Time travels so fast! Exactly to the date, three years ago, I started this blog and it has been a fantastic experience. Thanks all you faithful followers for your support and advice. Unfortunately, during 2018 other commitments made me write very few blog posts, I will try to improve on that in 2019.

For all of you who have struggled with the puzzles I gave in the most recent blog, 2018-12-31, here are the solutions.

Which is the number of the parking space hidden by the car?

Look at the following image and you will see it immediately!

The busy fly

John von Neumann did not waste any time. After having replied immediately to the student who had posed the question, the impressed student asked the master if he had found the solution without making any calculations. No, on the contrary, I summed all the partial distances, was von Neumann’s reply.

But if you are no master in making calculations in your head, you can even beat von Neumann in speed by using the following reasoning: Any physics students knows that s = v x t, i.e. distance equals speed times time. Each cyclist travels 25 km using a speed of 25 km/h, so they will meet after exactly 1 h. The fly travels at 50 km/h during the same 1 h, i.e. a total distance of 50 km. QED!

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.

Astronomers in Babylon used mathematical methods already during centuries before the start of the Christian era to track movements of Jupiter. In order to do so, they were taking the first steps from geometry toward calculus to figure out the distance it moved across the sky. Such methods with trapezoides were only used much later in Europe, during the 14th century AD.

Babylonian clay tablet. Photo: Trustees of the British Museum/Mathieu Ossendrijver

The proof for this can be found in four small clay tablets, which have been stored in the British museum for quite some time. However, it was only recently that Mathieu Ossendrijver from Humboldt University in Berlin was able to decypher their contents.

The interesting story can be found in the two references below.

The largest prime number ever found was announced last Wednesday, January 20, 2016. So, what is a prime number and what is it good for?

A prime number is defined as a natural number (a positive integer) greather than 1, which cannot be evenly divided by any other natural number than 1 and itself. Examples of such numbers are 2, 3, 5, 7, 11, etc. It has been proven that there are an infinite quantity of prime numbers.

Prime numbers have been known for a long time, e.g. Euclid’s Elements (300 years BC) already mentions them, and in the beginning of the 17th century, a French monk named Marin Mersenne devised a formula, of the form 2^{p} – 1, where p=1 is a prime number, to be used to check for unknown prime numbers. In fact, the largest prime number, which is exactly 2^{74.207.281} − 1, consists of more than 22 million digits, and the search for bigger numbers continue. Please see the three references below for more information about prime numbers in general and also about the discovery of the currently biggest prime number, including an interview with Curtis Cooper, the leader of the project that discovered it.

One of the practical usages of prime numbers is in public-key cryptography, where two large prime numbers are multiplied to obtain a product that it is extremely difficult to factorize and thus break the code. But the search for these very big numbers does not seem to have any major practical use today, although they are very well fitted to test the speed performance of computer hardware.

And speaking about cryptography, having means of obtaining secure data streams is of course essential when we want to communicate data from one point to another. The current standard for data communication in the world is based upon what is called fourth-generation (4G) technology, and although it offers very fast rates of data communication, there are applications that demand even faster data transmission speeds. Examples of such applications are some components of the ‘Internet of Things’ (e.g. driver-less cars), and also in remote surgery, when the patient is in a hospital somewhere in the world, and at the same time the head surgeon is in a totally different place, performing the surgery via advanced, fast video and manipulation technologies.

For that and other purposes, last Friday, January 22, 2016, TeliaSonera and Ericsson announced that in 2018, they will start 5G networks in Stockholm and Tallinn. The rest of Sweden should see 5G in use in 2020.

Update on 2016-01-27: Today’s program of ‘Vetenskapens värld’ on Radio Sweden’s domestic channel P1 penetrates into the 5G technology. It will be a standard mostly used for machine to machine communication, and there are good hopes that one standard will be used everywhere on Earth, with speeds 100 times higher than the current 4G standard. See reference 7 below.