Ducci sequence

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, the Italian mathematician credited with their discovery.

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]

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 mimimum 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] [2] [3]}

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 which contain only two different digits.

An obvious generalisation of Ducci sequences is to allow the members of the n-tuples to be any real numbers rather than just integers. 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).^[4]

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.^[5]

Modulo two form

When the Ducci sequences enter binary loops, it is possible to treat the sequence in modulo two. That is:^[6]

${\displaystyle (|a_{1}-a_{2}|,|a_{2}-a_{3}|,...,|a_{n}-a_{1}|)\ =(a_{1}+a_{2},a_{2}+a_{3},...,a_{n}+a_{1})\ mod2}$

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.^[7][8] Rule 102 reproduces the Sierpinski triangle.^[9]

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.^[10] While there are no practical applications of the Ducci map at present, its connection to the highly applied field of difference equations led ^[4] to conjecture that a form of the Ducci map may also find application in the future.

References

1. Chamberland, Marc; Thomas, Diana M. (2004). "The N-Number Ducci Game" (PDF). Journal of Difference Equations and Applications. London: Taylor & Francis. 10 (3): 33–36. Retrieved 2009-01-26.
2. Chamberland, Marc (2003). "Unbounded Ducci sequences" (PDF). Journal of Difference Equations and Applications. London: Taylor & Francis. 9 (10): 887–895. doi:10.1080/1023619021000041424. Retrieved 2009-01-26.
3. Andriychenko, Oleksiy; Chamberland, Marc (2000). "Iterated Strings and Cellular Automata". The Mathematical Intelligencer. New York, NY: Springer Verlag. 22 (4): 33–36. doi:10.1007/BF03026764.
4. Brockman, Greg (2007). "Asymptotic behaviour of certain Ducci sequences" (PDF). Fibonacci Quarterly.
5. Euich, Miztani; Akihiro, Nozaki.; Toru, Sawatari. (2013). "A Conjecture of Ducci Sequences and the Aspects" (PDF). RIMS Kokyuroku. Kyoto: RIMS. 1873: 88–97. Retrieved 2014-02-18.
6. Florian Breuer, "Ducci sequences in higher dimensions" in INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007)
7. S Lettieri, JG Stevens, DM Thomas, "Characteristic and minimal polynomials of linear cellular automata" in Rocky Mountain J. Math, 2006.
8. M Misiurewicz, JG Stevens, DM Thomas, "Iterations of linear maps over finite fields", Linear Algebra and Its Applications, 2006
9. Weisstein, Eric W. "Rule 102." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Rule102.html
10. 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