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 \( {\displaystyle (a_{1},a_{2},...,a_{n})} \), the next n-tuple in the sequence is formed by taking the absolute differences of neighbouring integers:
\( {\displaystyle (a_{1},a_{2},...,a_{n})\rightarrow (|a_{1}-a_{2}|,|a_{2}-a_{3}|,...,|a_{n}-a_{1}|)\,.} \)
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 this in the 1930s.
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]
Properties
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 \( {\displaystyle k(b_{1},b_{2},...b_{n})} \), k is a constant, and \( {\displaystyle b_{i}\in \{0,1\}} \).
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: \({\displaystyle (e,\pi ,{\sqrt {2}},1)\rightarrow (\pi -e,\pi -{\sqrt {2}},{\sqrt {2}}-1,e-1)\rightarrow (e-{\sqrt {2}},\pi -2{\sqrt {2}}+1,e-{\sqrt {2}},2e-\pi -1)\rightarrow } \) \( {\displaystyle (\pi -e-{\sqrt {2}}+1,\pi -e-{\sqrt {2}}+1,\pi -e-{\sqrt {2}}+1,\pi -e-{\sqrt {2}}+1)\rightarrow (0,0,0,0)} \)
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 \( {\displaystyle x^{3}-x^{2}-x-1=0} \)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]
Examples
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.
\( {\displaystyle (0,653,1854,4063)\rightarrow (653,1201,2209,4063)\rightarrow (548,1008,1854,3410)\rightarrow }\) \( {\displaystyle \cdots \rightarrow (0,0,128,128)\rightarrow (0,128,0,128)\rightarrow (128,128,128,128)\rightarrow (0,0,0,0)} \)
This 5-tuple sequence enters a period 15 binary 'loop' after 7 iterations.
\( {\displaystyle {\begin{matrix}15799\rightarrow &42208\rightarrow &20284\rightarrow &22642\rightarrow &04220\rightarrow &42020\rightarrow \\22224\rightarrow &00022\rightarrow &00202\rightarrow &02222\rightarrow &20002\rightarrow &20020\rightarrow \\20222\rightarrow &22000\rightarrow &02002\rightarrow &22022\rightarrow &02200\rightarrow &20200\rightarrow \\22202\rightarrow &00220\rightarrow &02020\rightarrow &22220\rightarrow &00022\rightarrow &\cdots \quad \quad \\\end{matrix}}} \)
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:
\( {\displaystyle {\begin{matrix}121210\rightarrow &111111\rightarrow &000000\\\end{matrix}}} \)
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]
Modulo two form
When the Ducci sequences enter binary loops, it is possible to treat the sequence in modulo two. That is:[7]
\( {\displaystyle (|a_{1}-a_{2}|,|a_{2}-a_{3}|,...,|a_{n}-a_{1}|)\ =(a_{1}+a_{2},a_{2}+a_{3},...,a_{n}+a_{1})\ mod2} \)
This forms the basis for proving when the sequence vanish to all zeros.
Cellular automata
The linear map in modulo 2 can further be identified as the cellular automata denoted as rule 102 in Wolfram code and related to rule 90 through the Martin-Odlyzko-Wolfram map.[8][9] Rule 102 reproduces the Sierpinski triangle.[10]
Other related topics
The Ducci map is an example of a difference equation, a category that also include non-linear dynamics, chaos theory and numerical analysis. Similarities to cyclotomic polynomials have also been pointed out.[11] While there are no practical applications of the Ducci map at present, its connection to the highly applied field of difference equations led [5] to conjecture that a form of the Ducci map may also find application in the future.
References
Chamberland, Marc; Thomas, Diana M. (2004). "The N-Number Ducci Game" (PDF). Journal of Difference Equations and Applications. 10 (3): 33–36. doi:10.1080/10236190410001647807. Retrieved 2009-01-26.
Clausing, Achim (2018). "Ducci matrices". The American Mathematical Monthly. 125 (10): 901–921. doi:10.1080/00029890.2018.1523661.
Chamberland, Marc (2003). "Unbounded Ducci sequences" (PDF). Journal of Difference Equations and Applications. 9 (10): 887–895. CiteSeerX 10.1.1.63.6652. doi:10.1080/1023619021000041424. Retrieved 2009-01-26.
Andriychenko, Oleksiy; Chamberland, Marc (2000). "Iterated Strings and Cellular Automata". The Mathematical Intelligencer. 22 (4): 33–36. doi:10.1007/BF03026764.
Brockman, Greg (2007). "Asymptotic behaviour of certain Ducci sequences" (PDF). Fibonacci Quarterly.
Euich, Miztani; Akihiro, Nozaki.; Toru, Sawatari. (2013). "A Conjecture of Ducci Sequences and the Aspects" (PDF). RIMS Kokyuroku. 1873: 88–97. Retrieved 2014-02-18.
Florian Breuer, "Ducci sequences in higher dimensions" in INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007) [1]
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
Weisstein, Eric W. "Rule 102." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Rule102.html
F. Breuer et al. 'Ducci-sequences and cyclotomic polynomials' in Finite Fields and Their Applications 13 (2007) 293–304
External links
Ducci Sequence
Integer Iterations on a Circle at Cut-the-Knot
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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