Volume XVI. April 2026

In this article we aim to answer the question: What natural numbers can be expressed as the sum of consecutive natural numbers?

Título: Sums of consecutive natural numbers Autor: Romina Vázquez Rodino, Duberly González Molinari

Morley’s theorem states that “the points of intersection of the adjacent interior trisectors of the angles of any triangle determine an equilateral triangle”. The first complete proof of Morley’s theorem is found in the article “The six trisectors of the angles each of a triangle”, which shows that there are 27 equilateral triangles, 18 of which are Morley triangles, and that the structure also has 9 Morley triangles that are not equilateral. The purpose of this work is to present a proof that the first Morley triangle is equilateral, as well as to develop, in GeoGebra, a geometric construction that illustrates these 36 triangles based on the article “Morley’s Triangle”, for which a basic knowledge of this software is required.

Título: A geometric construction of triangles related to Morley’s theorem Autor: Saulo Mosquera López

This paper presents the experience of a teaching innovation based on Google’s Artificial Intelligence Gemini in the subject of Strategic Management with students of 4th year 2024 Business Administration and Management at the University of Alcalá (Madrid). The main results reveal that students find the teaching and learning materials useful and relevant. In this sense, the process of interaction with Gemini to optimize the answers through the process of reformulating questions and contrasting them, may have provided a certain degree of motivation to the student that reinforces learning. The use of AI, in the academic field, is perhaps a disruptive change. It poses a major challenge in terms of effort and adaptation of teachers and students, which is not without uncertainty and involves reflection and planning, to face the immediate future.

Título: Teaching innovation using Artificial Intelligence Autor: Justo de Jorge-Moreno

The activation of motivation and the promotion of beliefs, attitudes and positive emotions are relevant in the teaching and learning process. This work proposes the design of mathematical routes from their configuration as training in centers to their implementation. A practical experience with students is collected with the results of previous questionnaires on knowledge and a subsequent questionnaire on the level of satisfaction and emotional performance of the participating students. The teacher training curricular design includes objectives, contents, methodology, the curricular framework as a complementary activity in school hours and outdoor space and its organization, as well as the materials necessary for the route and the data collection forms. The teaching reflection and the results of the questionnaires reflect that this type of proposal is useful to promote the cohesion of the teaching department and also to encourage the idea of enjoying mathematical work regardless of the level of knowledge. As a prospective, it would be interesting to promote the participation of other centers and to obtain more data on the level of student satisfaction.

Título: Study of mathematical routes as an element to improve the learning process Autor: Ana María Zarco y Adrián Monforte

Dichotomino —from “dichotomy” and “domino”— is a hands-on manipulative inspired by Dienes’ Logic Blocks and Touyarot’s KML blocks. It features six dichotomous variables (color, shape, size, height, interior, and texture) and a nesting design (small piece fitting into the corresponding large hollow piece) that expands the task space. The resource is intended to optimize the discovery and creation of similarity patterns and to foster algebraic thinking from early years through teacher education. After situating the notion of pattern in the curriculum, we present six classroom games (WODB, Quarto, a no-line “solitaire”, opposite pairs, n-difference series, and a Set-type game), together with STL files and templates for 3D printing and classroom use.

Título: Dichotomino: A resource for pattern discovery and the development of algebraic sense Autor: Carlos de Castro y Álvaro Nolla

This article is an introduction to the Riordan group. It presents, in a self-contained and coherent manner, the foundations for studying its algebraic properties and aims to convey the significance and appeal of such a study.

It includes the basic definitions and properties of formal power series, the statements and proofs of the two Fundamental Theorems of Riordan Matrices, and the proof that the set of all Riordan matrices indeed forms a group. Finally, the structure of the Riordan group is described as the inverse limit of an inverse sequence of finite-dimensional matrix groups together with a general strategy that made it possible to prove, in a unified way, many recent theorems concerning the algebraic structure of the Riordan group, some of which are listed at the end of the article.

Título: An introduction to the algebraic structure of the Riordan group Autor: Luis Felipe Prieto Martínez

In the nineteenth century, Georg Cantor opened an irreparable rift in the mathematical universe by conceiving the existence of infinities of different sizes. This revolutionary intuition, which defied the Aristotelian tradition, gave birth to set theory but also provoked a series of paradoxes that demanded a radical redefinition of the foundations of mathematics. This article traces that development: from Cantors transfinite arithmetic, through the logical paradoxes, Hilberts formalist programme, and Gödels incompleteness theorems, to the computational limits established by Turing and Church. Along this trajectory, tensions between intuition and formalism become evident, as does the role of human thought at the very heart of mathematical truth. Rather than offering a technical analysis, this essay proposes a philosophical reflection on the scope —and the limits— of rational thought.

Título: From the infinite to the undecidable: set theory and the foundational crisis in mathematics Autor: Daniel Martín-Cudero

This article presents an investigation into the behavior of the number 99 under a simple numeric transformation applied to non-palindromic two-digit numbers. Through digit inversion, calculation of absolute difference, and summation with its inverse, it is demonstrated that the resulting invariant of this process is consistently 99. A detailed algebraic analysis is provided, emphasizing its educational potential for introducing fundamental concepts in basic mathematics education.

Título: Formal Exploration of Number 99 as a Magic Number in Symmetric Two-Digit Transformations Autor: Daniel Andrés Martínez Robalino

We’ve all experienced the unpleasant sensation of our vehicle’s undercarriage scraping when entering a garage ramp with a certain incline. This article rigorously demonstrates the minimum ground clearance a vehicle must have (from the chassis to the ground) to avoid this scraping as soon as it enters the ramp.

Título: Critical height of a car chassis on a slope Autor: Elías Martínez Martínez

The following tale is one of the exercises completed in a creative writing workshop at the Fuentetaja School of Writing. A noir tale in which crimes occur with mathematical precision.

Título: Sequence of fortune and desire Autor: Eduardo Pedrero

This narrative encourages reflection on basic arithmetic, in particular it deals with even and odd numbers and some of their properties; It is focused on children between 9 and 12 years old.

Título: Math-adventures at the farm: even and odd Autor: Osvaldo Osuna y Berenice Reyes-Herrera

Matemápolis is an illustration carried out in the year 2022 by Lola Morales, mathematics teacher of Secondary and Baccalaureate.

The illustration, inspired by the wimmelbilderbuch style, shows, through drawings, simple (and complex) mathematical concepts and both old and current characters related to mathematics.

Título: Matemápolis Autor: Susana Merchán

This is the result of an idea that emerged some time ago as a project of the “Mathematical Thinking” Educational Innovation Group at the Polytechnic University of Madrid (UPM). The idea was to write a crime novel in which anyone interested was invited to participate. To do so, participants had to read the content of the story published up to that point and then contribute their own ideas, continuing the existing narrative. Everything was done through an online form, and together they shaped the story. The only rules were these: the novel had to fall within the “crime novel” genre and had to have as much mathematical content as possible; that is, this science had to be present, as much as possible, in a large part of the story’s development.

The plot developed, but at a certain point, it stalled until Asís Álvarez, a student at the School of Civil Engineering at UPM, took up the task and finished it.

A crime novel set in the School of Civil Engineering at the Polytechnic University of Madrid (UPM) is presented, with guaranteed mathematical content. Several professors from the Department of Applied Mathematics have died of natural causes. While the offices they left empty are being reorganized, a professor discovers a connection in the order of the deaths. After reading this novel, you’ll see the classrooms, hallways, laboratories, and even your math professors in a new light

Título: Professors who die, professors who kill Autor: Asís Álvarez Romero y varios autores