### Three cycling 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.

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

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**

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.

©2018 Alex Bellos – The Guardian