Lesson plan
Ducci sequence
A Ducci sequence is a sequence of n-tuples of integers, sometimes known as "the Diffy game", since it is based on differences (subtractions). Given an n-tuple of integers (a_1,a_2,...,a_n) , the next n-tuple in the sequence is formed by taking the absolute differences of neighbouring integers: : (a_1,a_2,...,a_n) \rightarrow (|a_1-a_2|, |a_2-a_3|, ..., |a_n-a_1|)\, . Another way of describing this is as follows. Arrange n integers in a circle and make a new circle by taking the difference between neighbours, ignoring any minus signs; Then repeat the operation. Ducci sequences are named after Enrico Ducci (1864–1940), the Italian mathematician who discovered in the 1930s that every such sequence eventually becomes periodic. Ducci sequences are also known as the Ducci map or the n-number game. Open problems in the study of these maps still remain.
Article summary
A Ducci sequence is a sequence of n-tuples of integers, sometimes known as "the Diffy game", since it is based on differences (subtractions). Given an n-tuple of integers (a_1,a_2,...,a_n) , the next n-tuple in the sequence is formed by taking the absolute differences of neighbouring integers: : (a_1,a_2,...,a_n) \rightarrow (|a_1-a_2|, |a_2-a_3|, ..., |a_n-a_1|)\, . Another way of describing this is as follows. Arrange n integers in a circle and make a new circle by taking the difference between neighbours, ignoring any minus signs; Then repeat the operation. Ducci sequences are named after Enrico Ducci (1864–1940), the Italian mathematician who discovered in the 1930s that every such sequence eventually becomes periodic. Ducci sequences are also known as the Ducci map or the n-number game. Open problems in the study of these maps still remain.
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Discussion questions
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- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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