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Moving Mountains

Here goes. The puzzles and brain-teasers that could land you a job at Microsoft.

MS puzzles: Spotting the best

"How Would You Move Mount Fuji?"

That is the title brain-teaser of William Poundstone's book (reviewed in Business Today issue dated June 8). Unfortunately, it has a profoundly disappointing answer - to do with truckload volume calculations. But hey, Microsoft is said to be reasonably receptive to creative responses to crazy questions, and mountain-moving is something that lends itself to assorted mythical, not to mention proverbial, interpretations.

Thankfully, Poundstone assures readers, Microsoft recruiters don't always ask questions designed to flummox you (just to gauge your response). The company's favourite question, ever since CEO Steve Ballmer asked himself this on a jog (as the legend goes), is deceptively simple. "Why are manhole covers round rather than square?"

It's a reasonable question, and it's also something you could expect people to have thought about. After all, the classic manhole is an object everybody sees and recognizes, and it was designed by somebody somewhere at some point, surely. So there ought to be a jolly good reason why it was designed that way. The most favoured answer, as the book tells it, is straightforward: a square cover could fall into its own hole. This is because the diagonal of a square is underroot-of-2 (1.414...) times its side. Hold a square manhole cover vertically, and turn it a little, plonk --- it falls straight into its own hole. A circle, in contrast, has the same diameter whichever way across you measure it. No matter how you tilt it around, it won't fall in.

Replies to that question, however, do tend to vary. Saying that holes are naturally round in the first place, for example, may be fair, but is too commonplace to impress anybody. A famous personality once even said that a round shield is better in a fight. And that a circle is symbolic of infinity, and thus a good reminder of the concept, like a dome.

Another interesting question is a variation of the old guess-the-colour-of-the-bear riddle. "How many points are there on the globe where, by walking one mile south, one mile east and one mile north, you reach the place where you started?"

A respondent unfamiliar with the old bear riddle would proceed to draw a mental map, and find it impossible to travel south, east and north (three sides of a square) and still end up at the point of origin. This is the 'flat earth society' trap.

Someone thinking globular instead of flat would quickly pinpoint the North Pole (yes, it's a white bear). Starting from this point on the globe, anywhere you walk would be south. Take a turn east, and you cut a one-mile arc with the North Pole as the center. Turn north and one mile ahead, you're back at the Pole. It's a pie-slice journey.

Needless to say, that cannot be done at the South pole. Unless you're Aussie, and insist on seeing the world upside down. But even then, there'll be just one such point on the planet. So, is the North Pole unique as an enabler of such a journey?

Er... no. Beware the 'unipolar' trap. How? Think unconventional. Think possibilities. Imagine starting out from a point a little more (say a foot) than a mile from the South Pole. Travel a mile south, take a 90-degree turn east, and execute a complete 360-degree circle about the South Pole of one-mile circumference - at each point travelling due east, of course - and then backtrack north a mile to the starting point. Voila! And just how many such points are there? An infinite number. That's the favoured answer.

For a slightly more complex brain teaser, try this one. "A train leaves LA for New York (3,500 miles apart) at a constant speed of 15 miles per hour. At the same moment, a train leaves New York for LA on the same track, at 20 miles per hour. At precisely that moment, a bird leaves LA station and flies along the track towards New York, at 25 miles per hour. When it reaches the train from New York, it instantly reverses direction, and, maintaining constant speed, reaches the other train, and then reverses direction again, and again, and again. The bird flies back and forth until the trains collide. How far will the bird have traveled?"

Quite a mouthful that question is, but once you've pictured what's happening out there, you can being structuring the arithmetic puzzle to be solved. If you've studied college-level math, you'd probably look at it all this back-and-forth flying as an infinite series problem (which has a particular formula for summation).

But uh-oh oh, be careful. Watch out for the 'linear track' trap. Think for a second, apply some regular common sense, and you'd be spared a lot of trouble. How? Change the way you look at the problem. Look at it this way. The two trains are approaching one another at a relative speed of 35 miles per hour (simple arithmetic: 15 plus 20). Given the distance, the collision should occur 100 hours later (3,500 divided by 35). Now, no matter how many times the poor bird had to change direction, it still maintains constant speed. For a projected 100 hours. At 25 miles per hour, that means 2,500 miles (25 times 100). That's the right answer.

Is it Microsoft's most favoured solution? Guess for yourself.

Meanwhile, try this - again, a variation of the classic heaven/hell puzzle. "In front of you are two doors. One leads to your interview, the other to an exit. Next to the door is a consultant. He may be from our firm or from a rival. The consultants from our firm always tell the truth. The consultants from the other firm always lie. You are allowed to ask the consultant one question to find out which is the door to your interview. What would you ask?"

This ranks as the software industry's hottest brain-teaser ever, some say. This is so for the simple force of logic that the solution demands. Think about it. The only way to get the correct door is to come up with a question that renders irrelevant whether the consultant tells the truth or lies. The way to do this is to use the power of logical negation.

Point to one door and ask, "If I asked you whether this is the way to my interview, would you say it is?" Now, the perfect liar is a consistent liar. So he would lie even about what he would say (a lie), if asked the straight question whether this is the correct door. So the liar would say the opposite of the originally intended lie (to the direct question). Since the reversal of falsehood yields the truth in this binary situation, that's what you get. The liar nods if and only if it is indeed the door to the interview.

As for a truthful consultant, the response to the same question would also reveal the correct door, since the truth teller is consistent as well. Another possible question to ask is, "If I asked a consultant of the other firm whether this is the door to my interview, would he or she say it is?" This involves both firms, but uses the same principle to obtain the outcome.

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