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Authors
Abstract(s)
Após uma análise da teoria e dos trabalho existente sobre o triângulo de rascal, apercebemo-nos que existiam lacunas na informação disponível, principalmente falta de informação
em português. Motivados pelo interesse e curiosidade sobre as propriedades, conjeturas e
aplicações deste triângulo (objeto de estudo recente), tendo em conta a escassa bibliografia
encontrada, fomos aprofundar o estudo investigando mais acerca do mesmo.
Nesta dissertação, começamos por apresentar a história do triângulo de rascal, as suas
propriedades e resultados existentes na bibliografia consultada. Na nossa investigação
foram analisadas conjeturas já conhecidas e outras resultantes das tarefas investigativas
que realizamos. Foi-nos, por isso, possível definir novos conceitos e novas propriedades
subjacentes ao tema principal desta dissertação. De salientar, que uma parte das propriedades
foi obtida por manipulação algébrica (formula algébrica dos números de rascal) e
outra através de interpretações geométricas no triângulo.
Apresentamos, ainda, duas generalizações para o triângulo que consistem na eliminação das restrições dos números de rascal e na interpretação combinatória dos mesmos.
Tendo em conta a escassez bibliográfica, como já referimos, foi-nos bastante trabalhoso
e complexo investigar o triângulo e analisar as suas propriedades. Tivemos necessidade
de compreender as propriedades fundamentais do triângulo de rascal e estudar a relação
entre estas e o já conhecido triângulo de Pascal (estabelecemos uma ligação entre os dois
triângulos porque o de Pascal já foi bastante estudado).
Apesar do trabalho árduo, foi-nos possível ultrapassar as dificuldades iniciais e aprofundar/
investigar este tema, descobrindo várias propriedades que iam sistematicamente
aparecendo. Isto deve motivar vivamente a continuação da investigação do tema da dissertação que apresentamos.
After an analysis of the theory and the existing works on the rascal triangle, we realized that there were gaps in the available information, mainly the lack of information in Portuguese. Motivated by interest and curiosity about the properties, conjectures and applications of this triangle (object of recent study), and taking into account the scarce bibliography found, we went deeper into the study by investigating more about it. In this dissertation, we begin by presenting the history of the rascal triangle, its properties and results existing in the consulted bibliography. In our investigation we analysed known conjectures and others resulting from the investigative tasks that we performed. It was, therefore, possible to define new concepts and new properties underlying the main theme of this dissertation. It should be noted that a part of the properties were obtained by algebraic manipulation (algebraic formula of the rascal numbers) and another through geometric interpretations in the triangle. We present also two generalizations for the triangle that consist in the elimination of the restrictions for the numbers of rascal and in the combinatorial interpretation of the same ones. Given the scarcity of bibliography, as we have already mentioned, it was quite difficult and complex to investigate the triangle and analyze its properties. We needed to understand the fundamental properties of the rascal triangle and study the relationship between them and the well-known triangle of Pascal (we have established a connection between the two triangles because Pascal's has been well studied). Despite the hard work, we were able to overcome the initial difficulties and to deepen/ investigate this theme, discovering several properties that were systematically appearing. This should strongly motivate the continuation of the investigation of the theme of the dissertation that we present.
After an analysis of the theory and the existing works on the rascal triangle, we realized that there were gaps in the available information, mainly the lack of information in Portuguese. Motivated by interest and curiosity about the properties, conjectures and applications of this triangle (object of recent study), and taking into account the scarce bibliography found, we went deeper into the study by investigating more about it. In this dissertation, we begin by presenting the history of the rascal triangle, its properties and results existing in the consulted bibliography. In our investigation we analysed known conjectures and others resulting from the investigative tasks that we performed. It was, therefore, possible to define new concepts and new properties underlying the main theme of this dissertation. It should be noted that a part of the properties were obtained by algebraic manipulation (algebraic formula of the rascal numbers) and another through geometric interpretations in the triangle. We present also two generalizations for the triangle that consist in the elimination of the restrictions for the numbers of rascal and in the combinatorial interpretation of the same ones. Given the scarcity of bibliography, as we have already mentioned, it was quite difficult and complex to investigate the triangle and analyze its properties. We needed to understand the fundamental properties of the rascal triangle and study the relationship between them and the well-known triangle of Pascal (we have established a connection between the two triangles because Pascal's has been well studied). Despite the hard work, we were able to overcome the initial difficulties and to deepen/ investigate this theme, discovering several properties that were systematically appearing. This should strongly motivate the continuation of the investigation of the theme of the dissertation that we present.
Description
Keywords
Sequências Tarefas Investigativas. Triângulo de Pascal Triângulo de Rascal