the reader understands (and hopefully agrees with) my most important premise – that most of the world’s myriad serious problems on the personal, interpersonal, group, enterprise, national, and even international level result from a single failing – an inability to adequately Reason objectively, unfettered by biases, prejudices, loyalties, and the “canned” prescriptions and proscriptions imposed by authority figures and institutions.

Although this precept seems to violate the principle that “Simplistic solutions to complex problems are almost invariably wrong,” I believe that this case constitutes one of the rare exceptions!

To be sure that there’s no ambiguity concerning what I mean by Reasoning, I conceive it to consist of:
1. The ability to objectively perceive and analyze an often complex problem situation, and then
2. Arrive logically and unemotionally at the course(s) of action required to best resolve the significant points of difficulty and/or contention involved.

When I arrived at this key realization of the almost universal difficulty most people experience in objectively solving real world problems, I believed that I had uncovered one of the central impediments to human progress throughout its history, and one whose conquest would rank among the most significant. And I also somewhat naively expected that this insight couldn’t possibly be one uniquely developed by me, but that surely some great thinker had long since both addressed this transcendentally important problem and solved it! But I discovered to my great surprise that not only was that not true, I was unable to find any reference to the fact that anyone had heretofore explicitly acknowledged that it was a even significant issue worthy of attention!

If this seems strange or improbable, I refer the reader to a bit of history in the field of mathematics by way of analogy (recognizing that, by their very nature, all analogies are necessarily imperfect). In this (possibly apocryphal) story, when John Napier, then an unknown Scot, published his treatise on Logarithms in 1614, it was so revolutionary that the head of the prestigious British Mathematical Society made a special trip from cosmopolitan London to semi-rural Scotland to meet him. As the story goes, on finally meeting Napier, the great man from London sat for perhaps a full half hour simply staring at him before finally saying something to the effect that “How can it be that this marvelous idea (of logarithms) escaped the best minds in all of humanity for thousands of years, yet was finally discovered by someone as ordinary as you?” I, of course, make no claim to intellectual equality with either Napier himself or his accomplishment, but mention this anecdote to emphasize my favorite dictum:

The validity and worth of an idea are unrelated to:
- Who proposed it.
- How long it has been believed.
- The number and importance of those who believe it.
- The vehemence with which they profess that belief.

History is replete with instances of the entire world believing something that was later acknowledged to be manifestly false (“The world is flat”), followed by a single man proposing an idea completely at variance with then conventional, accepted thinking, and ultimately prevailing. Galileo is perhaps the best known and most often cited example of this, and Einstein is another. But they were both operating in the world of physical science, where absolute proofs are possible. In the realm of ideas in which I’m operating no such absolute proofs exist, only opinion. Despite that, perhaps, just perhaps, it’s possible that, despite the uniqueness and novelty of my insight about Reasoning, I really might have discovered something that’s worth listening to!

Although I certainly wasn’t aware of it at the time, in retrospect it now seems that creating this new Reasoning paradigm was the goal toward which all of my training and life experiences had been pointing. How interestingly I describe those experiences and whether or not any of that really provides the reader with useful insights are crucial issues that only you can properly judge for yourself after you’ve perused what follows! Hopefully the result will satisfy us both.

After reading When Logic And Intuition Fail, someone asked me about a related paradox you may have heard about.

Let’s say you drive to work at 40 mph, and come back at 60 mph. What was your average speed?

It’s natural to think your average speed was 50 mph, but it was actually 48 mph. It would be 50 mph if you spent the same amount of time at both speeds. But since you’re spending more time at 40 mph than you are at 60 mph, your average speed has to be less than 50 mph.

If you’re driving to the beach, you might try to hold steady at 60 mph. But you won’t be able to stay exactly at that speed. You’ll sometimes be going a little faster, and sometimes be going a little slower. Even if the fast periods perfectly cancel out the slow periods, your average speed will still be less than 60 mph.

If you use cruise control, you not only save effort and gas, you also save time. That’s what slow and steady (emphasis on the steady) does for you.

And it’s one reason why you’re more productive when you do things at a steady pace, instead of slacking off and trying to make up for it later. It’s better to put your efforts on cruise control.

Seth Godin recently posted a wonderful brainteaser in Not so good at math:

Let’s say your goal is to reduce gasoline consumption.

And let’s say there are only two kinds of cars in the world. Half of them are Suburbans that get 10 miles to the gallon and half are Priuses that get 50.

If we assume that all the cars drive the same number of miles, which would be a better investment:

• Get new tires for all the Suburbans and increase their mileage a bit to 13 miles per gallon.
• Replace all the Priuses and rewire them to get 100 miles per gallon (doubling their average!)

Would you believe that you save more gas by putting new tires on the Suburbans? Because that’s the right answer.

What’s great about this problem is that it seems so simple, but the result is so astonishing. Even after you know the answer, it’s still hard to get your head around it.

Seth’s point was that we’re not wired for arithmetic. True, but I think what this problem really shows is that we’re wired for making faulty assumptions about numbers. It’s not our arithmetic that fails us in this case. It’s our logic and intuition that do.

How to solve it logically

In 6 Ways To Improve Your Telecommunication, Zack Grossbart shows a simple way to solve the problem by using pictures and plugging in real numbers. He crunches the numbers for one Suburban, one Prius, and a specific number of miles for the commute.

That’s one way I might have done it. Another way I might have done it is by flipping the miles per gallon (1/mpg) to get gallons per mile. When you want to see how much gas you’re burning, the relevant metric is gallons.

As you can see, upgrading a Suburban saves 2.3 times as much gas.

How to solve it intuitively

OK, we know how to arrive at the answer. But how can we resolve the paradox? How can getting 30% more mpg possibly be better than getting 100% more mpg? Here are two ways to understand it intuitively.

1. Consider a more extreme version of the problem.

You own a Hummer that gets 5 miles per gallon. You also own a futuristic supercar that can drive across the country on a single drop of gas. Would you rather get 1% better mileage on your Hummer, or 1,000,000% better mileage on your futuristic supercar?

Don’t turn this into a problem of comparing one percentage against another. There’s no point in upgrading the supercar. The percentage you improve it by is irrelevant, because nanodrops of gas don’t matter. But any improvement on the Hummer is huge because it burns a lot of gas.

2. Consider a reworded version of the problem.

All the Suburbans in the world burn 83% of the gas. All the Priuses in the world burn 17% of the gas. Which model should you upgrade?

When it’s phrased this way, the answer is obvious. The wording of the original problem distracted you from what was really important. Of course, problems aren’t always nice enough to phrase themselves in the way that is most convenient for you.

Why doesn’t common sense work?

There are several reasons why it’s so easy to be led astray.

1. We fail to spell out our objective. The wording is critically important, because it’s easy to solve the wrong problem.

Are we trying to maximize the mpg of the average car? No. (If we were, we should upgrade the Prius.) Our goal is to minimize the total amount of gas burned by all cars. So focus on that.

2. It might seem strange that we’re not trying to maximize the average miles per gallon. Isn’t that the same as reducing the total amount of gas burned? Well, it would be, if there was only one car. But averages can be tricky.

If we upgrade the Suburbans, the average car would get (13 + 50) / 2 = 31.5 miles per gallon.

If we upgrade the Priuses, the average car would get (10 + 100) / 2 = 55 miles per gallon.

55 is more than 31.5, so upgrading the Prius means burning less gas, right? It might seem like it should work that way, but there is no mathematical law that says so.

Looking at the average isn’t enough – you need to look at the distribution. Here are three pairs of cars, each pair averaging 10 mpg. The greater the variance within each pair, the more gas is needed to drive a fixed distance.

That’s what we get when we keep the distance fixed and look at how much gas we need, which is what we have to do for this problem. But just for fun, let’s keep the amount of gas fixed and look at how far we can drive with the same cars.

Is that surprising?

3. Reciprocals (mpg vs. gpm) are confusing. We’re trained to think in terms of miles per gallon. But gallons per mile is actually a much more natural unit to work with when you’re looking at how much gas you’re burning.

The Suburban gets 10 mpg, and the Prius gets 50 mpg. The Prius gets 400% better mileage (mpg), but it burns 80% less gas (gpm), so you have to be really clear on what you’re talking about.

The differences between reciprocals get more pronounced when you approach a singularity. 0/1 is very different from 1/0, which is what caused the difference in the previous two charts.

4. We’re told that there are equal numbers of Suburbans and Priuses, and we subconsciously think they should therefore be treated equally. But we need to discriminate. The Suburbans are burning 83% of the gas in the world, so they need to be given more weight. It doesn’t matter how many there are, only how much gas they’re all burning.

Of course, what people should really be doing is trading in their Suburbans for Priuses.

Someone who’s 100% certain is naive.
Someone who’s 0% certain is lazy.
Someone who’s 25% certain is ignorant.
Someone who’s 50% certain is either confused or enlightened.
Someone who’s 75% certain is either an idiot or a scholar.
Someone who’s 99% certain is overconfident.
Someone who’s 99.999% certain is dangerous.

I had a job interview today, where they gave me a logic test. I didn’t get to keep it, but I remembered most of the questions. Have fun!

1. Split 110 into two parts, so that one part is 150% of the other. What are the two numbers?

2. There are 100 people, and everyone is either a football player or a basketball player. There is at least one football player. For any two people, at least one of them is a basketball player. How many football players are there?

3. The number 8,549,176,320 is the only one of its kind. Can you figure out what’s so special about it?

4. There are 20 questions on a test. You gain 10 points for each correct answer, and lose 5 points for each incorrect answer. Someone answers all the questions and gets 125 points. How many questions did they get wrong?

5. Two coins add up to $0.55, and one of them is not a nickel. What are the two coins?

6. What is the biggest number you can make using two numbers? Just two numbers, no other mathematical symbols.
[When they say "just two numbers," I took that to mean two digits.]

7. The number of lilly pads in a pond doubles every day. Starting with just one lilly pad on the first day, the pond is completely covered with lilly pads after 60 days. How long did it take for the pond to be half covered?

8. An adult and two children need to cross a river. They have a boat that either child is able to handle by themselves. The boat can carry either the adult or both children, but not the adult and a child at the same time. How can they cross the river?

9. Someone introduces you to your mother’s only sister’s husband’s sister in law. He has no brothers. How do you address this person?
[This isn't stated too well, but I assume "he" refers to your mother's sister's husband.]

10. There are two different colors of socks in a drawer. Without looking at them, how many do you need to take out to ensure you have a matching pair?

11. According to someone’s will, $666,666 is to be divided between 2 fathers and 2 sons. They discuss it, and each person gets $222,222. Explain.

Photo by http2007

This is my 200th post. Thus, I can now say I’ve written hundreds of posts, instead of dozens. It will take 1,800 more posts before I can say I’ve written thousands, so this is a nice milestone!

I heard about this puzzle from @marelisa on Twitter, and I found a related but harder version on Wikipedia. It’s known as either Einstein’s Riddle, or The Zebra Puzzle. Albert Einstein allegedly created it as a boy, and he said that only 2% of the world’s population could solve it.

While it’s not clear whether it was actually created by Einstein, the 2% figure seems about right, especially because most people haven’t tried a puzzle like this before. But if you’re feeling up to it, I’ll give both versions of the puzzle, followed by some tips on how to do it.

Here’s the version of Einstein’s Riddle that Marelisa found. (Note there’s no zebra in this one.)

- In a street there are five houses, painted five different colors.

- In each house lives a person of different nationality.

- These five homeowners each drink a different kind of beverage, smoke a different brand of cigar, and keep a different pet.

Einstein’s riddle is: Who owns the fish?

Necessary clues:

1. The British man lives in a red house.
2. The Swedish man keeps dogs as pets.
3. The Danish man drinks tea.
4. The Green house is next to, and on the left of the White house.
5. The owner of the Green house drinks coffee.
6. The person who smokes Pall Mall rears birds.
7. The owner of the Yellow house smokes Dunhill.
8. The man living in the center house drinks milk.
9. The Norwegian lives in the first house.
10. The man who smokes Blends lives next to the one who keeps cats.
11. The man who keeps horses lives next to the man who smokes Dunhill.
12. The man who smokes Blue Master drinks beer.
13. The German smokes Prince.
14. The Norwegian lives next to the blue house.
15. The Blends smoker lives next to the one who drinks water.

Here’s the version of The Zebra Puzzle in Wikipedia. This is the first known publication of the puzzle, from 1962. While similar to the version above, it’s significantly harder. As far as I can tell, it requires making a guess from multiple possibilities, then looking ahead to see how it pans out, and backtracking if it doesn’t work.

1. There are five houses.
2. The Englishman lives in the red house.
3. The Spaniard owns the dog.
4. Coffee is drunk in the green house.
5. The Ukrainian drinks tea.
6. The green house is immediately to the right of the ivory house.
7. The Old Gold smoker owns snails.
8. Kools are smoked in the yellow house.
9. Milk is drunk in the middle house.
10. The Norwegian lives in the first house.
11. The man who smokes Chesterfields lives in a house next to the man with the fox.
12. Kools are smoked in a house next to the house where the horse is kept.
13. The Lucky Strike smoker drinks orange juice.
14. The Japanese smokes Parliaments.
15. The Norwegian lives next to the blue house.

Now, who drinks water? Who owns the zebra?

(In the interest of clarity, it must be added that the five houses are in a row, each is painted a different color, and their inhabitants are of different nationalities, own different pets, drink different beverages, and smoke different brands of cigarettes.)

Here are some tips.

I can’t imagine solving these puzzles without using a chart to keep track of what you know. Most people would use a chart like the following, where it starts off blank and you fill in the words as you learn what’s in each house.

This is how the chart would look near the very beginning, after applying the clue that milk is drunk in the middle house.

But this is what I did. I started with a chart listing all the possibilities. For example, each of the drink cells started off with “Water Tea Orange Milk Coffee,” which I abbreviated here as “WTOMC” so it fits. Then I began deleting options that were impossible.

Below you can see how it looks after learning that milk is drunk in the middle house.

This way is more cumbersome, and I would only do it on a computer, not on paper. But I think it makes it easier to keep track of what you know, and therefore easier to solve the puzzle. That’s because you can track every possibility you rule out, instead of only writing something down after you’ve ruled out all other possibilities.

One more tip: cross out clues after you no longer need them. As the list of clues shrinks, you fill in the details, and eventually find out who owns the fish or the zebra.

Photo by AMagill

The Matrix trilogy presents a number of yin yang pairs, one of which is seen in the Oracle and the Merovingian. Although they take on the form of humans in the matrix, they’re actually computer programs. And they’ve been designed to specialize in different kinds of logic.

The Oracle is gifted with foresight based on inductive reasoning. Although she’s not actually psychic, she was specifically created for the purpose of understanding humans, and this gives her amazing powers of prediction. She can effectively see into the future, up to the point where free will presents a choice. She says nobody can see past a choice they don’t understand, thus showing her limitations.

The Merovingian is gifted with hindsight based on deductive reasoning. Believing that everything is determined by cause and effect, he thinks that someone’s power is based on their understanding of why events unfolded the way they did, and he understands this quite well. But his ability makes him overconfident, because he really doesn’t know everything. For example, he didn’t know that his wife would turn against him, because he saw no cause that would create that effect.

The Merovingian and the Oracle are opposites in this regard, and he dismisses her as a silly fortune teller. He laughs at Neo and his friends for visiting him just because the Oracle advised them to, when they didn’t really know why they were there or what they expected to happen. They were just blindly following orders, without knowing the answer to that all-important question: why?

But despite his mocking, the Merovingian secretly desires the Oracle’s powers of induction to complement his powers of deduction. And understandably so, when you consider what one could do with both of them.

This post is fairly long and complicated, but it’s virtually guaranteed to boost your reasoning skills. Read it once, then enjoy the benefits of improved logic for a lifetime.

Inductive reasoning

The Oracle inductively knew that Neo was coming, and she was ready with cookies in The Matrix (1999).

With inductive reasoning, you reach a conclusion that is believed to be true but not guaranteed. Specifically, you use observations of particular cases to make a generalization. While this may not seem logical, we do it all the time.

Here’s a common example:

“The stock market has averaged 10% annual returns in the past, so it’s reasonable to expect that it will continue to do so.”

Statements like this are often followed with an admission that 10% returns are not guaranteed. As they say, past performance does not guarantee future results.

However, predicting 10% returns based on available data seems preferable to ignoring the past, and deciding that returns of 10%, -100%, or 1,000,000% are equally probable. In other words, we naturally want to make a prediction, so we should make one that fits the pattern.

The card game Mao is based on inductive reasoning. New players are not told what the rules are, because the point of the game is to figure out the rules. Players have no choice but to observe the game, make many mistakes, and slowly piece together the rules by induction.

Here are some examples of so-called strong induction:

“Mary always hates horror movies, so she’ll hate this horror movie.”

“I’ve never seen a green canary, so your canary is probably not green.”

“Technology has changed a lot in the last 100 years, so it will change a lot in the next 100 years.”

“White eggs have a hard shell, so brown eggs must have a hard shell.”

“Five channels are showing static, so the cable is out.”

“Penicillin killed these bacteria, so it will kill other bacteria.”

“Pi does not terminate or repeat after the first million digits, so it never terminates or repeats.”

This is called strong induction because the conclusion is likely to be true, assuming the premise is true.

However, these statements could be stronger if the wording were more specific. What does it mean that “Mary always hates horror movies?” Which ones, and how similar are they to the one you’re predicting she’ll hate? You could also say she’ll probably hate the movie, to acknowledge the possibility that she might not.

Here are some examples of what’s called weak induction:

“I could run fast 60 years ago, so I can run fast now.”

“I always sleep until noon, therefore everyone always sleeps until noon.”

“I made a wish and it came true, so all wishes come true.”

This is weak induction because the arguments aren’t very convincing at all. There’s a very weak link between the premise and the conclusion.

However, what one person considers weak induction, someone else may consider strong induction. For example, Isaac Newton induced his theory of gravity from observing the motions of planets and falling apples. This theory was undoubtedly met with varying degrees of resistance, depending on how strong someone considered the induction to be.

Any induction, particularly weak induction, carries the risk of overgeneralization, which can lead to prejudice and delusion.

Rejecting inductive reasoning

On the other hand, if you don’t generalize at all, that’s a problem too. If you know how to drive a Camry, it would be crazy to say that you don’t know how to drive a Corolla because it’s a different model. We have to generalize to survive.

I know a baby who rejects inductive reasoning. When he gets hungry, he cries. Most babies will stop crying when you start feeding them, but not him. He knows that just because baby food satisfied his hunger last time, there’s absolutely no guarantee that it will satisfy his hunger this time.

So he keeps crying, while mom shovels food into his mouth as fast as she can. He doesn’t stop crying until he actually feels full, and therefore has proof that his hunger was satisfied. And he’s a big baby, so this takes two jars.

You can imagine how his mom feels. She inductively reasons that since he’s done this every time, he’ll continue to do so for a while. Wouldn’t it be nice if her baby used inductive reasoning to determine what was likely to happen, instead of insisting on a guarantee?

Deductive reasoning

The Merovingian deductively believes that every cause and effect has already been determined, so he just sits back and enjoys himself in The Matrix Revolutions (2003).

With deductive reasoning, you apply known rules to given data to prove a conclusion. Unlike inductive reasoning, deductive reasoning lets you arrive at a guaranteed conclusion, as long as your reasoning is sound. This is what we usually think of as “logic.”

For example:

“The sum of the angles of every triangle is 180 degrees. In this triangle, two angles are 45 degrees, so the remaining angle must be 90 degrees.”

We like deductive reasoning for a couple of reasons. One, we’re so used to thinking of it as the definition of logic. Two, it’s a lot more certain than inductive reasoning, and people like certainty.

However, it also gives us false certainty. Suppose you’re looking at an animal, and want to prove that it can fly. So you use this logic:

“This animal is a bird. All birds can fly. Therefore, this animal can fly.”

There are a couple of problems here. Most importantly, the claim that all birds can fly is false. For example, penguins can’t fly. We call an argument like this valid because the deduction was logical, but not sound because it’s based on a false premise.

Also, how can you confirm that it’s a bird? On a math test, you’re given all the information you need. But in the real world, a problem won’t necessarily be set up so conveniently.

The problem with deductive reasoning is that it’s like an insurance policy that guarantees to pay off, but only if a particular set of conditions is met exactly. And you’d better read the fine print. It’s easy to be far more confident in the outcome than your logic warrants.

The deduction above is relatively simple, but verifying the preconditions is extremely difficult. How do you know it’s a bird? Because it looks like other birds you’ve seen? That’s inductive reasoning. What’s the definition of a bird anyway?

Is it true that all birds except penguins can fly? What about chickens? They can fly a little, or can they? What’s the definition of flight? What about injured birds? What about birds that are afraid to fly?

Math is logical, but filled with assumptions

Math makes heavy use of deductive reasoning, but it’s a lot less solid than we might think. Think back to your high school math. You learned how to deduce many things, but it was all based on a set of axioms that we just assumed to be true.

Some of these axioms seem obvious, like “all right angles are congruent.” Of course they are. How could they not be? But that’s an assumption that can’t be proven. Some axioms seem stupidly obvious, like “if A and B are true, then A is true,” or “x = x.” It hurts your head to even imagine them not being true, but we need to make these assumptions to support everything else.

In geometry, we even have three undefined terms: point, line, and plane. This goes beyond unproven – they’re not even defined!

For over 2,000 years, Euclid’s assumptions of geometry seemed so obvious that no one questioned them. Today, we need to specify “Euclidean geometry” when referring to the version that seems obvious to us, because there are different versions where these assumptions are violated.

Far from being an unquestionable universal truth, Einstein suggested that Euclidean geometry is a good model of physical reality only if the gravitational field is not too strong.

Yes, math is very logical, and you can say that A is definitely true, if you assume that B, C, D, E, F, G, and H are true. But in that case, what have you really proven?

An attempt at inductive reasoning in math

Say we have a function f, where

f(x) = x⁶ – 15x⁵ + 85x⁴ – 225x³ + 274x² – 120x

OK, but what is this, really? What’s f(0)? You plug it in and see that f(0) = 0. OK, what about f(1)? That’s also 0. You try f(2), f(3), and f(4), and they’re all 0 too. Looks like a pattern is emerging.

You try f(5) and get 0 again. Now you’re getting tired of this. Obviously, f(x) = 0 for all whole numbers x, right?

Nope, because f(6) = 720.

This is the problem with inductive reasoning. You can observe as many specific cases as you want, but you’ll never prove a generalization from observation unless you observe all possible cases.

However, our observations are still helpful, because we can apply deductive reasoning to them. The fact that f(0) = 0 means that x is a factor of f(x). The fact that f(1) = 0 means that x – 1 is a factor of f(x), and so on. By factoring everything out, we find that

f(x) = x(x – 1)(x – 2)(x – 3)(x – 4)(x – 5)

This form is much simpler, and now we can see exactly why f(x) = 0 only when x is 0, 1, 2, 3, 4, or 5.

So while our inductive reasoning failed, our observations turned out to be fuel for deductive reasoning.

Let’s try another one.

Mathematical induction

Let’s say you want to prove that 1 + 2 + 3 … + n = n(n + 1) / 2 for all natural numbers n. This seems far from obvious. How would you even get started?

Well, let’s plug in some numbers and check. It’s true for n = 1. Also 2, 3, 4, 5, 6, 7, 8, 9, 10…

But we saw in the last example that just because something is true for ten or a thousand or a million cases, that doesn’t guarantee that it’s true for all cases. So what can we do?

Think about when you set up a bunch of dominos to knock down in a chain reaction. It works like this:

If (1) all the dominos are set up in such a way that if one falls down, the next one falls down, and (2) the first domino falls down, then they all fall down, right?

Let’s use that same idea to prove that 1 + 2 + 3 … + n = n(n + 1) / 2 for all natural numbers n.

First, let’s check that it’s true for n = 1. Yes, it is.

Now, let’s check that if it’s true for some arbitrary number x, then it’s also true for x + 1, i.e., that 1 + 2 + 3 … + x + (x + 1) = (x + 1)(x + 2) / 2.

OK, if 1 + 2 + 3 … + x = x(x + 1) / 2, then 1 + 2 + 3 … + x + (x + 1) = x(x + 1) / 2 + (x + 1) = (x² + x + 2x + 2) / 2 = (x + 1)(x + 2) / 2. Done. And so the dominos all fall down.

This is called mathematical induction, yet it’s technically a form of deductive reasoning, because the conclusion is guaranteed if you do it right. However, it’s similar to inductive reasoning in that you’re taking a finite number of observations and generalizing them to an infinite number of cases.

But wait a minute. How do we know that the principle of mathematical induction, the “domino trick,” is true? Well, it’s usually taken as an assumption! It can only be proven if you make certain other assumptions, such as (1) the natural numbers are well-ordered, (2) every natural number is either 0 or the successor to another natural number, and (3) n + 1 > n for all natural numbers n.

Wow, we sure need to make a lot of assumptions in order to “prove” anything! Which brings up another point. Even if you manage to prove something, how can you prove the proof? Fermat’s Last Theorem was “proven” many times by reasoning that was ultimately revealed to be flawed.

Abductive reasoning

Deductive reasoning is often used in math, because they’re trying to prove things from a known starting point.

Inductive reasoning is often used in science, because they’re trying to discover things, not prove them.

And there’s another type of reasoning, called abductive reasoning, that’s often used by detective types, because they’re trying to explain things, not discover or prove them.

It works like this. A patient has certain symptoms, and goes to the doctor. The doctor knows that appendicitis will cause those particular symptoms. There are other possible causes, but appendicitis is far more likely, and therefore considered the best diagnosis based on the known information.

In some countries, doctors have been known to remove the appendix without actually testing for appendicitis. They thought it was better to occasionally be wrong than to consistently take the time and money to run a test to confirm a rather obvious diagnosis.

Now, using the method of your choice, possibly abductive reasoning, can you determine the point of this post?

Photo by stephend9

In the comments on The Birthday That Never Arrives, some people requested another lateral thinking puzzle. Here you go!

This one comes from Encyclopedia Brown, a series of children’s mystery books published from 1963 to as recently as 2007. These books star Leroy “Encyclopedia” Brown, the 10-year-old know-it-all supersleuth son of the chief of police. (For the members of generation Y, an encyclopedia was something like Wikipedia, but in books!)

This story has been reconstructed from what I remember.

Mr. Diamondthief is invited to a party at a friend’s house. Because the house contains things that people would like to steal, all arriving guests are frisked to make sure they aren’t carrying any weapons and such. Mr. Diamondthief is clean, so they let him in.

But Mr. Diamondthief has sinister motives. He’s really there to steal a diamond. And he’s been in this house before, so he knows exactly where it is. Upon entering the foyer, he goes up a narrow staircase, down a long hallway, into the fourth room on the left, where the diamond is. He takes the diamond.

Because he knows he’ll be frisked on the way out, he can’t just take the diamond out with him. So he looks around, trying to find some way to get the diamond out of the house. He comes across a bow and a quiver full of arrows. Perfect! He ties the diamond to an arrow, opens the window, and shoots the arrow into a tree down the street. He closes the window, puts the bow back where he found it, and wipes off his fingerprints.

He then walks out the front door. They frisk him, and he’s clean. The arrow with the diamond is stuck in a tree down the street, but he decides it’s too risky to get it now. Another party guest might see him walking down the street instead of getting into his car, or the cops might search the homes of all the party guests. He decides to come back for the diamond another time.

The next day, the owners of the house notice the missing diamond, and call the police. Police chief Brown calls all the party guests back to the house for questioning, and to search for the diamond. The assumption is that the diamond is probably still in the house, since all guests were frisked on the way out. Of course, Chief Brown takes his son Encyclopedia with him, since he’s the one with the best detective mind.

Encyclopedia thinks Mr. Diamondthief looks suspicious, but there’s no evidence against him. Encyclopedia starts exploring on his own, and finds the bow and quiver of arrows, noting that one arrow is missing. He sees the window, and realizes how the diamond was stolen.

As Encyclopedia comes down the stairs, he hears Mr. Diamondthief saying “Chief, this search is pointless. Even if the diamond is still here, this house is huge. We’ll never find it!”

Encyclopedia says, “Don’t worry, Mr. Diamondthief. The diamond is just an arrow flight away.”

Mr. Diamondthief says, “Well then, go outside and look for it.”

Encyclopedia says, “Dad, arrest him!”

How did he know Mr. Diamondthief was guilty?

As a kid, I thought the answer was a little cheesy. I actually like it much better now, though my preferred answer is slightly different from the official answer.

If you’ve heard it before, don’t blurt out the answer – let’s give a chance for other people to guess.