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Palindromes...semordnilaP


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I have a fascination with palindromes. Are there any intrinsic mathematical properties for them aside just from the visual symetry?
Hmmm. What makes you ask the question? And are you just referring to single word palindromes (e.g., "kayak") or are you including full sentences of palindromic words (e.g., "Did dad eye Ada?") or full sentences that are single palindrome as well (e.g., "Rise to vote sir!")?
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for numerical palindromes as 11, 272, one could write downsymmetry properties, however these properties will be dependent on the representation: a palindrome in decimal numbers won't be a palindrome in hexadecimal or binairy numbers. A lot of the beauty will be lost in my opinion...

 

Bo

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Obviously,

a palindrome in decimal numbers won't be a palindrome in hexadecimal or binairy numbers.
Changing the base would replace those 10s in Bumab's argument, which would however become much more complicated as there are more digits! I'm not at all sure that argument could lead to an algorithm that enumerates them exhaustively in increasing sequence.

 

In general, for n digits in base b, one can write the condition as a_i = a_(n - i) for each i and perhaps work something out.

 

One might even find something beautiful! ;)

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I'm not at all sure that argument could lead to an algorithm that enumerates them exhaustively in increasing sequence.
I wasn't sure that any algorithm could be designed that enumerates them exhaustively in increasing sequence but, after work, I had a look at it and it is possible! It's easy to see, once you think about it.
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Speaking about palindromes, you have the 196-algorithm that produce numerical palindromes from almost any number.

 

0 Start with a number with 2 or more integers, call it A

1. Reverse A, call it B

2. Do A+B, call it C, print C

3. let A=C

4. Go to 1

 

This algoritm will produce a sequence of numbers, of which several are palindromes. It may take many iterations to arrive at the first palindrome. Some numbers does not seem to generate palindromes at all, the first such number being 196.

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To get them all, and only them, in a given base b... just start counting!

 

For each number, having n digits ABCDEF.... consider it followed by all digits in reverse order, and also consider it followed by all except the unit digit in reverse order.

 

If you want to have them in increasing order then, for each n, you first do the counting and concatenate all but the unit digit, then repeat the counting and concatenate all digits.

 

This can be translated into sums of powers of b and hence into an arithmetical algorithm. In stretches of b consecutive results, the increment will be of the type 1000...0 or of the type 1000...0.

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___If I may interject a comment on the nature of the beast, ie. why we notice palindromes at all. It is largely a visual effect with little to do with number in the strictest sense. We only construct the algorthms after the fact; the fact of 'seeing' a pattern.

___Short number palindromes have auditory meaning, however longer ones loose meaning as the memory ability fades.

___What I mean to say is that a palindrome is no more or less than the tumbling glass in a kaleidoscope. :)

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Hi Good morning. I am a Doctorate in music. I have an excellent article on topic Palindrome. It covers 7th century poems in palindromic format (11 couplets).Added to it I have covered Thevaram. Thiruppugazh. Divya Prabandan. Maths. Physics.Chemistry. Archeology.Music and so on.I am Dr.S.Subbulakshmi.

I am interested in publishing the above article.

Thanks

 

 

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