The problem with the secret message raised last week has been, paradoxically, solved by making it more complex: the man adds a second lock to the chest and sends it back to the lady who removes the first lock and sends it to the man, now all you have to do is open your lock. A slow but secure system, because the message is out of reach of the mindless messenger all the time.

If we change the locks for large primes, we have an efficient cipher system that is widely spread. Let’s imagine that in pre-computer times, A wanted to share with B two secret numbers: 1901 and 2713 (or the difference between the two numbers: 2713 – 1901 = 812). He multiplies the two numbers and sends the result to B by mail or by phone without fear, because even if someone suspects that 5157413 is the product of two significant numbers, it is very difficult to find these factors (and with computer help too, if cousins are big enough). B, in turn, multiplies the received number by another four-digit prime number, for example, 1301, and sends the product, 6709794313, to A, which divides it by one of its two prime numbers, say, 1901, and sends the result to B, which When you divide it by its prime you get the other prime number of A: 3529613/1301 = 2713.

I’ve only used four-digit cousins to illustrate this cipher system (known as RSA by its developers: Rivest, Shamir, and Adleman), but for now, given the sheer computing power of computers, the primes used must be hundreds characters to ensure safety.

For other issues of the past week, see the corresponding comments section, where they have been extensively analyzed.

### From boxes to packages

And if not all: Problem 2 did not deserve the attention of savvy readers, who focused on the secret message and the traveling balls. However, it is an interesting packaging issue which I suggest again accompanied by a top view of the box (without lid):

In the orthodontic box there are three balls of 10 cm in diameter tangential to each other and tangential to the walls, base and lid of the box. How long are the sides of the box?

Stacking and filling balls of the same size is a question that has intrigued mathematicians and engineers since ancient times. British mathematician and astronomer Thomas Harriot (who, among other things, introduced the symbols > and < in mathematical notation) was the first to calculate, in the mid-16th century, the number of cannonballs in a stack in the form of a pyramid with a square base (how many?) , and Gauss showed that the maximum density that can be obtained by filling the three-dimensional space with equal spheres is about 3/4 (exactly π/3√2). The packing density is the fraction of the space occupied by the balls. What is this maximum packing density like?

Going from 3 dimensions to 2, the filling of balls in space becomes the filling of circles in the plane. And again, it was Gauss who showed that the maximum density is achieved, in this case, by the hexagonal arrangement, in which each circle is surrounded by six more tangents. Can you calculate the density of this packing? To avoid the distorting effect of the edges of the real surface, the problem is posed in an infinite plane.

If the circles packed in this way are flexible, grow and press against each other, they will form a hexagonal mesh. This is, in fact, what happens in the honeycomb.

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