Problems & Puzzles: Puzzles

 Puzzle 894. The first N integers X type arranged to form a Palindrome This is a follow-up to Puzzle 891. Here we pose the more general question: find palindromes concatenating an arrangement of the first N integers type X, for several X "values" as the following ones, "odd", "even", "perfect squares", and so on. Example for X="odd" by CR: If X="odd", the first palindrome solution comes for N=9, {1, 3,..., 17} 1.13.5.17.9.7.15.3.11 => 1135179715311 (palindrome, 13 digits, SOD=45). But if you are looking palprimes for X="odd", then the first solution comes for N=10, {1, 3,..., 19} 1.19.3.15.7.17.5.13.9.11 => 119315717513911 (palprime, 15 digits, SOD=55)  Q1. Find the smallest palindrome and the smallest palprime for X="even", "perfect square", "Fibonacci", etc. integers.

Contributions came from Emmanuel Vantieghem and Dmitry Kamenetsky, on set 2017

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Emmanuel wrote:

The first  X  I found that gave palindromic concatenations is the set of even composites : 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, ...
There are  24  concatentations, the smallest one is :
4&10&22&16&8&18&6&12&20&14   ->   41022168186122014
the biggest one is :
8&16&4&12&20&10&22&14&6&18   ->  81641220102214618
Obviously, no palprime is possible !

If you allow for  X  the set of powers  1, b, b^2, b^3, ...
then there are trivial concatenations that are palindromic.  For instance, take for  b  any number such that  1&b  is palindromic,as :
1&221, 1&2621, ...
Some palprimes :
1&31, 1&51, ... ,1&9391, ...

If  b = 15, we get the concatenation   b&(b^2)&1 = 152251.

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Dmitry wrote:

I couldn't find many interesting solutions. If we allow 0 then here is a solution for the first 9 even numbers:

0.12.4.16.|8|.6.14.2.10

If we use non-prime numbers, then here is a solution for the first 61:

18.57.54.51.10.39.44.80.46.14.78.36.77.25.52.48.64.20.66.58.28.9.63.32.65.34.12.6.22.70.|5|0.72.26.21.4.35.62.33.69.
82.8.56.60.24.68.42.55.27.76.38.74.16.40.84.49.30.1.15.45.75.81

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