A 'Carrollian' blackboard

Last week's 6 full contact cigarettes can be improved with the arrangement shown in the figure, in which a seventh cigarette is inserted into a central hollow formed by the others.

What has not been demonstrated (as far as I know) is that 7 is the maximum possible (although it has been demonstrated that the solution to the figure is not unique, as we will see later).

As for cigarettes—or cylinders—of infinite length (this variant was proposed by the British mathematician John Littlewood), they haven't merited the attention of my astute readers, as they haven't provided any solution. Although perhaps it wasn't a question of attention, but of difficulty, since it must be recognized that it's not at all easy : the figure shows a solution with 6 cylinders, considered insurmountable until, recently, two different solutions were found with 7 cylinders (which, obviously, are also true for the case of finite cigarettes). By carefully observing the figure, and from it, you might find one of these solutions; but arm yourself with patience, because, although knowing its existence makes things easier, it's still very difficult.

And moving from rigid to flexible cylinders, a simple problem to compensate for the difficulty of the previous ones:

We have a 1-meter-long elastic band with a circular cross-section of 2 mm in diameter. If we stretch it until its cross-section is 1 mm in diameter, and assuming the deformation is uniform along the entire length of the band, how long is it now? Once you've found the solution, try to "cook" it.

The relationship between stretching and narrowing is called Poisson's ratio, and for most materials it is positive (the greater the stretch, the greater the narrowing); however, some materials, called auxetic materials, have a negative Poisson's ratio, meaning they widen when stretched.

Synthetic auxetic materials, such as expanded polytetrafluoroethylene (marketed as Gore-Tex, among other brands), have many applications, and, coincidentally (although in mathematics and macroscopic physics nothing is coincidental), a maximum number of threads (infinite cylinders, for practical purposes) in mutual contact is the basis for the manufacture of auxetic meshes, which led researchers to search for that maximum possible number. Which turned out to be 7, as they discovered before the mathematicians.

A non-Euclidean blackboard?

The two girls in last week's main illustration don't seem very good at math, judging by what they've written on the board. But let's not be too quick to judge: they may be expressing themselves in unconventional terms, but they're not necessarily wrong. Or perhaps they've been the target of a mathematical joke concocted by Lewis Carroll himself, so fond of baffling young girls with his apparent nonsense . Is there any possibility that the equation 180 = 90 + 30 makes sense?

On the other hand, the angles of a triangle only add up to 180º in the Euclidean plane; on another surface, they can add up to more or less than two right angles (can you draw—or describe—a triangle whose angles add up to 270º?).

As for the triangle on the blackboard, we know it's right-angled because we see the usual hidden angle marking, though not its value (only the final 0 is visible); but we do see the values ​​of the acute angles: 40° and 30° respectively. Is it possible to give the hidden right angle a value that makes the figure meaningful ?

And from last week's illustration, we can move on to the text to pose a small meta-problem: What logical reason could there be for the automatic spell checker to italicize the name of Poisson, the famous French mathematician and physicist?