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I changed the original "one tuple per line" format because it took up way too much space on the screen and looked unbalanced. But I agree there is now an ambiguity in the order of reading. Maybe an explanatory note would help. Alternatively, arrows could show the sequence of iterations like this:
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No, what I am saying is that, even for N which is not a power of 2, there are sequences that converge to 0. The article says "It has been proven that for n not a power of two, the Ducci sequence will settle on a loop with 'binary' sequences. That is, with elements composed of only two different
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Very good. Now please give credit where credit is due. An external link to the page suggested above will do the job. Besides a Java simulation, the page contains additional references including to a recent paper by Greg
Brockman that brought him the 6th place at the Intel Science Talent Search
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The common claim that Ducci iterations converge to a zero sequence iff the length of the sequence is a power of 2 is incorrect. The correct formulation is "Ducci iterations converge for any starting sequence iff the length of the sequence is a power of 2. For example, for N = 6, one gets to 0s
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Thanks for the feedback. I didn't quite understand the difference between the common claim and the correct formulation. Are you saying that there are sequences that converge but not to zeros? As your link states, it's possible to converge to e.g 333333 but this gives 000000 in the next step.
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Florian Breuer clearly states: 'Every Ducci sequence of n-tuples of integers vanishes if and only if n is a power of two.' Other sources are also clear on the if and only if part, but we've got an example of a 6-tuple that vanishes. Where's the misunderstanding?
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It looks harder to read the sequences with the new format. If I was unfamiliar with the topic, I wouldn't know at first if I should read them line by line or column by column. Do others have the same impression or are they better in a multi-column format?
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Yes those arrows could help, though it looks a bit cluttered. But it might work better on the simpler sequences. If the big 4-tuple follows as the last example, it would be enough to write the starting numbers and simply state that it ends after 24
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digits." This is incorrect. Again, for some starting sequences the iterations will converge to 0 even for, say, N = 6, as the three examples 101010, 123210, or 121210 show.
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I've added arrows to the examples, and shortened the first one by only showing first and last few iterations.
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397:{\displaystyle (0,653,1854,4063)\rightarrow (653,1201,2209,4063)\rightarrow (548,1008,1854,3410)\rightarrow }
505:{\displaystyle (460,846,1556,2862)\rightarrow (386,710,1306,2402)\rightarrow (324,596,1096,2016)\cdots }
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Aha! yes 101010 do indeed converge to zeros. I've changed the text accordingly.
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