Illustration

The bridges of Königsberg.

The atypical successions of last week could continue in these ways (but also in others):

2, 10, 12, 16, 17, 18, 19, 200, 201, 202…

1, 2, 3, 4, 7, 10, 17, 24, 41, 58…

Egg, chicken, cow, fly, spider...

Sponge, ostrich, kangaroo, dingo, starfish...

Strawberry, tangerine, lemon, grape, blueberry…

January, March, April, June, July

Do, fa, la, mi, re, si, sol

Fa, la, re, mi, si, do, sol

I leave it to the sagacity of my readers to discover the criterion of continuity in each case.

Two clues: last week's illustration refers to one of the sequences, and the ellipses indicate that the sequence could continue (while no ellipsis means the list is complete).

With regard to the minor prophets, no one has ventured any hypothesis regarding their two different ordinations (the first corresponds to the Jewish and Catholic Bible, and the second to the Septuagint or Greek Bible), so the question remains open.

The lost and found graph

Regarding the problem taken from Clara Grima's wonderful book

(a book that you start thinking: "Why didn't I write it?" and you end up recognizing that it is better that she did it), the solution is that Alice has shaken hands with 4 people, as is easy to see by drawing the corresponding graph.

Recall that a graph is a set of points, called vertices or nodes, joined by a series of lines, called sides or edges, that represent binary relationships between the elements of a set. If from any of the points of the graph you can go to any other along edges, it is a connected graph. And if there are no closed circuits in a connected graph, it is called a tree, due to its resemblance to the trees of nature (Ramón Llull's Tree of Science and Porphyry's Tree are illustrious precursors of tree graphs).

It can be considered that graph theory was inaugurated by Euler in 1736 with an article on the famous problem of the bridges of Königsberg (now Kaliningrad), which we have dealt with on occasion, in which he showed that it was impossible to traverse the four areas of the city passing through its seven bridges only once and returning to the starting point.

Another well-known graph problem is drawing an open envelope without lifting the pencil from the paper or tracing the same path twice.

It's possible?

What if the envelope is sealed?

Going back to Clara Grima's book, in one of its chapters she talks about the controversial (some consider that it has been proven and others think that it has not been completely proven) topological theorem of the four colors, which states that four colors are enough to color any map in a correct way. that never two bordering zones are of the same color.

Four colors are enough, but not always necessary;

for example, to color a chessboard in this way, two colors are enough.

And to color the map of Andalusia so that its eight provinces are clearly differentiated (that is, without two border provinces being the same colour), are four colors necessary or can it be achieved with less?

And to color the map of peninsular Spain?

(Easy version: divided into autonomous communities; less easy version: divided into provinces).

Carlo Frabetti

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