In this paper, we give examples that improve the lower bound on the maximum number of halving lines for sets in the plane with 35, 59, 95, and 97 points and, as a consequence, we improve the current best upper bound of the rectilinear crossing number for sets in the plane with 35, 59, 95, and 97 points, provided that a conjecture included in the literature is true. As another consequence, we also improve the lower bound on the maximum number of halving pseudolines for sets in the plane with 35 points. These examples, and the recursive bounds for the maximum number of halving lines for sets with an odd number of points achieved, give a new insight in the study of the rectilinear crossing number problem, one of the most challenging tasks in Discrete Geometry. With respect to this problem, it is conjectured that, for all n multiples of 3, there are 3-symmetric sets of n points for which the rectilinear crossing number is attained.

We give two new ways of calculating the probability of a chord of circumference randomly selected being larger than the side of an equilateral triangle inscribed in the circumference (this problem is known as the Bertrand paradox). The first one employs an immersion in R4, and the second one uses a direct probability measure over the set of chords.

In this paper, we provide improvements in the additive constant of the current best asymptotic upper bound for the maximum number of halving lines for planar sets of n points, where n is an even number. We also improve this current best upper bound for small values of n, namely, 106≤n≤336. To obtain this enhancements, we provide lower bounds for the sum of the squares of the degrees of the vertices of a graph related to the halving lines.

Este trabajo presenta un proyecto cuyo eje central es la realización de una competición educativa, adecuada para la enseñanza y aprendizaje de los estudiantes de diferentes etapas formativas, a través de un concurso que aúna la ciencia y el arte. El concurso Art & Science ha sido una actividad del Aula Taller Museo de las Matemáticas π-ensa de la Universidad Politécnica de Madrid (UPM) y de la comunidad SSERIES de la Alianza de Universidades Europeas EELISA, organizada conjuntamente con el Instituto Nacional de Física Nuclear (Italia) y el CERN en Ginebra, para promover la cultura científica entre los estudiantes combinando los lenguajes del arte y la ciencia, dos herramientas de conocimiento que se encuentran entre las más altas expresiones de la creatividad humana. El proyecto ha estado dirigido a estudiantes de secundaria y bachillerato que han sido mentorizados por estudiantes universitarios de la Alianza EELISA. El objetivo principal ha sido la divulgación científica a través del arte. Se ha realizado entre los meses de diciembre de 2022 y abril de 2023. Este capítulo es una ampliación de la comunicación publicada en el Libro de Actas del Congreso CIVAE 2023.

Art & Science contest has been an activity of the Museum of Mathematics pi-ensa (Polytechnic University of Madrid) and SSERIES community of the EELISA European University Alliance, organized jointly with the National Institute of Nuclear Physics (Italy) and the CERN in Geneva. Its main objective has been to promote scientific culture among students, combining the languages of art and science, two tools of knowledge among the highest expressions of human creativity. The project is aimed at secondary and high school students from the Madrid region (Spain), who are mentored by university students from the EELISA Alliance. The aim: the dissemination of science through art. It has been carried out between the months of December 2022 and April 2023.

1 Universidad Politécnica de Madrid (SPAIN)2 Student in the dual degree program in Physical Activity and Sport Sciences and Primary Education at the Comillas Pontifical University in Madrid. (SPAIN)3 Comillas Pontifical University (SPAIN)4 Polytechnic University of Madrid (SPAIN)5 Professional dancer and student of the Higher Degree in Dance at the Superior Conservatory of Dance of Madrid “María de Ávila (SPAIN)

This article discusses a theoretical construction based on the graph theory to rework the space of potential partitions in envy-free distribution. This work has the objective of applying Sperner’s lemma to the distribution of three rotating shifts for three workers who are to cover a 24 h job position in a company. As a novel feature, worker’s preferences have been modeled as functions of probability for the three shifts, according to salary offers for said shifts. Envy-free allocation was achieved, since each worker received their preferred shift without the need for negotiation between agents in conflict. Adaptation to the type of dynamic situations that arise with rotating shifts, as well as the consideration of probabilistic preferences by workers are some of the main novelties of this work.

In this paper, we improve the lower bound on the minimum number of  ≤k-edges in sets of n points in general position in the plane when k is close to n2. As a consequence, we improve the current best lower bound of the rectilinear crossing number of the complete graph Kn for some values of n.

The study presented makes an original, new and exhaustive analysis of the adaptation of a classical board game which has been named Binary Who is Who? This proposal shows a very useful tool for the consolidation of mathematical concepts related to the study of real-valued functions that are treated in the different levels of the teaching of mathematics (first and second year of superior secondary studies and the first years of some university degrees). The use of games as a means for learning is the authors’ proposal. The aim is to offer teachers the chance of using the games as a method of teaching mathematical concepts, as well as a motivating instrument for them. This game has been created to be played face-to-face in the classroom and it has also been programmed to create a video game which allows the students to play virtually.

The incorporation of game elements on behalf of the teachers in order to improve the interest and the motivation of the students is having a quick rise in every educational level. This work is a reflection about gamification as a strategy of enrichment and approach to mathematic topics. Through the paper there are described some of the activities which the Educational Innovation Group "Pensamiento Matematico" of the Universidad Politecnica de Madrid (UPM), is carrying out in the context of the Aula Taller Museo de las Matematicas pi-ensa. The gamification and the impact this technique is having on the students of the Civil and Territorial Engineering degree of the UPM, as well as on the students of Secondary Education and Bachillerato who visit pi-ensa, are analysed. The paper tries to be an essential basis for starting processes of research and application of the gamification in the teaching of Mathematics at every level, especially in the students of new entry in the different university degrees.