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So how long does it take to move a mountain? – Solving ‘hard-to-solve’ problems

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We work hard to recruit the right people to our management consultancy practice. And one thing we’re looking for here is the ability to tackle ‘hard-to-solve’ problems which are often part of the projects we have to tackle. We test for this ability using ‘estimating questions.’ This blog explains one of these ‘hard-to-solve’ problems expressed as an estimating question- and shows you how to solve it. You’ll find it interesting if you’re keen to think like a consultant.

BTW it’s the ability to tackle these estimating questions that seems to illustrate the difference between excellent consultants and very average ones generally. Not only =mc uses them but also PWC, McKinsey, Boston Group and indeed all the major commercial consultancies.

Let’s be clear with what we’re talking about. Estimating questions are typically of the form:

  • How many barbers are there in Chicago?
  • How many golf balls are used annually in Japan?
  • To make most money with least risk, which should a thief rob – baker, jewellers or off license?

We’re often asked why we pose such ‘silly’ questions. Our overall argument is that in general they help to show three things:

  • First how you can tackle a challenge where there isn’t a right answer
  • Second how systematic you are when tackling problems
  • Third how able you are to express complex problems simply

These kinds of questions surprisingly have an important role in our ‘real world’ work. So as consultants we are often asked to tackle real questions to which we need to apply the same kinds of skills. Recent examples of projects we’ve been asked to tackle that fit in this ‘hard to solve category’ include:

  • How much would it cost to make a market entry for face-to-face fundraising in Indonesia?
  • How many volunteer counsellors should the Samaritans have available for Xmas?
  • How many phone lines are needed to respond to an appeal for funds after an earthquake?

The challenge in each case is that often there is no solid dataset to work out a ‘right’ answer. So to complete the assignment we need to work on shaping an estimated answer with a clear set of assumptions built in so that we, and possibly the customer or other experts, can test the answer out. If you like we’re doing the thinking out loud for the customer.

So the question I’d like you to work on for this blog is “how long does it take to move an ‘average’ mountain 10 miles using a single average-size lorry?”

(The moving a mountain example is a famous one. There are many different answers. Note much of the workings for this particular answer are taken for from Victor Cheng’s work who has a great site a

To be acceptable to us in the interview, and to meet the customer’s need, your answer must involve a specific response in the form of a specific time- so X hours, Y days, and Z years. We’re really, as indicated above, interested if you can demonstrate how systematic your approach has been rather than how exactly right the answer is. The answer must also be ‘acceptable’ and ‘interrogable’; i.e. there isn’t really a ‘correct’ answer unless a lot more detail is given so we need to be able to follow your thinking and challenge it.  Note that we don’t have any more detail to give here- so you need to make some key assumptions and make them explicit.

Among the answers that could be “acceptable” are those that work out at a few hundred hours to a few hundred years. The size of the range emphasizes the importance of making explicit and interrogable your underlying assumptions. So one implication is if you put more resource in what difference would it make? Let’s tackle the problem as is.

First you need to work out what the shape of the solution us. So the ultimate answer will be given by this formula, which anyone who’s done secondary school maths could work out:

volume of mountain/volume of lorry * time per lorry trip = total time to move the mountain

Victor Cheng says it’s a good idea to write that down. We agree. Simply, that formula expresses the amount of time it would take one lorry to move an average size mountain 10 miles.

To answer this question in more detail you’ll have to make some further assumptions and some estimates. 

So you need to ‘nest’ some calculations to drive the estimates. Specifically you need to estimate:

  • the cubic volume of an average mountain
  • the cubic volume of an average lorry load- the amount it can move
  • the average length of time to make a round trip of 20 miles

For the purpose of the calculation we’re not going to worry about, say, how often the lorry would need to be refuelled or the challenges of working at night. Those things may be important. But it’s fine if you show that you’ve excluded them- you can always factor them in later.

Let’s work out some of those key elements:

The volume of the mountain
: Let’s get back to secondary school maths to help work out the mountain’s volume. A mountain is more or less a big cone, and there is a formula for working out the volume of a cone: v = hπr2/3. But maybe in the heat of the moment you forget it. One option is use the much simpler formula for a cube then divide it by 50%. (So a half a cube is like a cone really.)

Let’s also assume the average size mountain is 1 mile tall, 1 mile wide, and cone-shaped. That’s approximately 5,000 ft in height and base. The volume of a cube that’s 5,000 ft tall, 5,000 ft wide, and 5,000 ft deep is 125,000,000,000 cubic ft. Since we’re trying to estimate a cone, and not a cube, let’s take that down by 50%. So with some slight rounding, that gets us 60,000,000,000 cubic ft.

Hurrah we can now fill in the first part of the formula: 

Update 1: 60,000,000,000 cubic ft / volume of a lorry * time per lorry trip = total time to move mountain.

Next task is to estimate the carrying volume- or cargo capacity load- of a lorry.

The lorry capacity: The carrying capacity of a lorry is the width x length x height of the load container at the back. But how big is our lorry?

Lorries are a wider than a car, but not by much since they still must be able to fit into a lane on the motorway. So if a car sits 3 people across, assuming 2 ft shoulder width per person, that’s 6 ft of interior space. If we add on a little more we can assume bigg-ish lorries are around 8 ft in width.

Lorries must be at least twice the length of most passenger cars. (Think about lying down beside a car. It might be twice as long as a 6ft tall man including the engine and boot.) So that makes a car around 12 ft long. So if you double that to scale up for a lorry, then our hypothetical ‘average’ lorry is 24 ft long. You need to take out 4 ft for the driver’s compartment, and that leaves about 20 ft in length for the load.

If you imagine a worker standing at the back of a lorry the load area is normally taller than the worker.  So let’s assume that the load area is about 8 ft tall- assume the tyres are about 4ft high.

So the volume of the load area of an earth moving lorry is 8 ft wide x 20 ft long x 8 ft tall = 1,280 cubic feet.

Again for simplicity, round that down to 1,250 cubic feet. Now put that number into the formula.

Update 2: 60,000,000,000 cubic foot mountain / 1,250 cubic foot lorry capacity * time for lorry trip = total time to move a mountain

trip time: the final factor missing for our estimate is working out the round trip time for a lorry to move its load 10 miles, drop it and return for the next load. Let’s assume the lorry travels at a constant 60 miles per hour. 10 miles takes 10 minutes. So the total round trip is 20 minutes. Lets also assume that the loading and off loading process have been designed to be quick- 5 minutes each. (The load is “dropped” and then repositioned while the lorry is on its return.) That means each round trip takes 30 minutes or 0.5 hours.

So let’s go back to our formula again and update it.

Update 3: 60,000,000,000 cubic ft mountain / 1,250 cubic foot lorry capacity * 0.5 hours per lorry trip = total time to move a mountain

Last stretch. For the first two elements of the formula, that works out to about 50,000,000 (50 million lorry loads).

So 50 million lorry loads x 0.5 hours, is a total of 25 million hours to move our mountain.

If we assume a typical day has 25 hours, just to make our calculations a little simpler, that’s means a million days to move the mountain using only one lorry. So the total time works out to a little less than 3,000 years. (More lorries or bigger lorries are needed if the customer needs a faster shifted mountain.)

This answer would most pass most estimation question interviews including the =mc one. If you got as far as this well done. If you gave up think about why you stopped when probably you could have solved it. And if you think you’d like to have a go working on problems like this for a living, have a look at our recruitment pages.

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