The sequence of the number of different trees that can be formed with n nodes is: 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235, 551 … (The first 1 corresponds to the elementary graph of a single node).
We asked ourselves last week if there is a simple formula that gives the number of different trees depending on the number of nodes. The answer is no. The sequence begins resembling that of Fibonacci (1, 1, 2, 3, 5, 8, 13, 21 …), and it is tempting to relate them, since it often appears in the trees of nature; but the growth of the "arboreal sequence" is increasingly faster than that of Fibonacci.
As we saw, in the case of three points that are vertices of an equilateral triangle, the Steiner tree is obtained by taking the center of the triangle as "Steiner point". What if the triangle is not equilateral? In that case, look for the point whose sum of distances to the three vertices is minimal. It is the Fermat point or Torricelli point of the triangle, named because the first one posed the problem to the second and this solved it.
In a triangle that does not have an angle greater than 120º, the Fermat point, which coincides with the Sreiner point of the nodes located in the vertices, is constructing two equilateral triangles in two of the sides and joining their outer vertices with the opposite vertices of the triangle in question, as indicated in the figure.
In the case of a square, the minimum length tree is obtained with the configuration of the figure, in which the three angles that converge in each of the two Steiner points are 120º. If we take the side of the square as a unit, what is the length of this minimum tree? What savings do you make with respect to the most obvious tree, formed by three sides of the square? And with respect to the tree formed by the two diagonals of the square?
And a harder one still: how is the Steiner tree from the vertices of a regular pentagon?
The woman who calculated
This delivery of The game of science it is published on March 8, International Women's Day, and as a modest contribution to its celebration I propose three logical riddles of one of the few women who, in the wake of Ada Lovelace, have devoted themselves to studying and refining forms of calculation. I refer to Angela Foxx Dunn, of whom I have already spoken on some occasion, and who in the sixties of the last century he made, for a couple of technical magazines, an excellent weekly section of mathematical riddles, almost always related to the calculus, entitled Problematical Recreations, and also published several books on the subject (there is at least one in Spanish, although difficult to find: Grandpa ready, RBA, 2008). Here are three of his riddles:
In a 100-piece puzzle, how many moves are necessary to complete it? A movement consists of assembling two sets of pieces (including one-piece sets).
A drawer contains an odd number of brown socks and an even number of black socks. What is the smallest number of brown and black socks that the drawer must contain so that when removing two socks at random the probability that both are brown is 1/2?
What is the highest result that can be obtained as a product of natural numbers (whole and positive) whose sum is 100?
Carlo Frabetti He is a writer and mathematician, a member of the New York Academy of Sciences. He has published more than 50 scientific dissemination works for adults, children and young people, among them Damn physics, Damn mathematics or The big game. He was a screenwriter The Cristal ball.