Bonjour guzzleurs,

As we are almost midway through the Tour de France, I thought it would be a good moment for some bicycle puzzles.

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?

2) Xavier and Yves must make a 20 km journey. They are to start together, and must arrive at the destination at 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?

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

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?

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I’ll be back with the answers at 5pm UK time.

UPDATE: The solutions are now posted here.

Meanwhile, NO SPOILERS. Suggest your other favourite puzzles about bicycles.

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.

Today’s second puzzle is adapted from an old puzzle by Henry Ernest Dudeney. The third has a long history, most recently analysed in detail by George Hart, about whom more later.

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

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


Contributor

Alex Bellos

The GuardianTramp

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