The cuboid with the highest V/S ratio (volume divided by total surface) is the cube, and therefore the one-liter tetrabrik that would optimize – that is, minimize – the material used in its manufacture, for which we asked ourselves the last week, is a cube of side 10 cm, whose total area is 6 x 10² = 600 cm², while the total area of a 20x10x5 cuboid is 2 (20×10 + 20×5 + 10×5) = 700 cm². Going from the conventional to the cubic tetrabrik would mean a not inconsiderable material saving of almost 15%, why not do it? I invite my astute readers to give some reasonable answer.
As the less young will surely remember, the tetrabrik, honoring its name, began as a tetrahedron (called Tetra Classic); but it was a less easy way to handle and stack, so in the 1960s it was replaced by the current orthohedral model, which should strictly have been called hexabrik or ortobrik, but kept the name for marketing reasons. And in addition to the aforementioned handling and storage difficulties, there was another compelling reason (pun intended) to abandon the tetrahedron in favor of the cuboid; which is?
As for the integer-sided cuboids that meet the volume + perimeter = area condition, here is the elegant solution by Julio Díaz-Laviada:
It is about solving: abc + 4(a+b+c) – 2(ab+ac+bc) = 0
You can put this: (a-2)(b-2)(c-2) + 8 = 0; (a-2)(b-2)(c-2)= -8
And resolving -8 into three factors in all possible ways, we get a = 1, b = 6, c = 4 and a = 1, b = 3, c = 10.
The same reader complains (not without reason) that the problems of coating a chessboard with tetrabriks are excessive in terms of the number of possible sub-problems. Obviously, I do not pretend that all of them are resolved, but rather those variants that each one considers more interesting. To simplify, the coatings that use the same combination of faces could be considered equivalent, so the question would be: with how many different combinations of the three faces of the tetrabrik can the chessboard be covered?
What is a brick for?
The familiar tetrabrik has little tetra, as we have just seen, but a lot of brik (or brick, which is brick in English), since each side is half of the previous one. These doubly “brick-like” dimensions make it a very versatile building block that allows interesting problems to be posed in three dimensions. For example, what is the smallest cube that we can form with our ideal 20x10x5 cartons? And the minor without fracture planes? What condition must a cuboid meet in order to be formed by coupling tetrabriks? What other interesting problems and constructions occur to my astute readers when they have a tetrabrik in their hand?
Going from the purely geometric to the physical brick, its versatility made it the subject of a well-known test to select creative people. A brick, in addition to its conventional function as a construction material, can serve as a throwing weapon, as a paperweight, to hold a door that tends to close due to air currents… I invite you to take a pencil and paper and write down all the possible applications of a brick that you can think of in three minutes.
Carlo Frabetti is a writer and mathematician, member of the New York Academy of Sciences. He has published more than 50 popular science works for adults, children and young people, including ‘Damn Physics’, ‘Damn Mathematics’ or ‘The Great Game’. He was a screenwriter for ‘The Crystal Ball’.
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