# Bill Gates' bathroom

Answers to the logic questions posed in the May 17, 2004, Guide to Recruitment & Staffing

Were you stumped by the questions in the May 17, 2004, Guide to Recruitment & Staffing from William Poundstone’s book How Would You Move Mount Fuji?. Never fear. Here are the answers:

Suppose you had eight billiard balls. One of them is slightly heavier, but the only way to tell is by putting it on a scale against the others. What’s the fewest number of times you’d have to use the scale to find the heavier ball

The answer is two. Here’s how to arrive at it:

A balance is the simple two-pan setup, like the one the blindfolded figure of justice holds. It tells you which of the two pans is heavier, though not by how much. It can also tell you when the two pans are of equal weight.

The obvious approach will not work. That would be to weigh four balls against four balls. The heavier pan would have to contain the defective ball. Split that group’s balls into two pairs and weigh the pairs against each other. Again, the heavier pan has to contain the defective ball. You’re then out of your two allotted weighings. There is no way to decide which of the two suspect balls is heavier.

The solution is to make full use of the fact that the balance can tell you two pans are equal. Whenever the two pans are of equal weight, you can conclude that the defective ball is not in either pan.

For the first weighing, pick any three balls and weigh them against any other three balls. This has two possible outcomes.

One is that the balance finds the two pans equal. In that case, the defective ball must be one of the two you didn’t weigh. For the next and last weighing, just compare the two still-untested balls. The heavier one has to be the defective ball.

The other possible outcome of the first weighing is that the balance finds one of the pans heavier. The defective ball must be in the heavier pan. For the final weighing, pick any two of the balls in the heavier pan and compare them. If one is heavier, it’s the defective ball. If both are equal, the defective ball has to be the third ball that you didn’t weigh this time.

How would you weigh a jet plane without using scales?

Some candidates propose to look up the specs from Boeing’s website. They’re crushed when the interviewer brushes this aside (not allowed to use the Internet!?!). A traditional version of the puzzle asks you to weigh an elephant without scales. Elephant or jet, you aren’t allowed to cut it into manageable pieces.

The intended answer is that you taxi or fly the jet onto an aircraft carrier, ferry or other ship big enough to hold it. Paint a mark on the hull of the ship showing the water level. Then remove the jet. The ship will rise in the water.

Now load the ship with items of known weight (100-pound bales of cotton, whatever) until it sinks to exactly the line you painted on the hull. The total weight of the items will equal the weight of the jet.

If you’re more work-phobic than math-phobic, you can save effort by computing the volume of the ship between the two water levels, and multiplying that by the density of water. That too will give you the jet’s weight.

Why are manhole covers round instead of square?

The answer interviewers consider the best is that a square cover would fall into its hole, injuring someone or getting lost underwater. This is because the diagonal of a square is 1.414 times its side. Should you hold a square manhole cover nearly vertical and turn it a little, it falls easily into its hole. In contrast, a circle has the same diameter in all directions. The slight recess in the lower part of the cover prevents it from ever falling in, no matter how it’s held.

A more flippant answer (not that this question merits any other kind) is “because the holes are round.” Maybe that’s not so flippant. Holes are round, you might claim, because it’s easier to dig a round hole than a square one.

Another answer is that a person can roll a circular cover when it needs to be transported a short distance. A square cover would require a dolly or two persons. Perhaps a lesser reason is that a round cover need not be rotated to fit the hole.

Why do mirrors reverse right and left instead of up and down?

When you first start thinking about this question, you may feel cut adrift from everything you learned in school. You can’t apply arithmetic, physics or psychology. It is not even a logic puzzle in the usual sense.

In outline the two most popular responses are the following:

(a) denying that the mirror does reverse right and left; and
(b) insisting that mirrors can reverse up and down (as for instance when the mirror is on the ceiling or floor.)

Start with (a). When you hold up the front page of a newspaper to a mirror, the reflection is reversed and hard to read. Imagine the text is printed on a transparent plastic sheet. You can press the sheet up against the mirror and see that the text exactly coincides with its reflection. The mirror is not “flipping” the image underneath the sheet.

This is even clearer when you hold an arrow up to the mirror. Hold the arrow horizontally and point it toward the left. The arrow’s reflection also points left. Nothing is reversed. Point the arrow right, and the reflected arrow points right.

These are valid points, as far as they go. We still know that some kind of quote-unquote reversal is going on, even if this reversal is not quite what people imagine. Your interviewer will come back with “Yes, but if there’s no reversal, why can’t you read a newspaper’s mirror image? Why do you have to flip the transparent plastic sheet left for right, and not up for down, in order to read the reflection?”

Answer (b) takes the opposite tack. Mirrors reverse in every direction. When the mirror is on the floor — pointing “up” you might say — it reverses up and down. A mirror pointed north by northeast reverses north by northeast and south by southwest. A mirror pointed left reverses left and right. There are no “favoured” directions. Nothing in the physics of mirrors tells them to reverse left and right.

The interviewer will probably then want to know why we have this popular misconception that mirrors reverse right and left. You might argue that it all comes down to culture, to conventions of architecture and interior design. Mirrors are not positioned at random orientations in space. They almost always hang on vertical walls of rectilinear rooms. They consequently almost always reverse horizontally (north and south or east and west), not vertically (up and down). This horizontal reversal is conventionally described as a reversal right and left. It makes more sense than saying that a particular mirror reverses north and south, for these are geographic absolutes, and there is nothing absolute about a mirror’s reversal. It is all relative to the way the mirror is pointed.

Explanation (b) makes some good points. It still skirts the main issue. In a Las Vegas hotel suite with mirrors on the ceiling; in an igloo, a yurt or a carnival hall of mirrors; in the most unconventional or un-Western environment you can devise — you still can’t read a newspaper in a mirror because it’s reversed right and left. Why is that?

Let’s go back to square one. Which way do mirrors reverse? This time, let’s be careful about language and decide what is the best, most general description of how mirrors reverse.

Explanation (a) is accurate in maintaining that mirrors do not (necessarily) reverse right and left. It would be an old thing if they did. “Right” and “left” are defined relative to a human observer’s body. How can a dumb piece of glass know that somebody’s looking at it and from what orientation? It would have to “know” that in order to reverse right and left consistently.

It is much more plausible, and correct, to think that a mirror’s direction of reversal depends on its own orientation in space. This is what explanation (b) is driving at. By pointing a mirror in any direction, you can cause it to reverse in any direction.

How does this work exactly? The arrow experiment demonstrates that an arrow parallel to the mirror’s surface is not reversed at all. Is there any way of holding an arrow so that its reflection points in a different direction? There is. Just point the arrow straight at the mirror (away from you). The reflected arrow points in the opposite direction, out of the mirror (toward you). Or point the arrow toward you/away from the mirror. Then the reflection points away from you/deeper “into” the mirror.

A wordy but accurate way to describe what a mirror does is to say that it reverses directions “into the mirror” and “out of the mirror.” When you look into a mirror, some things that appear to be in front of you — behind the glass of the mirror — are really behind you. Directions parallel to the mirror’s surface, such as left and right or up and down, are not reversed.

This fact is so totally obvious that we almost always ignore it. For the most part, you don’t perceive a mirror image as a looking-glass world behind the mirror. You know you are seeing your own face in your own room. The brain filters out the into-the-mirror/out-of-the-mirror reversal and interprets the mirror’s reversed image as the real world.

This normally useful deception fails only with certain asymmetrical objects and actions. Screw threads, snail shells, knots and scissors can exist in one of two otherwise identical forms. The most familiar of all such asymmetrical objects are our hands. A right hand is similar to a left hand in all details, yet it is also completely different.

Your right and left hands are “mirror images” of each other. Touch the tips of your fingers together, as if there were an invisible pane of glass between them. It seems like the right hand has undergone an into-the-glass/out-of-the-glass reversal to become the left hand.

Here is where the greatest confusion comes in. It is conventional in English and many other languages to describe the two mirror-image forms of asymmetric objects as “right-handed” and “left-handed.” This is a figure of speech. It has nothing to do with right and left directions. We could just as well call the two forms of a screw thread “A” and “B” or “plus” and “minus” or “normal” and “reversed.”

One consequence of a mirror’s reversal is that the reflection of any of these asymmetrical objects is transformed into the object’s “other” form. Evidently the brain is not very good at filtering out this difference. Because most text is strongly asymmetrical, its mirror image becomes strange and unreadable. Asymmetrical actions, such as using scissors, can be frustratingly difficult when working from a mirror image.

We struggle also to put these difficulties into words. A careful speaker could say that the mirror reverses the so-called right- and left-handedness of asymmetric objects. That is usually shortened to the statement that a mirror “reverses right and left” — a statement that is actually quite different and actually quite wrong. We all accept and pass on a lot of statements we’ve heard without thinking them through. This is one of those statements.

The 15-second version of all this: A mirror doesn’t necessarily reverse left and right or up and down. It reverses only in the directions into and out of the mirror. This changes the so-called handedness of asymmetric objects, so that a right hand’s reflection looks like a left hand, and reflected text is unreadable.

Answering this question well demands a paradigm shift. You are asked to explain why mirrors reverse right and left, when really they don’t. Many interviewees never dig themselves out of that hole. The question tests a willingness to challenge assumptions, including those that come from a superior.

Which way should the key turn in a car door to unlock it?

There is a school of thought among interviewees that it doesn’t matter whether you pick right or left. The real agenda is to see whether you can make a fundamentally meaningless A or B decision without getting hung up over it.

Actually, there is a preferred answer. Here’s why: Hold you right hand out, pretend you’re holding a key. (Also, pretend you’re right-handed if necessary.) Your hand is a fist, palm-side down, with the imaginary key between your thumb and the curled side of your index finger.

Turn your hand clockwise, as far as it can go without discomfort. You can probably turn your fist a full 180 degrees, easily. The palm side is now up.

Try it again, turning the hand counterclockwise. It’s tough to turn it just 90 degrees. You have less strength near the limit of the turn.

The design of the hand, wrist and arm thus makes it easier for a right-handed person to turn a key clockwise (so that the top of the key turns to the right). It is the opposite with left-handed people. There are fewer left-handed people, though, creating a true asymmetry. That provides a basis for saying that the key should turn one way or another.

Why is it that, when you turn on the hot water in a hotel, the hot water comes out instantly?

In most people’s homes, the hot-water heater is dozens of feet from the hot water taps. “Hot water” pipes themselves are not actually heated, of course. When no water is flowing, the water in the pipes cools to ambient temperature. When you turn on the hot water, the line pressure has to flush a volume of now-cool water out of the pipes before you get hot water.

It’s possible to brainstorm several ways of getting instant hot water. You could have a mini-hot water heater for each tap, as close to the tap as possible. You could have a system for heating the hot water pipes. These are acceptable (though wrong) guesses in an interview.

The real answer is this: Hotels and some homes have a hot water recirculating system. It consists of a pump attached to an extra line that runs “backward.” This line goes from near the hot water tap farthest from the hot water heater, all the way back to the heater. The pump slowly circulates hot water through the hot water lines so that the water in the lines never gets cold. When you turn on the tap, the line water is already hot.

How do they make M&Ms?

The main issue is how they get a perfectly smooth, layered candy shell on a mass-produced product that never knows the touch of a human hand until the bag is opened. Dipping the chocolate in a liquid candy that hardens seems unsatisfactory. You would have to place the candy somewhere while waiting for the shell to harden. If you did that, you’d expect the candy to have a flat bottom, like hand-dipped chocolates do. On ingenious (wrong) answer: “There’s a sheet of hot boiling chocolate, and they freeze the peanuts and fire them through it so it instantly freezes and the chocolate is hard by the time it hits the ground.”

The actual method used by the Mars Company is both clever and simple. Unfortunately, it’s hard to guess, and no one expects you to be a candy-trivia buff. The chocolate centers of “plain” M&Ms are cast in little molds. The chocolate ellipsoids are then put in a big rotating drum, something like a cement mixer. While jostling in the drum, they are sprayed with a sugary liquid that hardens into a white candy shell. The constant movement prevents the candies from congealing into a big lump. The motion also smoothes out any rough edges. In concept, the rotating drum is like that used to polish gemstones.

The candies are then squirted with a second, coloured sugar liquid. This hardens into the colored coating on top of the white shell.

If you are on a boat and toss a suitcase overboard, will the water level rise or fall?

This question is easy, provided you know the role that floating objects displace a weight of water equal to their own. That’s the catch. Probably, most technically trained people have heard of that rule somewhere along the line. Most are shaky on how well they remember it. (Uh, was it the weight or the volume that objects displace?.)

Here’s a virtually math-free way of working it out for yourself. Start with the basics. Any time you throw weight out of a floating boat, the boat gets a little lighter and therefore rises. Okay?

Unfortunately, that’s not what this question asks. It asks whether the water level rises or falls, not the boat.

Normally you don’t pay attention to the water level in a body of water large enough to permit boating. Tossing a suitcase out of boat is not going to change the water level of a lake or ocean perceptibly. The question is whether it changes the level in principle.

The only thing that will change the water level is a change in the volume of submerged objects. The fancy term for that is displacement. Picture a toy boat in a bathtub. The submerged hull of the boat occupies a certain volume that, consequently, isn’t occupied by water. That volume is called the boat’s displacement. It is not the total volume of the boat but only the part that is below the waterline.

The tub’s water level depends on the total displacement of all the toy boats, rubber ducks and other objects that are floating or submerged in it. Add more toy boats, and that increases the displacement. The displaced water has to go somewhere, so the water level rises. Take toy boats out, decreasing the displacement, and the water level falls.

All this applies to a lake or ocean too. It’s just that the irregular shape of a lake bed or ocean basin makes it harder to visualize the effect. Any water-level changes in the ocean are going to be microscopic anyway.

The question becomes “how does tossing a suitcase out of a boat change the displacement?” We know that tossing the suitcase out makes the boat lighter. That, in turn, makes the boat sit a little higher in the water, decreasing the boat’s displacement. But once the suitcase splashes back into the water, it displaces water too. Is the net effect positive, negative or zero?

To answer that, we need to establish a relationship between displacement and weight. Here’s a “mental physics” experiment to do so:

Picture an inflated beach ball floating in a bathtub. A beach ball normally does not have much weight, being mostly air and thin plastic. In your mind’s eye, it sits practically on top of the water surface, barely displacing any water — right? Conclusion: Near-zero weight means near-zero displacement.

Therefore, tossing a suitcase out of a boat makes no difference in the total displacement (or water level) assuming the suitcase floats.

If the suitcase sinks, the water level will fall.

What does all the ice in a hockey ring weigh?

A hockey rink is maybe 100 by 200 feet. The ice is about an inch thick. Let’s put everything in inches: around 1,000 by 2,000 by one, or two million cubic inches of ice.

How much does a cubic inch of ice weigh? A little less than what a cubic inch of water does. It’s probably easier to fall back on the metric system. A cubic centimetre of water weighs a gram. An inch is about 2.5 centimetres, so a cubic inch is about (simplifying it for mental math) 2 x 3 x 2.5, or 3 x 15, or 15 cubic centimetres. That means a hockey rink’s ice comes to 15 x 2 million = 30 million cubic centimetres and weighs as many grams. Or 30,000 kilograms. Or about 60,000 pounds.

An NHL regulation rink is an oval, 85 x 200 feet, with a corner radius of 28 feet. One inch is a common ice thickness in rinks. Use these accurate figures, and the weight of that volume of liquid water would be 38,500 kilograms. Allowing for the lower density of ice, the weight should be about 35,200 kilograms.

How many times a day do a clock’s hands overlap?

Most people quickly realize that the answer has to be 24, give or take. The issue is nailing down that “give or take” part.

Recognize, first of all, that there is nothing capricious about the overlaps. Both hands move at fixed speeds. Therefore, the time interval between overlaps is constant.

This constant interval is a little more than an hour. At midnight, the hour and minute hands are exactly superimposed. It takes an hour for the minute hand to make a complete circuit. In that same time, the hour hand has move 1/12 of a circuit to the numeral one. It then takes another five minutes for the minute hand to catch up to where the hour hand was, in which time the hour hand has crept a bit farther.

Before getting sucked into a Zeno’s Paradox, let’s settle for the moment by saying that the interval is a little more than 65 minutes. We also know that the exact interval has to divide evenly into 24 hours, since the day ends as it started, with both hands up and overlapping. In fact, it has to divide evenly into 12 hours. The way the hands move in the P.M. in an exact replica of the way they move in the A.M.

Focus on the 12-hour period from midnight to noon. The hands cannot overlap 12 times in that period, for if they did, it would mean that the interval between overlaps was 12/12, or exactly one hour — and we know it’s a bit more than 65 minutes. No, there must be 11 overlaps in a 12-hour period. That means the interval between overlaps is 12/11 hour, which comes to 65.45 minutes. This must be the precise interval that we balked at calculating a moment ago.

Doubling 11 gives 22 overlaps in a 24-hour period. Twenty-two is the answer, unless you want to split hairs. Should you count the overlap at the midnight that begins the day, and also at the midnight that ends the day, the answer is 23.

Mike and Todd have \$21 between them. Mike has \$20 more than Todd. How much does each have? You can’t use fractions in the answer.

A trick question incorporating a challenge. The basic problem is straightforward. You may be tempted to say that Mike has \$21 and Todd has \$1. But no, that adds up to \$22. The real answer has to be that Mike has \$20.50 and Todd has 50 cents. If that’s not obvious, you can write out the equations and use algebra. You can also prove that this is the only answer. But the interviewer insists there can be no fractions in the answer.

The interviewer is wrong (or hiding behind the technicality that whole cents aren’t “fractions”). You’re supposed to stand your ground and defend the \$20.50/50 cent answer. That’s life in a big organization.

On average, how many times would you have to flip open the Manhattan phone book to find a specific name?

By “flip open”, the interviewer means that you open the book to a random two-page spread. (You don’t try to open it to the right place based on the letter of the alphabet.) Should the desired name be anywhere on the two visible pages, you’ve found it.

There is a simple answer and a more sophisticated one. The simple answer is this: Say the Manhattan white-page directory has 1,000 pages. That means the book has 500 openings. The chance of flipping the book open to any specific name on your first (or any subsequent) try is therefore about one in 500. On the average, it takes about 500 random flips to a particular name.

The more sophisticated answer, detailed in the book, is that the original estimate of 500 random openings would give you only a 63 per cent chance of finding the name, because in random flipping the same pages are often revisited. To achieve 90 per cent confidence, you would need 1,150 flips.

How would you design Bill Gates’ bathroom?

There are two key points in answering this question. One is that Bill Gates gets what Bill Gates wants. The other is that you’re supposed to come up with at least some ideas that Gates wants but wouldn’t have thought of on his own (otherwise, what’s the point of hiring you to design his bathroom?)

You’re supposed to start off saying that you’d sit down with Gates and listen to what he wants his bathroom to be like. You’d get the budget and deadlines up front. You’d suggest a lot of ideas and see which ones he likes. Then you’d make a plan and show it to Gates for feedback. The plan would go through many cycles of revision. Meanwhile, you’d make sure the project came in on time and within budget. This much applies to any design question.

As to the ideas you come up with, be warned that it’s tough to top the reality. Gates’ bathtub has a feature that lets him fill it to desired temperature from his car. For real.

Putting computer technology throughout the home, bathroom included, is something that Microsoft people take seriously. Microsoft’s research divisions are pursuing things such as smart medicine cabinets and cupboards that tell you when your prescription needs refilling or you’re out of toilet paper.

So if you want to impress Microsoft’s interviewers, you’re not going to get much mileage out of talk of electrically warmed toilets. Here are a few ideas of the slant they are looking for:

•A feature that automatically locks medicine cabinets or cupboards containing household chemicals when an unaccompanied child enters the bathroom. Gates’ house already has rudimentary ways of “knowing” when someone is in a room, and who what someone is. There’s talk of iris scanners and such that would monitor people’s identities more accurately and unobtrusively.

•A hands-free notepad. Everyone gets great ideas in the bathroom. You don’t want to use a PDA when your hands are wet, and if there’s one room in the house that isn’t going to have a PC in it, it’s the bathroom. All you need is a voice-recognition device that can record a spoken message after you say a code phrase such as “memo for Bill.” The device automatically e-mails the message to your mailbox so it’s ready for you to work.

•A mirror that doesn’t reverse left and right. It’s a video screen with a hidden camera, showing your own image the way other people see you. It makes it much easier to use scissors to trim a stray hair.

Design a spice rack for a blind person.

You are allowed to intelligently redefine the question. Should you have a strong reason for devising a spice-lazy Susan rather than a rack, then that’s what you should do. So, all right, we are devising an integrated system for storing and dispensing spices for a blind person.

The feature common to almost all answers is Braille labels. There is something to be said for putting Braille on the lids. It is hard to read Braille on a curved surface. Since most spice jars are cylindrical, the lid is the one accessible flat surface. You could run your fingers over a row of lids to find the one you want.

There are several disadvantages to this approach. It’s easy to read the lids only if you’ve made meticulously sure that every jar is returned to the rack with the lid in the right orientation. Otherwise the labels will be in all possible states of rotation. That makes them hard to read.

A second problem is that lids easily get separated from the jars while cooking. A third is that you have to transfer spices from the supermarket’s jars to the special jars and lids used with the spice rack.

How about putting the Braille on the jar? Then it doesn’t matter whether the lids get swapped. But, as we’ve already noted, it’s hard to read a curved Braille label, and the label might not be facing outward.

Making the jars square in cross section, with identical Braille labels on all four sides, can solve these problems. The spice rack itself should be designed so that jars are squared up when returned. It should also have an opening where the labels are so you can efficiently slide your finger across them to find the right one.

Poundstone’s book goes into much more detail about various designs for spice racks for blind people.

Why are beer cans tapered at the top and bottom?

If you guess it’s to make the can stronger, you’re sort of right. The tapered ends are an architectural issue. Cans, like suspension bridges, work together as a whole. That often means it’s difficult to supply a simple explanation for a particular feature.

From a historic perspective, the tapers were not added to make cans stronger. Beer cans were already strong enough to hold beer. What more can you ask of a beer can? The tapers are instead a feature of a design that minimizes the amount of metal used. That may not seem like such a big deal, but it is when you consider how many cans are produced and recycled each year.

There was a time when beer and carbonated soft drinks came in heavy steel cans whose cross sections were nearly rectangular. The steel had to be fairly thick to keep a carbonated beverage under pressure. These cans were three-part, meaning that a circular top and bottom were attached by a crimp to a cylindrical middle.

As can companies became more cost and environment conscious, they figured out ways to switch to thinner aluminium cans The thin aluminium is less strong. Like eggshells, today’s cans are just about as thin as they can be and still reliably enclose their contents. This demands architectural tricks that weren’t necessary with the steel cans.

The thickest and strongest part of the can is the top, attached separately with a crimp. The top has to stand the stress of someone ripping opening the flip top. Because the top is thicker metal, the manufacturers found it desirable to minimize its diameter. So they shrunk the top a little. This meant adding a bevel at the top to connect it to the rest of the can. (They couldn’t shrink the diameter of the whole can, or it would hold less beer.) Once you shrink the top, you also have to shrink the bottom, for the cans are supposed to stack. Both top and bottom are tapered.

The above text is excerpted and edited from William Poundstone’s book How Would You Move Mount Fuji? Microsoft’s Cult of the Puzzle — How the World’s Smartest Companies Select the Most Creative Thinkers. Poundstone is the author of nine books, including the bestselling Big Secrets series and Carl Sagan: A Life in the Cosmos. He has written for The Economist, Esquire, Harper’s and the New York Times Book Review among other publications. His science writing has been nominated twice for the Pulitzer Prize.