If you look at a
clock and the time is 3:15, what is the angle between the hour and the minute hands?
The answer to this is not zero! The hour hand, remember, moves as well. The hour
hand moves a quarter of the way between three and four, so it moves a quarter of
a twelfth (1/48) of 360 degrees. So the answer is seven and a half degrees, to be
You have a five-gallon jug and a three-gallon
jug. You must obtain exactly four gallons of water. How will you do it?
You should find this brainteaser fairly simple. If you were to think out loud, you
might begin by examining the ways in which combination of five and three can come
up to be four. For example: (5 - 3) + (5 - 3) = 4. This path does not actually lead
to the right answer, but it is a fruitful way to begin thinking about the question.
Here's the solution: fill the three-gallon jug with water and pour it into
the five-gallon jug. Repeat. Because you can only put two more gallons into the
five-gallon jug, one gallon will be left over in the three-gallon jug. Empty out
the five-gallon jug and pour in the one gallon. Now just fill the three-gallon jug
again and pour it into the five-gallon jug. Ta-da. (Mathematically, this can be
represented 3 + 3 - 5 + 3 = 4)
You are faced with two doors. One door leads
to your job offer (that's the one you want!), and the other leads to the exit.
In front of each door is a guard. One guard always tells the truth. The other always
lies. You can ask one question to decide which door is the correct one. What will
The way to logically attack this question is to ask how you can construct a question
that provides the same answer (either a true statement or a lie), no matter who
There are two simple answers. Ask a guard: "If I were to ask you if this door
were the correct one, what would you say?" The truthful consultant would answer
yes (if it's the correct one), or no (if it's not). Now take the lying
consultant. If you asked the liar if the correct door is the right way, he would
answer no. But if you ask him: "If I were to ask you if this door were the
correct one, what would you say," he would be forced to lie about how he would
answer, and say yes. Alternately, ask a guard: "If I were to ask the other
guard which way is correct, what would he say?" Here, the truthful guard would
tell you the wrong way (because he is truthfully reporting what the liar would say),
while the lying guard would also tell you the wrong way (because he is lying about
what the truthful guard would say).
If you want to think of this question more mathematically, think of lying as represented
by -1, and telling the truth as represented by +1. The first solution provides you
with a consistently truthful answer because (-1)(-1) = 1, while (1)(1) = 1. The
second solution provides you with a consistently false answer because (1)(-1) =
-1, and (-1)(1) = -1.
How many gallons
of white house paint are sold in the U.S. every year?
THE "START BIG" APPROACH: If you're not sure where to begin, start
with the basic assumption that there are 270 million people in the U.S. (or 25 million
businesses, depending on the question). If there are 270 million people in the United
States, perhaps half of them live in houses (or 135 million people). The average
family size is about three people, so there would be 45 million houses in the United
States. Let's add another 10 percent to that for second houses and houses used
for other purposes besides residential. So there are about 50 million houses.
If houses are painted every 10 years, on average (notice how we deftly make that
number easy to work with), then there are 5 million houses painted every year. Assuming
that one gallon of paint covers 100 square feet of wall, and that the average house
has 2,000 square feet of wall to cover, then each house needs 20 gallons of paint.
So 100 million gallons of paint are sold per year (5 million houses x 20 gallons).
(Note: If you want to be fancy, you can ask your interviewer whether you should
include inner walls as well!) If 80 percent of all houses are white, then 80 million
gallons of white house paint are sold each year. (Don't forget that last step!)
What is the size of the market for disposable diapers in China?
Here's a good example of a market sizing. How many people live in China? A
billion. Because the population of China is young, a full 600 million of those inhabitants
might be of child-bearing age. Half are women, so there are about 300 million Chinese
women of childbearing age. Now, the average family size in China is restricted,
so it might be 1.5 children, on average, per family. Let's say two-thirds of
Chinese women have children. That means that there are about 200 million children
in China. How many of those kids are under the age of two? About a tenth, or 20
million. So there are at least 20 million possible consumers of disposable diapers.
1 billion people x 60% childbearing age = 600,000,000 people
600,000,000 people x 1/2 are women = 300,000,000 women of childbearing age
300,000,000 women x 2/3 have children = 200,000,000 women with children
200,000,000 women x 1.5 children each = 300,000,000 children
300,000,000 children x 1/10 under age 2 = 30 million
How many square feet of pizza are eaten in the United States each month?
Take your figure of 300 million people in America. How many people eat pizza? Let's
say 200 million. Now let's say the average pizza-eating person eats pizza twice
a month, and eats two slices at a time. That's four slices a month. If the
average slice of pizza is perhaps six inches at the base and 10 inches long, then
the slice is 30 square inches of pizza. So four pizza slices would be 120 square
inches. Therefore, there are a billion square feet of pizza eaten every month.
300 million people in America
200 million eat pizza
Average slice of pizza is six inches at the base and 10 inches long = 30 square
inches (height x half the base)
Average American eats four slices of pizza a month
Four pieces x 30 square inches = 120 square inches (one square foot is 144 inches),
so let's assume one square foot per person
200 million square feet a month
How would you estimate the weight of the Chrysler building?
This is a process guesstimate - the interviewer wants to know if you know what questions
to ask. First, you would find out the dimensions of the building (height, weight,
depth). This will allow you to determine the volume of the building. Does it taper
at the top? (Yes.) Then, you need to estimate the composition of the Chrysler building.
Is it mostly steel? Concrete? How much would those components weigh per square inch?
Remember the extra step - find out whether you're considering the building
totally empty or with office furniture, people, etc.? (If you're including
the contents, you might have to add 20 percent or so to the building's weight.)
Why are manhole covers round?
The classic brainteaser, straight to you via Microsoft (the originator). Even though
this question has been around for years, interviewees still encounter it.
Here's how to "solve" this brainteaser. Remember to speak and reason
out loud while solving this brainteaser!
Why are manhole covers round? Could there be a structural reason? Why aren't
manhole covers square? It would make it harder to fit with a cover. You'd have
to rotate it exactly the right way. So many manhole covers are round because they
don't need to be rotated. There are no corners to deal with. Also, a round
manhole cover won't fall into a hole because it was rotated the wrong way,
so it's safer.
Looking at this, it seems corners are a problem. You can't cut yourself on
a round manhole cover. And because it's round, it can be more easily transported.
One person can roll it.
You have 12 balls. All of them are identical except one, which is either heavier
or lighter than the rest. The odd ball is either hollow while the rest are solid,
or solid while the rest are hollow. You have a scale, and are permitted three weighing.
Can you identify the odd ball, and determine whether it is hollow or solid?
This is a pretty complex question, and there are actually multiple solutions. First,
we'll examine what thought processes an interviewer is looking for, and then
we'll discuss one solution.
Start with the simplest of observations. The number of balls you weigh against each
other must be equal. Yeah, it's obvious, but why? Because if you weigh, say
three balls against five, you are not receiving any information. In a problem like
this, you are trying to receive as much information as possible with each weighing.
For example, one of the first mistakes people make when examining this problem is
that they believe the first weighing should involve all of the balls (six against
six). This weighing involves all of the balls, but what type of information does
this give you? It actually gives you no new information. You already know that one
of the sides will be heavier than the other, and by weighing six against six, you
will simply confirm this knowledge. Still, you want to gain information about as
many balls as possible (so weighing one against one is obviously not a good idea).
Thus the best first weighing is four against four.
A company has 10 machines that produce gold coins. One of the machines is producing
coins that are a gram light. How do you tell which machine is making the defective
coins with only one weighing?
Think this through - clearly, every machine will have to produce a sample coin or
coins, and you must weigh all these coins together. How can you somehow indicate
which coins came from which machine? The best way to do it is to have every machine
crank a different number of coins, so that machine 1 will make one coin, machine
2 will make two coins, and so on. Take all the coins, weigh them together, and consider
their weight against the total theoretical weight. If you're four grams short,
for example, you'll know that machine 4 is defective.
The four members of U2 (Bono,
the Edge, Larry and Adam) need to get across a narrow bridge to play a concert.
Since it's dark, a flashlight is required to cross, but the band has only one
flashlight, and only two people can cross the bridge at a time. (This is not to
say, of course, that if one of the members of the band has crossed the bridge, he
can't come back by himself with the flashlight.) Adam takes only a minute to
get across, Larry takes two minutes, the Edge takes five minutes, and slowpoke Bono
takes 10 minutes. A pair can only go as fast as the slowest member. They have 17
minutes to get across. How should they do it?
The key to attacking this question is to understand that Bono and the Edge are major
liabilities and must be grouped together. In other words, if you sent them across
separately, you'd already be using 15 minutes. This won't do. What does
this mean? That Bono and the Edge must go across together. But they can not be the
first pair (or one of them will have to transport the flashlight back).
Instead, you send Larry and Adam over first, taking two minutes. Adam comes back,
taking another minute, for a total of three minutes. Bono and the Edge then go over,
taking 10 minutes, and bringing the total to 13. Larry comes back, taking another
two minutes, for a total of 15. Adam and Larry go back over, bringing the total
time to 17 minutes.
What is the decimal equivalent of 3/16 and 7/16?
A commonly-used Wall Street interview question, this one isn't just an attempt
to stress you out or see how quick your mind works. This question also has practical
banking applications. Stocks often are traded at prices reported in 1/16s of a dollar.
(Each 1/16 = .0625, so 3/16 = .1875 and 7/16 = .4375).
What is the sum of the numbers from one to 50?
Another question that recent analyst hires often report receiving. This is a relatively
easy one: pair up the numbers into groups of 51 (1 + 50 = 51; 2 + 49 = 51; etc.).
Twenty-five pairs of 51 equals 1275.
You have a painting that is $320 that is
selling for 20 percent off. How much is the discounted price?
Calculate quickly: What's 80 percent of $320? The answer's $256. Even
in a question like this, if you are good with numbers and use shortcuts, don't
be afraid to talk aloud. For example: 80 percent of $320 can be broken down to a
calculation like 80 percent of $80 x $4, or 162.
playing three-card monte. Two cards are red, one is black. (Note: In three-card
monte, the three cards are face down and you try to pick the black card in order
to win.) You pick the middle card. After you pick, the dealer shows that one of
the cards you have not chosen is red. You are given the chance to switch your selection.
The short answer is yes. By switching, you are betting that the card you initially
chose was red. By not switching, you are betting that the card you initially chose
was black. And because two out of three cards are red, of course, betting on red
is the way to go.
Let's break it down, starting with the not switching case. Say the first card
you chose was the black one. This happens one-third of the time. If you do not switch
your choice, you win. Needless to say, the other two-thirds of the time, having
picked a red card, and deciding not to switch, you lose. In other words, if you
do not switch, you win a third of the time.
Now let's examine what happens when you switch cards. Say the first card you
chose was the black one. Again, this would happen one-third of the time. If, after
being shown a red card, you switch, you lose. The other two-thirds of the time,
if you switch, you win because the dealer has already shown you that one of the
cards you did not pick is red. Given the premise that your original pick was a red
card, the card you are switching to must be the black one. You will win two-thirds
of the time.
A straight flush beats a four-of-a-kind
in poker because it is more unlikely. But think about how many straight flushes
there are - if you don't count wraparound straights, you can have a straight
flush starting on any card from two to 10 in any suit (nine per suit). That means
there are 36 straight flushes possible. But how many four of a kinds are there -
only 13. What's wrong with this reasoning?
Immediately, you should think about what the difference is between a straight flush
and a four-of-a-kind. One involves five cards, and the other involves four. Intuitively,
that's what should strike you as the problem with the line of reasoning. Look
closer and you'll see what that means: for every four of a kind, there are
actually a whole bunch of five-card hands: 48 (52 - 4) in fact. There are actually
624 (48 x 13) of them in all.
If you have seven white socks and nine black socks in a drawer, how many do you
have to pull out blindly in order to ensure that you have a matching pair?
Three. Let's see - if the first one is one color, and the second one is the
other color, the third one, no matter what the color, will make a matching pair.
Sometimes you're not supposed to think that hard.
Tell me a good joke that is neither sexist nor racist.
If you can't think of any, you're in the same boat as the unfortunately
tongue-tied recent candidate at Salomon Smith Barney. Find one and remember it.
I were to fill this room with pennies, how many pennies would fit in?
A literally in-your-face guesstimate.
Say you are driving on a one-mile track. You do one lap at 30 miles an hour. How
fast do you have to go to average 60 miles an hour?
This is something of a trick question, and was recently received by a Goldman candidate.
The first thought of many people is to say 90 miles an hour, but consider: If you
have done a lap at 30 miles an hour, you have already taken two minutes. Two minutes
is the total amount of time you would have to take in order to average 60 miles
an hour. Therefore, you can not average 60 miles an hour over the two laps.