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# Here's a mind-boggler

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A positive whole number that is the same when reading from right to left and left to right is called a palindromic number. If all of them were put into a sequence that is from small to big:1,2...11,22...,101,111...,10001,10101,.....

Then which term is the number 78987 in this sequence?  For example : 1 is the 1st term,2 is the second....etc

yomyhomies  Oct 22, 2017
#2
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Starting with 0,  78987  is the 889th palindrome

[BTW...889 is known as the palindrome's rank ]

Starting with 1, it is the 888th palindrome

This can be verified with the calculator here :

http://rhyscitlema.com/algorithms/generating-palindromic-numbers/

[ The calculator is about half-way down the page.....there are also other intems of interest related to palindromes on this page ]

A procedure is desribed for finding the nth palindrome, but I don't see one that tells us how to determine the rank of any particular palindrome  [ although I did not look at the page with a fine tooth comb]

Maybe heureka knows of a proceedure to produce this???

CPhill  Oct 22, 2017
#3
+20024
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which term is the number 78987 in this sequence?

Here 0 is the 1st term:

The position of a palindrome within the sequence can be determined almost without calculation:

If the palindrome has an even number of digits,

prepend a 1 to the front half of the palindrome's digits.

Examples: 98766789=a(19876)

If the number of digits is odd, prepend the value of front digit + 1 to the digits from position 2 ... central digit.

Examples: 515=a(61), 8206028=a(9206), 9230329=a(10230).

which term is the number 78987 in this sequence?

The number of digits is 5 is odd, prepend the value of front digit + 1 to the digits from position 2 ... central digit.

$$\begin{array}{|rrrrll|} \hline & & & \Rsh & & \text{ until central digit} \\ &7 & 8 & 9 & 8 & 7 \\ &| & | & | \\ &+1 & | & | \\ &\downarrow & \downarrow & \downarrow \\ \text{term is } & \color{red}8 &\color{red}8 &\color{red} 9 \\ \hline \end{array}$$

Starting with 1, it is the 888th palindrome

heureka  Oct 23, 2017
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Wow!!!!....thanks, heureka.....that's almost too easy....!!!

CPhill  Oct 23, 2017