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Still from the series 'The Simpsons'.
Still from the series ‘The Simpsons’.

We asked ourselves last week if to color the map of Andalusia so that its provinces are clearly differentiated —that is, without two bordering provinces being the same color— four colors are necessary or can it be achieved with fewer. And we asked ourselves the same question regarding peninsular Spain divided into communities. Well, three colors is enough both in the case of the eight Andalusian provinces and in the case of the 15 peninsular communities, as can be seen in the figure.

However, to color the map of the peninsular provinces in this way, four colors are necessary. To deduce it, it is not necessary to examine the entire map: it is enough to realize that there are provinces that are surrounded by an odd number of neighbors. If the number were even, two alternate colors would suffice to color the surrounding provinces plus one for the circled one, three in total; but if the number is odd, the alternation ends with two surrounding provinces of the same color together; therefore, three colors are needed for these and another for the central one, which borders all of them. In the figure we see the case of Valladolid, surrounded by seven other provinces.

As for the envelopes, the open one can be drawn in a single stroke, but the closed one cannot, because three edges meet at all its vertices. As the pencil starts from one vertex, it has to reach the other three, leave and return (and it could not leave again without going through a path already covered), so it would have to finish its itinerary in three points at a time, which is obviously impossible. In the case of the open envelope, however, there are only two vertices (the lower ones) where three edges meet, so the path of the pencil can start at one of them and end at the other. Generalizing, when an even number of edges concur in a node, if the route starts from it, it must end in it; and vice versa: when the number of concurrent edges at a node is odd, if the path starts from it, it cannot end at it. As in the case of maps, parity is the key, since, although they seem to be very different problems, both can be schematized using similar graphs.

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From envelope to box

And from rectangular envelopes, the flat containers par excellence, we can jump to orthodontic boxes, the most common three-dimensional containers. Among them, the familiar one-liter tetrabrik, whose dimensions are approximately 20x10x5 cm (more like 19x9x5.9, but ours is the platonic ideal tetrabrik). An elegant shape (each side is half of the previous one) and manageable; but is it optimal from the point of view of the use of the material? What savings would be possible, in square centimeters of wrapping, if we optimized the volume/surface ratio?

Both the bases (10×5) and the larger lateral faces (20×10) of our tetrabrik are dominoes, and the smaller lateral faces (20×5) are right tetrominoes (rectangles whose larger side is four times the smaller). Bearing this in mind, in how many different ways can we cover a standard chessboard with squares of 5 cm on each side with tetrabriks -supporting them on any of their faces? And a 10×10 checkerboard? Is it possible, in either case, a coating without fracture lines? (See the article “Without fracture lines”, from 10 12 21).

And, finally, a problem proposed by our assiduous commentator Luca Tanganelli:

Find all the trigons with integer sides such that volume + perimeter = area.

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, among them ‘Maldita physics’, ‘Malditas Matematicas’ or ‘El Gran Juego’. He was a screenwriter for ‘The Crystal Ball’.

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George Holan

George Holan is chief editor at Plainsmen Post and has articles published in many notable publications in the last decade.

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