France, Bicycle and Eclair.

A treat math story today shared here with the permission of the author: Lawrence J. Krakauer. You can find more of his blog posts on his Memoirs and Musings blog.




During my French study summer in Pau, France, in 1961, I found myself one afternoon in the company of one of the girls in our group, Diane. Diane and I wanted to get to some event that was scheduled to take place in Pau's Place Clemenceau, and we were a bit late. I had a bicycle, which I was renting from the entrepreneurial son of the family I was staying with.

Unfortunately, the bicycle was not one that two people could comfortably ride. We started out walking together, with me pushing the bicycle, but then I had a better idea - a way we could get there faster, while still arriving at the same time. We could do that if we each bicycled half way, and walked half way.

To do that, I consulted my map of the city of Pau, and picked a point along our route that seemed to be half way to the Place Clemenceau. We agreed that Diane would ride the bicycle to that point, lock it up at an agreed-upon corner, and walk the rest of the way. I would walk to that point, pick up the bicycle, and ride the second half. We should both arrive at our destination at about the same time.

So Diane set off, carrying the bicycle lock, but leaving me the key. I proceeded to walk to the agreed upon corner, and as I arrived there, I was pleased to see the bicycle, chained to a pole. My brilliant idea was working! But just as I unlocked the bicycle, who should step out of a nearby doorway but Diane. I looked at the sign above the door - it was a patisserie (a pastry shop). In one hand, Diane held an éclair.

It seemed that upon arriving at the designated corner, Diane had noticed the patisserie, and had found it irresistible. So there we were, now half way to the Place Clemenceau, together again. Since the past is past, I guess the logical thing to do would have been to have picked another location halfway from our new starting point. But I recall being so annoyed at Diane for ruining my wonderful idea that I got on the bicycle and rode off, making her walk the rest of the way (as we had originally planned).

Let me add that Diane was not alone in being magnetically drawn to Pau's patisseries. All the girls in our group seemed to spend a lot of their time in these French gardens of Eden (for some reason, the girls much more than the boys). Once, during a morning French lesson, the professor decided to expand our vocabularies by asking the students to name a few French pastries. Overwhelmed by the response, he called a halt to the exercise after the students, the girls mostly, had named about fifty or sixty, some of which the professor had never heard of. Thus I discovered where many of the girls had been spending their free afternoons.

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Of course we all expected some romance to appear in this story of American students spending a summer in France. But assuming no romantic involvement between Lawrence and Diane, the half bike/half walk idea looks brilliant. How much time do you think could be saved by such an arrangement?

Assume that the distance they need to cover is S km. Both walk at 5 km/hour and both bike at 15 km/hour. We use km as we are in France.
In a half bike and half walk arrangement suggested by Lawrence, it will take:
0.5 x S /5 + 0.5 x S / 15 = S/10 + S/30 = 4S/30 = 2S/15

Should they politely continue walking and pushing the bicycle, it will take them S/5 hours. Or 3S/15.

It is easy to see that the half-and-half arrangement is S/15 faster than all walk or takes 2/3 of the walking time.
For a distance of 1km, it will be 1/15 hour = 4 min faster. The walk will take 12min and half-and-half only 8min.

For a distance of 5km, it will be 5/15 hour = 20 min faster. The walk will take 60 min and half-and-half arrangement 40 min.

I love this story, the clever trick and the unexpected patisserie surprise. Life is much more than a set of equations, however these equations do come handy from time to time.

Top image by mhall209, distributed under CCL.