Brainteaser - ClockA key part of a consultancy interview is the case study. Below is an example of the kind of case study question you might be asked.

At 3.25 pm, what is the angle between the hands on a clock?

Answer

This question can be solved in four steps:

Step 1: Choose a point to anchor your angle calculations, say, 12 o’clock.

Step 2:  Calculate the angle between the minute hand at 3:25 and 12 o’clock:    (25-0)/60*360˚ = 150˚

Step 3:  Calculate the angle between the hour hand at 3:25 and 12 o’clock:   3/12*360+25/60* 1/12*360˚ = 102.5˚

Step 4: At 3.25pm, the angle between the hour hand and minute hand is the difference between step 2 and step 3:   150˚-102.5˚ = 47.5˚

Note: It is not uncommon for people forget to take account of step 3!

Tube TrainsA key part of a consultancy interview is the case study. Below is an example of the kind of case study question you might be asked.

How many trains are there on the London Underground?

Answer:

For this type of structured estimation question, it is especially important to make your assumptions and thought process clear.  You will be expected to make some simplifications as you go along.  Getting the answer exactly right isn’t required.

You should start by stating any restrictions you are going to make to simplify the problem or make the scope more manageable. For example, you might want to consider operation during peak times only, and only cover trains actually in operation – excluding trains not in service, sitting in sidings etc. In addition, you might want to clarify that the London Underground excludes the DLR.

Next break the problem into sub-questions or a logic flow to show the interviewer how you are thinking and that you grasp the key principles required to solve the problem. For example, for this question one key principle to establish at the start is the idea of “routes” or branches, rather than individual lines (a complex line such as the Northern line consists of several branches, and each train serves a particular branch rather than the entire line).

One possible flow is shown below.  We answer the questions as numbered.

 

Question 1: How many individual routes are there on the underground?

Your interviewer would expect you to make a sensible guess on the number of lines and routes, but would not expect you to get the answer spot on – or would provide a Tube map for reference.

One might assume there are 10-12 lines and on average two routes or branches per line.

We estimate 25 distinct routes.

Question 2: What is the average number of stations on a route?

Again, this is the type of question where a sensible guess should be sanity checked, with the interviewer if needed.  We estimate there are 25 stations for the average line.  25 stations on 12 lines gives 300 stations in total. This sounds reasonable.  

However, to determine average number of stations on a route, we must be careful not to under-weight stations shared by multiple routes. One way to solve this problem is to consider each route to be isolated, and that every station that is shared between say 2 routes, counts as 2 stations. Now we must estimate how many stations are ‘doubles’ – say half for simplicity,  given that there are on average two routes per line.

This means the effective number of stations is (300 + (1/2 x 300)) = 450 effective stations.

The average number of stations on a route is then 450/25 = 18 stations per route.

Question 3: What is the average journey time per station? A reasonable estimate without any calculation would be to allow 4 minutes travel time per station (including time spent stood at platform): you could estimate this based on your own experiences using the Underground.

Question 4: What is the average round trip time for a route?

From the above, an average round trip on our average route would take 18 x 4 x 2 = 2hr 24mins.

Allowing some time at the terminal, an average round trip might take 2hrs 30mins.

Question 5: How frequently do trains stop at each station?

Train frequencies at busy, central stations are every 2-3 minutes, often due to overlapping routes.  Further out train frequencies can be as low as every 15 minutes on some routes Again, this estimate would be based on your own experiences. A reasonable approximation for train frequency on a single route would be somewhere between the two.

We assume every 7.5 minutes (equivalent to 8 trains per hour) during peak time.

Question 6: How many trains are required on an average route?

To maintain a frequency of 8 trains per hour (given that the time taken for one round trip is 2 hrs 30 mins) would require 20 trains on each route (i.e. 8+8+8/2 )

Question 7: How many trains on the underground?

There are 25 routes and 20 trains per route, so a very approximate figure for the number of trains on the London Underground at peak time is 20 x 25 = about 500 trains. (This is very close to the actual figure!)

Newspaper

A key part of a consultancy interview is the case study. Below is an example of the kind of case study question you might be asked.

How many words are there in the daily edition of The Times?

Answer:

When faced with a structured estimation question such as this one, it is essential to make your assumptions and thought-process clear at the beginning. You should start by laying out any key principles which will help you simplify the problem – getting the answer exactly right isn’t required. In this example, the key principle is to come up with the number of text-only pages in a standard copy, and combine this with an estimate of words per text-only page. One way of simplifying the problem is to exclude text in adverts and diagrams, to make it easier to covert a standard page into a fraction of text-only page. Another simplification is to exclude any additional supplements.

Once you have set out any key principles and simplifying assumptions, break the problem into sub-questions or a logic flow to show the interviewer how you are thinking. One possible set of sub-questions is shown below:

1. On average, how pages are there in the daily Times, excluding supplements?2. How much of the paper is taken up by adverts?3. What fraction of news story pages is taken up with images or diagrams?4. What is the number of words on a text-only page?

Question 1You would be expected to answer this question with a reasonable estimate based on knowledge, for example by breaking down the paper into its constituent sections (UK news, World news, Business, Sport, Court & Social etc). The interviewer would also help guide you in the right direction if needed. The answer is approximately 65 pages.

Question 2The two most common advertisement sizes in the Times are full page and 1/3 page.Estimating the share of advertisements is not trivial but a good approximation would be that 20% of the paper is taken up by adverts with minimal text. You could estimate this number based on common sense / experience, and then check with the interviewer to make sure. Given the low word density of advertisements compared to text, it would be a reasonable approximation to discount their word contribution, and therefore assume that, of the 65 pages, 20% are non-text pages. This is equivalent to 65 / 5 =  13 pages of adverts.

This leaves 65 – 13 = 52 pages of news stories.

Question 3In addition to adverts, news stories are often illustrated with images or diagrams that take up a large amount of space on the page. This would vary across different sections of the paper (for

example, the main News section will have more images than the Business section), so you would be expected to take an average across the whole paper. A reasonable approximation is that 25% of news space is taken by diagrams and images.

Remembering that you have calculated that there are 52 pages excluding adverts, 0.25 x  52 pages = 13 pages of news related images and diagrams.

This leaves 52 – 13 = 39 text-only page equivalents.

Question 4One way of working out the number of words on a text page is to break it down into (number of columns) x (number of words per column line) x (number of lines per column).From experience, you might know that a typical compact broadsheet format like The Times has five columns of text on a page. On average, it is reasonable to expect 6 words per line in a column. This totals 30 words across each horizontal line of page.

A guess of the number of lines down a page based on the experience of reading The Times is more difficult, so you might want to compare it to a more familiar paper size, such as an A4 sheet. You might estimate that the length of a compact broadsheet is equal to roughly 1.5 A4 sheets (whereas a traditional broadsheet is over 2 times longer than an A4 sheet). Wide ruled A4 paper has about 35 lines per page – so 1.5 wide-ruled A4 pages would hold 52.5 lines – you could round this to 50 to make later calculations easier. You would also want to take into account the fact that newspapers typically use much smaller type fonts than hand-writing, allowing them to fit more lines to the page. Assuming that newspaper line spacing is about half that of wide ruled A4, a typical newspaper may have 2 x 50 = 100 lines per page.

This would make 100 lines * 30 horizontal words = 3000 words per page.

An extra step to the analysis would be to account for the space taken up by headlines, which typically take up around 1/3rd of the page.Accounting for headlines, the number of words on would be 2/3 x 3000 = 2000 words per page.

We calculated previously that there were 39 pages of text in the average copy of The Times. This implies that there are 39 x 2000 = 78 000 words in the average copy of The Times. Given that the number of calculation steps involved, you might want to conclude by expressing your final estimate as a range, rather than a precise number: e.g using a confidence interval of +/- 5%.

Therefore, a copy of the Times might typically contain 75 000 – 80 000 words.

Brainteaser - iPodA key part of a consultancy interview is the case study. Below is an example of the kind of case study question you might be asked.

How many songs are stored on iPods in the UK?

Answer:

A guesstimate case question should be solved by posing and answering a number of sub-questions. To estimate how many songs that are stored on iPods in the UK, the following sub-questions and answers could be used (the numbers within brackets below are for illustrative purposes only. Note that this question requires you to have some background knowledge about digital music):

1. How many people live in the UK?2. What is the MP3 player penetration in the UK? 3. What is Apple’s share of the MP3 player market in the UK?4. What is the average storage space on an iPod?5. How large is the proportion of storage space used for music?6. What is the average size of song?

SQ1: How many people live in the UK?You don’t have to be exactly right, but you should know that the population is around 60 million people.

SQ2: What is the MP3 player penetration in the UK?When possible, try and relate the question to something you know, such as how many of all the people you know have an MP3 player (say, 25% across all ages). This will give you a starting point, but remember that the people you know might not be perfectly representative of the entire UK population, e.g. in terms of disposable income (assuming the people you know are a representative sample of the UK population, the penetration rate would remain 25% of the market).

SQ3:  What is Apple’s share of the MP3 player market in the UK?iPods are the dominant MP3 players and the key competitors SanDisk, Microsoft and Creative only have small fractions of the market. You could estimate a number based on the people you know who have an MP3 player (say, 70% of them have iPods. Note that you could have estimated the iPod penetration directly by combining questions 1 and 2, but splitting them into penetration and market share shows that you know these 2 factors are key revenue drivers).

SQ4: What is the average storage space on an iPod?Consider the relative mix of budget versions with less storage space and more expensive versions with more storage space (say, 25% of the market has 1 GB, 50% has 4GB and 25% has 40GB weighted average of 12.25 GB ≈ 12 GB).

SQ5: How large is the proportion of storage space used for music?iPods are not only used for music, some versions can for example also store videos and pictures. Don’t forget to consider empty disk space. (Assume that 33% of the average storage space is used for music).

SQ6: What is the average size of song?This is something you essentially need to know, or at least you need to know how many songs that can be stored on an iPod with a given size (assume that the average size is 4 MB).

Finally, you need to combine the answers from all the sub-questions in order to come up with the estimated number of songs stored on iPods in the UK. (60 million in population * 25% market penetration * 70% market share gives you 10.5 million iPods. 12 GB per iPod * 33% music / 4 MB per song gives you 1000 songs per iPod. In total, there should be around 10.5 million iPods * 1000 songs per iPod ≈ 10 billion songs stored on iPods in the UK).

Extra:In case there is time left of you interview, the interviewer might follow up with some additional questions, such as: What is the yearly market value of digital music in the UK? How much money is Apple making by selling music through iTunes in the UK?

Sources:http://www.bridgeratings.com/press.05.23.07.CompMediaUse.htmhttp://blog.wired.com/music/2008/05/ipod-loses-mark.html

Brainteaser - BirthdaysA key part of a consultancy interview is the case study. Below is an example of the kind of case study question you might be asked.

What's the probability of at least 2 players on a football field of 22 sharing the same birthday?

Answer:

This is a classic problem in probability and statistics, often called the Birthday Problem. It is usually simplified by assuming 1) nobody was born on February 29 and 2) people's birthdays are equally distributed over the other 365 days of the year.

The easiest way to approach this problem is to calculate the probability of the complementary event, i.e. when none of the players have the same birthday. This probability can be calculated or approximated in several ways and one of these methods is outlined below.

Start with showing that you understand the problem by breaking it down in smaller components, for example: The second player can’t have the same birthday as the first, i.e. there are 364 out of 365 days that are OK. The third player can’t have the same birthday as player 1 or 2, so then

there are 363 days that are OK, and so on... Finally, all these probabilities need to be multiplied with each other to get the combined probability. Formally, it looks like this:

Obviously, you are not expected to calculate the exact answer in an interview situation, but showing that you understand how to approach and solve the problem are the key qualities the interviewer will look for. Once you have stated the formula above, you would probably want to give an estimate of the probability. One way to do this would be by drawing a graph with the probability on the vertical axis and the number of players on the horizontal axis. By definition, you know that if there is only 1 player, the probability is 0 and if there are 365 players, the probability would be 1. This gives you the starting and ending point of the graph. To come up with the shape of the graph, you can think in terms of: What is the difference between having 364 and 365 players? Probably very small. However, the difference between 2 and 3 players is large (the probability almost triples since there are now 3 combinations instead of 1). Hence, the marginal effect of adding an extra player will decrease as the number of players increase. The graph should look something like the one below and would allow you to estimate to probability to just below 50%.

Brainteaser - Telephone BoxHow much interest is lost on money sitting in phone boxes in England each year? Calculate for 2006 and 1985.

Answer

Answering this question requires answering a number of sub-questions (SQ):– How many telephone boxes are there in England?– How much money is in each telephone box?

SQ1: How many telephone boxes are there in England?– Start by considering how many people there are per phone box - There are two telephone boxes in my village and it has around 200 houses - Assuming that the average household has four people living in it (parents plus two children), then there are: 200 houses * 4 people / 2 boxes = 400 people per phone box in rural areas - Now consider urban areas, given the higher density of housing, say double, there will be 400 houses per two phone boxes - However, occupancy rates are likely to be lower in urban areas e.g. more flats. Assuming, that half are family houses with four people and half are flats with two people, the average household will have three people living in it - There are: 400 houses * 3 people / 2 boxes = 600 people per phone box in urban areas - Finally, assume that in England 25% of the population live in rural areas and 75% live in urban areas, then the average number of people per phone box is: 600 people * 75% + 400 people * 25% = 550 people per telephone box in England

– Use the population of England to estimate the number of boxes - Total number of people in England is ~50 million - Total number of telephone boxes is: 50 million / 550 people ~= 90,000

SQ2: How much money is in each phone box?– Under the assumption that telephone boxes will be emptied when they are full and not before, we do not need to consider frequency of use and instead require an estimate of coinage capacity– Phone calls cost 20p in England so the majority of coins will be 20p– The collection trays in phone boxes are approximately 20cm high, 20cm wide and 10cm deep, giving a total volume of 20 * 20 * 10 = 4,000 cm3– A 20p coin is about 2cm in diameter and about 1/4cm in depth, giving a total space taken up of 2 * 2 * 1/4 = 1cm3– Therefore, the collection tray can hold 4,000 20p pieces when full, or £800 – this assumes that the shoot into the tray maintains an orderly coin arrangement– Assuming that there is a consistent use of phone boxes through time, the average amount in a phone box is £400

Final Step: How much interest is lost on money sitting in phone boxes in England each year?– The average total amount in all English phone boxes is: £400 * 90,000 = £36 million– Assuming an annual interest rate of 5% on deposited money, the total amount of interest lost in a year is £1.8 million

Extra Question: Calculate for 1985.– Under the assumption that the phone box is emptied when it is full, the average number of coins in the box remains at 2,000– In 1985 a call cost 10p rather than 20p so the average amount in the phone box was £200 (ignore the slightly different sizes of the coins)– We used to have four boxes in our village instead of the two we have now – Assume there were twice as many boxes in England in 1985 i.e. 180,000 boxes– The average total amount in all English phone boxes would have been: £200 * 180,000 = £36 million– Assuming an annual interest rate of about 10% in 1985, the total amount of interest lost in a year was £3.6 million

Brainteaser - HedgehogHow many spikes are there on the back of a hedgehog?

Answer

Answering this question requires answering a number of sub-questions (SQ):– How do you define the ‘back’ of the hedgehog?– What is the area of its back?– How many spikes are there in a specific area?

SQ1: How do you define the ‘back’ of a hedgehog?– Possible solutions:- Could define this as the whole area upon which spikes can be found- Could be the whole area that is not in contact with the ground- Could be an area in a similar proportion to the circumference of the hedgehog (when viewed from above) but of smaller size i.e. the area around it can be described as the sides rather than the back– Let’s go with the latter option and try to determine the area of that

SQ2: What is the area of the back?– Estimating the size/shape of a hedgehog, you could say it could be held comfortably in your hands:- Ellipsoid in shape so Elliptic when viewed from above- 20cm in length- 16cm in width– Assuming that the curvature of the hedgehog is ‘smooth’ between the back and sides and so the

back ends and side starts halfway between centre and outside of the ellipse:- the length of the back is 10cm (radius L = 5cm)- the wide of the back is 8cm (radius W = 4cm)– For simplicity, assume the surface area of the back when flattened is of similar size– Using the standard formulae for the area of an ellipse, the area of the back of the hedgehog = Pi * radius L * radius W = 3.14 * 5 * 4 ~= 63cm2

SQ3: How many spikes are in a specific area of the back?– Let’s consider an area of 1cm2 as the specific area– Spikes are much thinker than human hairs so let’s assume that the width of a spike is 1mm– Assuming that the spikes are packed together and each takes up an area of about 1mm2, then the total number of spikes in 1cm2 is 100

Final Step: How many spikes are there on the back of a hedgehog?– Let’s assume that the density of spikes is uniform across the back– Given a total area of 63cm2 on the back and 100 spikes within each 1cm2, the total number of spikes is approximately 6,300

Brainteaser - Racing CarIf a car does one lap of a race track at a speed of 75mph, how fast must it go on lap two to average 150mph?

Answer

Suggested approach:

To answer this you should consider the equation: Speed = Distance / Time

Step 1: Choose a distance to represent one lap of the track

The best distance to use here, for ease of calculations, is that which the required average speed would cover in 1 hour. Therefore, we set the track length to be 150 miles.

Step 2: Calculate total time available to complete both laps

Two laps of the track is 300 miles. At 150mph (average speed) this would take 2 hours

Step 3: Calculate the time the car has taken in completing lap 1:

Time = Distance / SpeedTime = 150 / 75Time = 2 hours

So the car has taken 2 hours to do one lap of the circuit.

In order to average 150mph over two laps of the (150 mile) circuit the car must complete both laps in two hours. However, the car has already taken two hours to do one lap of the circuit so the task is impossible, as the car would need to complete the second lap in zero time!.

Note: If lap 1 had instead been completed at an average of 100mph, we would proceed in the following way:

Time to complete both laps is still 2 hoursTime taken for lap 1 = distance / speed   = 150 / 100 = 1.5 hoursWe therefore have 0.5 hours remaining for the car to complete lap 2Speed = Distance / TimeSpeed = 150 / 0.5 = 300 mph

So if the car averaged 100mph on lap 1, the car would need to travel at an average speed of 300mph on lap 2 to average 150mph over both laps.

Playing and Watching FootballA key part of a consultancy interview is the case study. Below is an example of the kind of case study question you might be asked.

In an average week, do more people play football, or watch a football match?

Answer:

More people play football than watch a football match.

The key to approaching market sizing type questions is: to start by clarifying the question; then to identify any assumptions you can make to simplify the problem; next to break down the problem logically, before finally getting into the detail of the numbers. Interviewers will be impressed with people who can approach a problem conceptually, and also who can find ways of getting to a roughly right or “80/20” version of the truth quickly, which can be refined later.

Start by raising any questions you have to clarify the parameters of the analysis. In this case, you might want to confirm that “watching a match” means watching a match in person, not on TV; and that “playing in a match” means playing in an organized manner with 2 full teams and a defined pitch, rather than a kick-about with friends in the local park, or with your children in the garden. Football could include 5-a-side as well as full XIs, but does not include non-standard football, e.g. Eton Fives or other variants.

There are normally some simplifying assumptions to make up front – and you should always lay these out to the interviewer and confirm s/he is happy with them, before proceeding. The assumptions in this example could be:

1. Assume we are talking about people in England – and that the conclusion for England will be the same as for the rest of the U.K, as there is no structural reason to believe the other nations behave differently

2. Assume we are talking about distinct individuals rather than played occasions – so people playing > once a week or watching > once a week does not count

3. Assume you only need to prove that the number playing is greater / less than the number watching – rather than the absolute numbers involved in each.

Once you’ve clarified the scope and agreed your simplifying assumptions, you can start laying out a logical structure to breaking down the problem. For this question, one approach would be to start by listing out all the different playing and watching “occasions”. This allows you to show your interviewer that you’ve considered the problem as a whole, rather than diving down a particular line of argument – and also allows them to alert you to anything you’ve missed. Don’t be afraid to use a piece of paper to write these down.

For example, an occasion list might look like:

Children

School PE lessons Inter-school matches Other children’s matches (county or other local leagues)

 

Adults

Adult amateur: 5-a-side, local Sunday leagues, company teams etc Adult professional: Premier League etc.

 

Once you laid out the occasions, you should then prioritise which ones to analyse, in this case, based on:

a). Do you believe the numbers of players vs spectators will be asymmetric, or do the players and spectators cancel each other out? If canceling out, you can ignore the occasion, for the purposes of the analysis (remember, you don’t need to know how many, just more or less)

b). Which occasions contribute the most to total player or spectator numbers: you should always focus your efforts on the “big ticket” items which will have the most impact on the final solution.

A “prioritization matrix” might look like:

 Occasion   Number Playing

  Number Watching

  Priority to analyse

 School PE lessons

   Significant    None    High

 School matches    Cancel out    Cancel out    Low

 Other children    Cancel out    Cancel out    Low

 Adult amateur    Unknown    None    Low

 Adult professional

  Likely to be low

   Significant    High

 

The logic behind this prioritization is as follows:

Most children (especially boys) will play football at school, but there are rarely any non-playing spectators other than the PE teacher

A child playing in a match (school or other league) will on average bring 1 parent or other friend to watch – so the number playing and the number watching will cancel out

Practically no-one watches adults play amateur football on a regular basis Professional, i.e. League, football players are a small proportion of all adults playing

football – and certainly in proportion to the number of spectators

This leaves us with two occasions in favour of players – School PE (significant) and Adult Amateur (unknown); and one occasion in favour of spectators – Adult Professional (significant). Therefore, if you can prove that School PE is greater than Adult Professional, you can answer the question, without needing to work out the number of Adult Amateur players (currently unknown).

First, start by working out how many children play football in school

a) How many school-age children are there in England?

English population = 50m (you would be expected to know this / know that the total UK population is around 60m, and most of this is in England)

Population in each age bracket (i.e. one year) = 50m / 75 (average life expectancy)                                                                         = 2m / 3  

Years of school playing regular matches = 9

(Ages 7 – 16, as most don’t start PE in their first few years at school, and tend to stop playing in Sixth Form, or leave school)

Therefore, total children  = 2m / 3 * 9                                        = 2m *3                                        = 6m

There are actually 7.5m children in school in the UK, so 6m of match-playing age in England seems sensible.

b) Of school-age children, how many play football?

Remember, you are looking for the minimum likely number, to prove it is more than people watching football, rather than the actual.You should also bear in mind that the proportion of children playing is very different for girls than boys.A cautious minimum estimate could be

Proportion of girls                   = 0%Proportion of boys                  = 25%Therefore, total players          = 0%*(6m/2) + 25%*(6m/3)                                               = 25%*(6m/2)                                               = 750,000

Therefore, the minimum likely number of people playing football in an average week is 750,000.

Next, work out how many people watch professional football

There are two dimensions to the total number of spectators:

number of matches * number of spectators per match.

a) How many professional matches are there per week?

Assume that each professional team plays each week.

Number of matches        = number of professional teams / 2                                       = 92 / 2                                      = 46

You may know the number of professional teams already – but don’t worry if you don’t. The best approach is to break it down further – into number of leagues * teams per league. If you assume there are 4 leagues, and 20 – 25 teams per league, this gives you a range of 80 – 100 teams, or an average of 90 teams therefore 45 matches.

The interviewer will let you know if you are completely wrong on the number: the key thing is to show a logical approach to estimation.

b) What is the attendance at each match?

The first think to bear in mind is that attendance will vary significantly between the different leagues – so the Premier League would have much higher attendance than Division 2. You could start by assuming that each League has half the attendance per match than the previous league. This means you only need to estimate the average attendance at a Premier League match.

Premier League total      = 30k  (this is an estimate)Other Leagues total       = 30/2 + 30/4 + 30/8                                          = 15k + 8k + 4k (round up)All Leagues  total            = 30k + 15k + 8k + 4k                                       = 57kAll Leagues average       = 57k / 4                                       = approx 15kThe actual number for average attendance is 15.4k, so very close to this number.

c) What is the total average attendance?

Total attendance           = matches * spectators per match                                      = 45 * 15k                                      = 675k (you can multiply this out longhand)  

Therefore, the likely number of people watching professional football is 675k.

675k is less than the 750k minimum estimate for children playing football, which does not include the number of adults playing amateur football.

Therefore, more people play football than watch football in an average week. 

Water and WineA key part of a consultancy interview is the case study. Below is an example of the kind of case study question you might be asked.

Imagine you have a glass of wine and a glass of water. If you move 50ml from the water into the wine and stir the wine, and then move 50ml from the wine into the water…. is there more water in the wine, or wine in the water?

Answer:

There is the same amount of water in the wine, as there is wine in the water.

This is because the volume contained in the glass has not changed (we moved 50ml in each case). For this to be true, the amount of water in the wine has to be the same as the amount of wine in the water – otherwise one glass would have more volume in than the other.