If you cycle at 20 km an hour, how long will it take you to cycle to the centre of the Earth

In my puzzle blog earlier today I set you the following three challenges:

1) The King of the Mountains went up the col at 15 km an hour and down it at 45 km an hour. It took him two hours in total. Assuming that the distance he travelled up and down are the same, how far is it from the bottom to the top of the col?

Solution

The only technical knowledge we need to know here is the equation

speed = distance/time

On the way up the col, therefore 15 = d/t, where d is the distance up and t is the time it takes to get there.

On the way down, 45 = d/T, where T is the time it takes to get back.

In other words, d = 15t, and d = 45T. Putting these together, 15t = 45T or t = 3T.

We also know that t + T = 2. So, 3T + T = 2, which reduces to T= 1/2. The distance from the bottom to the top of the col, d, is 45/2 = 22.5 km.

2) Xavier and Yves must make a 20 km journey and arrive at the same time. They have a single bicycle, which they ride in turns. Each time a rider dismounts he leaves the bike by the side of the road, and walks on, while the other one eventually arrives at the bike and jumps on it. Xavier walks at 4 kmh and cycles at 10kmh, while Yves walks at 5kmh and cycles at 8kmh.

Assuming that both men are either walking or cycling at those speeds, and never stop to rest, how do they arrange their journey?

Solution

Here’s how I solved it. Let Xavier ride for an hour, after which he will have covered 10 km. Let him drop the bike and carry on walking. Yves, who walks at 5kmh, will get to the bike after two hours, at which stage Xavier will already be at km 14. If Yves rides the bike now, an hour later he will be at km 18, coinciding with the arrival of Xavier.

However, our aim is for the pair to get to km 20 at the same time, not km 18. If they have to cover the remaining 2km such that they start and arrive at the same time, they need to both walk and ride for a ninth the time that they did before. So, let Xavier ride for 1/9 of an hour (covering 10/9 km), and then walk for 2/9 of an hour. Yves will walk for 2/9 of an hour and cycle the rest. The total journey is thus 3 hours and three ninths of an hour, or 3 hours 20 mins.

In my solution Xavier rides for 10 km, walks for 8km, then rides for 1 and 1/9 km, then walks the rest, while Yves rides 8km when he gets the bike the first time. This answer is, however, one of several solutions. The key point is that Xavier must cycle first, and that the total distance he cycles is 11 and 1/9 km. He could cycle the whole lot in one go if he fancied, or in shorter segments.

Harder extra question: Xavier and Yves are joined by Zoe, who walks at 3kmh but cycles at 12kmh. How do the three of them now arrange the journey so that they arrive at the same time?

Solution

The answer is that in the journey Xavier must ride for 7 and 11/27km, Yves for 1 and 13/27 km and Zoe for 11 and 3/27 km. In order to save space here, and for you to showcase your problem-solving skills, please post a full answer with workings below the line. To all those people who cannot resist posting solutions when I say NO SPOILERS, this one is for you!

3) Here’s a picture of a bicycle. If you attach a piece of string to the bottom pedal and pull it backwards (as illustrated by the red arrow), does the bike move backwards or forwards?

Solution

The bike moves backwards (while the pedal moves anti-clockwise). Or at least, it will do so in almost all bikes, apart from those set in an extremely low gear. It is a counter intuitive answer, and looks surprising when you try it out yourself. This puzzle got a lot of traction (sorry!) a couple of years ago thanks to the following video by the brilliant George Hart. Here he explains the mechanics of what is going on a lot more eloquently than I ever could.

I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.

I’m the author of several books of popular maths, including the puzzle books Can You Solve My Problems? and Puzzle Ninja, which is out in paperback this week.

I also co-write the children’s book series Football School.

The forces of gravity are pulling equally in all directions and you are now weightless.

But don't get too comfortable.
It’s over 6,000C here - you were cooked long ago.

The distance to the centre of the Earth has been taken as the volumetric mean radius of the Earth: 6,371km. Note that this distance and the depth of geological layers varies across geographical locations. Depths shown are maxima, except where stated. Sea creature depths are lower limit of normal depth range, except where stated. Oil well and borehole values are vertical, not measured depths. Vehicles are not to scale.

Do you ride your bike enough to maintain good health ? I suspect that many people actually do not. A surprising amount of cycling is required if that's the main thing you do for exercise

Many sources suggest that half an hour a day of exercise is around the minimum to maintain good health.

My racing bike. A left-over from the 1980s, steel frame and non-indexed gearing. Not the most comfy choice for me for

long rides, ideal for a quick spin around town & countryside

Because I work from home, I don't have a commute forced upon me, as was the case in the past. That means I don't automatically get a daily dose of cycling by going to work and back. Our deliveries are all made at least partially by bike, but while that means I'm riding a relatively heavily loaded cargo bike, the short round-trip distance of about 3 km doesn't take long enough. For that reason, I go out for a "commute" on at least some mornings even though I don't have to. This morning I rode a little over 16 km, going through the town, stopping for traffic lights, heading out into the countryside against a stiff headwind and returning home again with a tailwind which of course can never add as much as the headwind already took away. It took me 31 minutes to cover the distance. I'm not about to set any records at that speed. That's not the point. This is moderate exercise, taking it fairly easy, expending enough effort to get slightly out of breath sometimes, enough to raise a bit of a sweat but certainly not enough to make my heart pound as if is going to pop out of my chest.

Recommended Minimum Daily Allowance - 15 km

At a moderate rate of exertion, 30 minutes of cycling at a rate conducive to maintaining health equates to covering about 15 km at an average speed of about 30 km/h. 15 km per day equates to about 100 km per week or about 5500 km a year. i.e. it takes about 5500 km of cycling each year to do enough exercise to maintain health.

Another day, with company, I rode another trusted old friend on which I've ridden many thousands of kilometres over the

last 20 years.

If you ride a round-trip commute of around 21 km (13 miles) each day five days a week then that's just about perfect and will add up to about the annual target figure. If your commute is shorter, take a detour sometimes. People like me who don't have a regular commute to account for 5500 km (3300 miles) each year probably ought to get out on their bikes more often, which is what I do these days.

An inexpensive bike computer helps keep track of the total. It's December and we're heading into winter, but that's no reason to stop: Cycling in winter is particularly rewarding.

How this relates to other means of transport

If cycling is to benefit us as exercise as well as a form of transport it should be our first choice for at the first 5000 km that we travel each year. Until we've covered that distance we can consider the cost of the "fuel" used to cycle to be zero as the alternative would be to burn away the same amount of energy by wasting it at a gym or in some other sporting activity.

Motor vehicle enthusiasts occasionally make an absurd claim that the environmental cost of providing food to humans riding bicycles makes us so much less efficient than modern motor vehicles that we'd be better off driving a car than riding a bike. It's nonsense of course. Bicycles are the most efficient vehicles on the planet by some margin. This goes double for the first 5000 km because expending that energy through our bodies isn't optional. We don't need to eat extra to be able to ride those kilometres, we merely need to eat what is required to maintain a healthy weight while also taking a healthy amount of exercise.

Faster than light travel

Further to the above, cycling is the only means of travel which allows you to cover distances at infinite speed, taking no time at all to make your journey. No time at all needs to be allocated for most cycling journeys. How so ? Let's begin by assuming that we will actually do that 30 minutes a day of exercise which is required to remain healthy. By exercising while you cycle you can fit in two 15 minute journeys each day (e.g. to work and back) without allocating any specific "cycling" time. It's just your minimum exercise. By comparison, driving a car might appear to get you to work in about half the time, taking 8 minutes each way, but afterward you'll also need to drive to a gym and back (5 minutes each way) and you'll also still still have to spend 30 minutes exercising, but in this case pointlessly staring into space on a machine in the gym instead of watching the world go by as you cycle. So overall, driving takes as much time out of your day as the actual travel time in the car, in this case 26 minutes (8+8+5+5), while cycling anything up to thirty minutes a day actually takes no time at all.

Disclaimer

I sometimes write about health and cycling, but I am not a doctor. In particular, I am not your doctor and I certainly cannot offer health advice to you personally. If you need health advice or you're thinking about changing your exercise habits please see a professional.

Update: What about the Dutch average ? What about e-bikes ?

It's often quoted that the average Dutch person cycles less than 1000 km a year. If so, they're obviously not doing nearly enough exercise on their bikes to keep themselves healthy. I'm not sure I know anyone who really cycles just 1000 km a year, though. It's almost impossible to ride less than a thousand kilometres a year if you use a bike for everyday journeys. I find I often cover twice that distance on my town bike each year even though most of the distance that I cover is on other bicycles, and I don't even have a regular commute these days.

Research has shown that average e-bike riders don't get much exercise at all. If you are trying to maintain your health by riding an assisted bicycle then you will possibly not get much benefit unless you turn the assistance down and also ride much more than 5000 km a year. If you ride less than that distance then you could consider turning off the motor altogether. Otherwise you can expect less of a health benefit than would result from riding a 100% human powered bike.