MAJA

Mathematical algorithms analyzed for everybody

The MAJA project was dedicated to the analysis of mathematical algorithms during its three-year period from October 2022 to December 2025. According to the basic thesis of the project, it is essential that the widespread use of mathematical algorithms goes hand in hand with raising public awareness of how they work, their limitations and the resulting problems. A team of researchers from the University of Salzburg and the Salzburg University of Education worked together with teachers and pupils from the Salzburg Academic Gymnasium and the HTL Braunau, to develop concepts with the aim of achieving this goal.

Multiplicatively dependent tuples

In one of the MAJA workshops, participating students succeeded in discovering new multiplicatively dependent tuples with the help of self-written computer programs. A tuple is an ordered sequence of numbers. It is called multiplicatively dependent if the numbers can be multiplied or divided with each other (the tuple numbers may be used more than once) so that they cancel out and finally result in the number 1. A simple example of such a tuple is: (16,3,2,18), because (16 ∙3^2)/(2^3∙ 18)=(16 ∙ 9)/(8 ∙ 18)=144/144=1.

Towers of Hanoi

While writing his bachelor's thesis as part of the MAJA project, a student teacher developed the idea that this well-known puzzle is particularly suitable for illustrating the analysis of a mathematical algorithm. The task is to transfer a tower of increasing discs consisting of n parts, for example n=3,4,5 or 2026, from place A to place C with the help of place B. Only one disc may be moved per step, and a disc can never be placed on a smaller disc. This can be solved algorithmically, and it becomes apparent that quickest way this can be done is in 2^n-1 steps.

Patchable meshes

The construction of basic functions that are smooth in all directions (they can be pictured as gentle hills) on adaptable base grids plays an important role in the finite element method for solving differential equations and thus in many physical applications. As part of the MAJA project, a design for the generation of patchable meshes was developed. This involved assigning a bundle of basic functions to each constraining grid point in such a way that its surface area matches the patch assigned to that grid point.

The project concentrated on recursive algorithms in discrete mathematics and adaptive algorithms in numerical mathematics. Additionally, a survey and evaluation of the mathematical worldviews of the participating pupils was carried out, which formed the math-education-specific research content of the project. Cooperation with the partner schools was possible via an optional subject or an elective subject in the module system and multi-day workshops were also held regularly. In addition to specialist publications, learning materials were created and a MAJA app was programmed, which, together with the learning materials, will continue to be available for use in investigations and in the classroom.

This project is already completed.