2018-12-31 (Réveillon)

Today, I learned that:

There are so many beautiful natural scenes around us, if we only take the time to go looking for them. This month, I had the the joy of taking my family on a road trip to the southernmost states in Brazil, Santa Catarina and Rio Grande do Sul.

And it was really worth the while. Our main interest was to know the highlands in those states. In the latter one, we concentrated on “serra gaúcha” with the twin cities Canela and Gramado, and although they now are highly commercialised, they still offer quite some entertainment.

But the highlight of the whole trip was no doubt the highlands of Santa Catarina. In winter, many tourists go their to enjoy (?) snow in São Joaquim, but since that is something I have had way too much of in my life, I prefer the summer that has so much more to offer! Not far away from São Joaquim is one of the most breathtaking views in the world, namely the highway that serpentines Serra do Rio do Rastro down in 284 curves, from an altitude of 1 421 m to 220 m in a distance of a mere 12 km. Of course it is impossible to capture all the excitement in a photo, but I hope that today’s header photo, taken from the belvedere overlooking the abyss, can give you a hint. On a totally clear day, one can even see the Atlantic ocean, 100 km away!

See also references # 1 and 2 below.

Some things to think deeply about

I am sure that more than once in your life, you have been challenged to solve a mathematical puzzle that involves discovering the missing term in a sequence.

One such sequence is called an arithmetic sequence, where the difference between the consecutive terms is constant, e.g. 2, 5, 8, 11, 14, … Anyone promptly says 17 when asked of the upcoming term.

It gets a bit more complicated if I give you a geometric progression, such as 2, 6, 18, 54, … ; or 10, 5, 2,5 , 1,25, …; or even 1, 8, 27, 64, 125, …

But there is also a sequence of numbers that can really trick us, until we discover the underlying fact. Here is one of those, which is the number of the parking space hidden by the car in the following image?

Was that too easy? Try the next one then:

Two cyclists have decided to meet half way between their cities. The distance between the cities is 50 km. Both cycle at a constant speed of 25 km/h. At the very moment they begin their trips, a fly takes off from one of the cyclists and when it reaches the other one, it inverts its trajectory and flies back to the cyclist from where it started. When it arrives there, once more it inverts its trajectory and keeps on repeating the process until it has comes to a stop when the cyclists finally meet. If the fly holds a constant speed of 50 km/h, how far has it flown when the cyclists meet?

The brilliant mathematician John von Neumann, who proposed the computer architecture that now bears his name, was once asked the same question. One of his early skills was to make very complex calculations in his head, so he answered the question in a snap. Can you?

The solution will be published in my next blog post.

I wish you an EXCELLENT YEAR of 2019 !

That’s what I learned in school !

Refs.:

1: Serra do Rio do Rastro live

2: Serra do Rio do Rastro

*: What did you learn in school today ?

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