A Ducci sequence is a sequence of n-tuples of integers, sometimes known as "the Diffy game", because it is based on sequences.
Given an n-tuple of integers , the next n-tuple in the sequence is formed by taking the absolute differences of neighbouring integers:
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.[1]
[2]
From the second n-tuple onwards, it is clear that every integer in each n-tuple in a Ducci sequence is greater than or equal to 0 and is less than or equal to the difference between the maximum and minimum members of the first n-tuple. As there are only a finite number of possible n-tuples with these constraints, the sequence of n-tuples must sooner or later repeat itself. Every Ducci sequence therefore eventually becomes periodic.
If n is a power of 2 every Ducci sequence eventually reaches the n-tuple (0,0,...,0) in a finite number of steps.[1]
[3]
[4]
If n is not a power of two, a Ducci sequence will either eventually reach an n-tuple of zeros or will settle into a periodic loop of 'binary' n-tuples; that is, n-tuples of form , is a constant, and .
An obvious generalisation of Ducci sequences is to allow the members of the n-tuples to be any real numbers rather than just integers. For example,
[2] this 4-tuple converges to (0, 0, 0, 0) in four iterations:
The properties presented here do not always hold for these generalisations. For example, a Ducci sequence starting with the n-tuple (1, q, q2, q3) where q is the (irrational) positive root of the cubic does not reach (0,0,0,0) in a finite number of steps, although in the limit it converges to (0,0,0,0).[5]
Ducci sequences may be arbitrarily long before they reach a tuple of zeros or a periodic loop. The 4-tuple sequence starting with (0, 653, 1854, 4063) takes 24 iterations to reach the zeros tuple.
This 5-tuple sequence enters a period 15 binary 'loop' after 7 iterations.
The following 6-tuple sequence shows that sequences of tuples whose length is not a power of two may still reach a tuple of zeros:
If some conditions are imposed on any "power of two"-tuple Ducci sequence, it would take that power of two or lesser iterations to reach the zeros tuple. It is hypothesized that these sequences conform to a rule.[6]
When the Ducci sequences enter binary loops, it is possible to treat the sequence in modulo two. That is:[7]
This forms the basis for proving when the sequence vanish to all zeros.
Florian Breuer, "Ducci sequences in higher dimensions" in INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007) S Lettieri, JG Stevens, DM Thomas, "Characteristic and minimal polynomials of linear cellular automata" in Rocky Mountain J. Math, 2006.
M Misiurewicz, JG Stevens, DM Thomas, "Iterations of linear maps over finite fields", Linear Algebra and Its Applications, 2006
F. Breuer et al. 'Ducci-sequences and cyclotomic polynomials' in Finite Fields and Their Applications 13 (2007) 293–304