Problems & Puzzles: Puzzles

Puzzle 661 Carpet palindrome Persian art style Regarding the palindrome sandwiched by primes solution given to the puzzle 659 by Giovanni Resta: 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8 8 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 8 8 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 8 8 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 8 8 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 8 8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8 8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8 8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8 8 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 8 8 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 8 8 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 8 8 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 8 8 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 8 8 1 0 0 0 0 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 1 1 1 0 0 0 0 1 8 8 1 0 0 0 0 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 1 1 1 0 0 0 0 1 8 8 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 8 8 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 8 8 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 8 8 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 8 8 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 8 8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8 8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8 8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8 8 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 8 8 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 8 8 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 8 8 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 8 8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 with a kind of cross of zeros in the center over a bottom of ones and a frame of eights. Q. Find other more complex or simple but equally beauty/aesthetical solutions (exactly 30x30 digits) _____________ Note: You are able to explore other arithmetical conditions, not necessarily a palindrome sandwiched by primes.

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Contributions came from W. Edwin Clark, Emmanuel Vantieghem & Jan van Delden

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Edwin wrote

Back in 2003 I started a thread in the math-fun mailing list concerning what 
I called "Contact primes".  The reason for the term Contact is that Carl Sagan
wrote a novel called "Contact". In the novel he mentioned that somewhere in the
base 11 expansion of Pi one might find a sequence of 0s and 1s which when arranged 
in a matrix would form a circle. The implication was that this was a message 
placed there by the creators of the universe. See the last  paragraph of the plot
summary in http://en.wikipedia.org/wiki/Contact_%28novel%29

Also I rasised the idea on the Yahoo group primenumbers.  The discussion
can be found here: Contact Primes (messages in reverse order):  The idea is
to take a prime number and write it in binary. If the length of the number in binary
is a product nm, then write it in n rows and m columns and see the picture so 
formed (if any). 

Here's one example I found that is a little better than my first try:

 

If one converts the prime:

14120070003959365676883255292421778218144809818877732767398\
181721742371443508001854208548588362622544357892897188147925\
7136218675419345650954881192658725502209920390914577986254098\
0335509923043069239333581971669175500801

to binary, arranges the bits  into a 27 x 27 matrix, and then replaces
each 1 by * and each 0 by a blank one obtains the following pattern:

(with the proper font it is a circle inside a square with * at the four corners

of the square)

 *                                                   *
                           *                          
                   *               *                  
             *                           *            
                                                      
                                                      
         *                                   *        
                                                      
                                                      
     *                                           *    
                                                      
                                                      
   *                                               *  
                                                      
                                                      
                                                      
   *                                               *  
                                                      
                                                      
       *                                       *      
                                                      
                                                      
           *                               *          
                                                      
                 *                   *                
                       *       *                      
 *                                                   *

 

The most amazing example was the following found by Renaud Lifchitz
which is based on the 378 digit prime N =
3101805506546616280631635616871932001139678275905519848096343683700\
1064413299689909982581233599912690445747213000469488219983668395902\
39921127145134275349012365903656875021343855075758594511157511105331\
56859616185181236406115162160900508184097411406718499740643043787761\
84058000031904561144039147499937678209426297296078944799076442958087\
6218511506376879633780612363524021157887
 

If we write its base-2 digits in a 19x66 matrix, replace each 1 by X and each 0 by _
we get the following. Which justifies calling this a self-proclaimed prime. (It spells
out "a prime" in X's with a border of X's.




I hope these come through so that they are clearly visible. If not perhaps I can send them
as gif files.

John McCarthy suggested that perhaps one could draw any picture one wished with 0's and 1's and then by jiggling a few bits so  as not  to spoil the picture too much,  eventually
one should get a prime. Although Renaud Lifchitz said he didn't  find his prime that way.

 

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Emmanuel wrote

This is my first solution :

 

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

8 8 0 1 0 1 1 0 1 1 0 1 1 8 8 8 8 1 1 0 1 1 0 1 1 0 1 0 8 8

8 0 8 1 1 0 1 1 1 1 1 1 8 1 8 8 1 8 1 1 1 1 1 1 0 1 1 8 0 8

8 1 1 8 0 1 1 1 1 1 1 8 1 1 8 8 1 1 8 1 1 1 1 1 1 0 8 1 1 8

8 0 1 0 8 1 1 1 1 1 8 1 1 0 8 8 0 1 1 8 1 1 1 1 1 8 0 1 0 8

8 1 0 1 1 8 1 1 1 8 1 1 1 1 8 8 1 1 1 1 8 1 1 1 8 1 1 0 1 8

8 1 1 1 1 1 8 1 8 1 1 1 1 1 8 8 1 1 1 1 1 8 1 8 1 1 1 1 1 8

8 0 1 1 1 1 1 8 1 1 1 1 1 0 8 8 0 1 1 1 1 1 8 1 1 1 1 1 0 8

8 1 1 1 1 1 8 1 8 1 1 1 1 1 8 8 1 1 1 1 1 8 1 8 1 1 1 1 1 8

8 1 1 1 1 8 1 1 1 8 1 1 0 1 8 8 1 0 1 1 8 1 1 1 8 1 1 1 1 8

8 0 1 1 8 1 1 1 1 1 8 0 1 0 8 8 0 1 0 8 1 1 1 1 1 8 1 1 0 8

8 1 1 8 1 1 1 1 1 1 0 8 1 1 8 8 1 1 8 0 1 1 1 1 1 1 8 1 1 8

8 1 8 1 1 1 1 1 1 0 1 1 8 0 8 8 0 8 1 1 0 1 1 1 1 1 1 8 1 8

8 8 1 1 0 1 1 0 1 1 0 1 0 8 8 8 8 0 1 0 1 1 0 1 1 0 1 1 8 8

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

8 8 1 1 0 1 1 0 1 1 0 1 0 8 8 8 8 0 1 0 1 1 0 1 1 0 1 1 8 8

8 1 8 1 1 1 1 1 1 0 1 1 8 0 8 8 0 8 1 1 0 1 1 1 1 1 1 8 1 8

8 1 1 8 1 1 1 1 1 1 0 8 1 1 8 8 1 1 8 0 1 1 1 1 1 1 8 1 1 8

8 0 1 1 8 1 1 1 1 1 8 0 1 0 8 8 0 1 0 8 1 1 1 1 1 8 1 1 0 8

8 1 1 1 1 8 1 1 1 8 1 1 0 1 8 8 1 0 1 1 8 1 1 1 8 1 1 1 1 8

8 1 1 1 1 1 8 1 8 1 1 1 1 1 8 8 1 1 1 1 1 8 1 8 1 1 1 1 1 8

8 0 1 1 1 1 1 8 1 1 1 1 1 0 8 8 0 1 1 1 1 1 8 1 1 1 1 1 0 8

8 1 1 1 1 1 8 1 8 1 1 1 1 1 8 8 1 1 1 1 1 8 1 8 1 1 1 1 1 8

8 1 0 1 1 8 1 1 1 8 1 1 1 1 8 8 1 1 1 1 8 1 1 1 8 1 1 0 1 8

8 0 1 0 8 1 1 1 1 1 8 1 1 0 8 8 0 1 1 8 1 1 1 1 1 8 0 1 0 8

8 1 1 8 0 1 1 1 1 1 1 8 1 1 8 8 1 1 8 1 1 1 1 1 1 0 8 1 1 8

8 0 8 1 1 0 1 1 1 1 1 1 8 1 8 8 1 8 1 1 1 1 1 1 0 1 1 8 0 8

8 8 0 1 0 1 1 0 1 1 0 1 1 8 8 8 8 1 1 0 1 1 0 1 1 0 1 0 8 8

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

 

I found a few other ones, but I could not increase symmetry.

 

It might be a littlbit naughty, but I could not resist the temptation to search for a titanic example.

Working in a 32x32 grid gave me a supplementary symmetry.  There are four identical 16x16 grids in which there are four symmetry axes :

 

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

8 8 0 1 1 0 1 1 1 1 0 1 1 0 8 8 8 8 0 1 1 0 1 1 1 1 0 1 1 0 8 8

8 0 8 0 0 1 1 0 0 1 1 0 0 8 0 8 8 0 8 0 0 1 1 0 0 1 1 0 0 8 0 8

8 1 0 8 1 1 1 1 1 1 1 1 8 0 1 8 8 1 0 8 1 1 1 1 1 1 1 1 8 0 1 8

8 1 0 1 8 1 1 1 1 1 1 8 1 0 1 8 8 1 0 1 8 1 1 1 1 1 1 8 1 0 1 8

8 0 1 1 1 8 1 0 0 1 8 1 1 1 0 8 8 0 1 1 1 8 1 0 0 1 8 1 1 1 0 8

8 1 1 1 1 1 8 0 0 8 1 1 1 1 1 8 8 1 1 1 1 1 8 0 0 8 1 1 1 1 1 8

8 1 0 1 1 0 0 8 8 0 0 1 1 0 1 8 8 1 0 1 1 0 0 8 8 0 0 1 1 0 1 8

8 1 0 1 1 0 0 8 8 0 0 1 1 0 1 8 8 1 0 1 1 0 0 8 8 0 0 1 1 0 1 8

8 1 1 1 1 1 8 0 0 8 1 1 1 1 1 8 8 1 1 1 1 1 8 0 0 8 1 1 1 1 1 8

8 0 1 1 1 8 1 0 0 1 8 1 1 1 0 8 8 0 1 1 1 8 1 0 0 1 8 1 1 1 0 8

8 1 0 1 8 1 1 1 1 1 1 8 1 0 1 8 8 1 0 1 8 1 1 1 1 1 1 8 1 0 1 8

8 1 0 8 1 1 1 1 1 1 1 1 8 0 1 8 8 1 0 8 1 1 1 1 1 1 1 1 8 0 1 8

8 0 8 0 0 1 1 0 0 1 1 0 0 8 0 8 8 0 8 0 0 1 1 0 0 1 1 0 0 8 0 8

8 8 0 1 1 0 1 1 1 1 0 1 1 0 8 8 8 8 0 1 1 0 1 1 1 1 0 1 1 0 8 8

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

8 8 0 1 1 0 1 1 1 1 0 1 1 0 8 8 8 8 0 1 1 0 1 1 1 1 0 1 1 0 8 8

8 0 8 0 0 1 1 0 0 1 1 0 0 8 0 8 8 0 8 0 0 1 1 0 0 1 1 0 0 8 0 8

8 1 0 8 1 1 1 1 1 1 1 1 8 0 1 8 8 1 0 8 1 1 1 1 1 1 1 1 8 0 1 8

8 1 0 1 8 1 1 1 1 1 1 8 1 0 1 8 8 1 0 1 8 1 1 1 1 1 1 8 1 0 1 8

8 0 1 1 1 8 1 0 0 1 8 1 1 1 0 8 8 0 1 1 1 8 1 0 0 1 8 1 1 1 0 8

8 1 1 1 1 1 8 0 0 8 1 1 1 1 1 8 8 1 1 1 1 1 8 0 0 8 1 1 1 1 1 8

8 1 0 1 1 0 0 8 8 0 0 1 1 0 1 8 8 1 0 1 1 0 0 8 8 0 0 1 1 0 1 8

8 1 0 1 1 0 0 8 8 0 0 1 1 0 1 8 8 1 0 1 1 0 0 8 8 0 0 1 1 0 1 8

8 1 1 1 1 1 8 0 0 8 1 1 1 1 1 8 8 1 1 1 1 1 8 0 0 8 1 1 1 1 1 8

8 0 1 1 1 8 1 0 0 1 8 1 1 1 0 8 8 0 1 1 1 8 1 0 0 1 8 1 1 1 0 8

8 1 0 1 8 1 1 1 1 1 1 8 1 0 1 8 8 1 0 1 8 1 1 1 1 1 1 8 1 0 1 8

8 1 0 8 1 1 1 1 1 1 1 1 8 0 1 8 8 1 0 8 1 1 1 1 1 1 1 1 8 0 1 8

8 0 8 0 0 1 1 0 0 1 1 0 0 8 0 8 8 0 8 0 0 1 1 0 0 1 1 0 0 8 0 8

8 8 0 1 1 0 1 1 1 1 0 1 1 0 8 8 8 8 0 1 1 0 1 1 1 1 0 1 1 0 8 8

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

 

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Jan wrote

I found the following prime Sierpinsky Carpet (27x27).
It’s not 30x30, but nice enough I think.

 

111111111111111111111111111

131131131131131131131131131

111111111111111111111111111

111555111111555111111555111

131555131131555131131555131

111555111111555111111555111

111111111111111111111111111

131131131131131131131131131

111111111111111111111111111

111111111444444444111111111

131131131444444444131131131

111111111444444444111111111

111555111444444444111555111

131555131444444444131555131

111555111444444444111555111

111111111444444444111111111

131131131444444444131131131

111111111444444444111111111

111111111111111111111111111

131131131131131131131131131

111111111111111111111111111

111555111111555111111555111

131555131131555131131555131

111555111111555111111555111

111111111111111111111111111

131131131131131131131131131

111111111111111111111111111

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On July 29,2017, Jan van Delden sent the following 4 graphs:

Your example (0=blue,1=white,8=red):

The first solution by Emmanuel:

 

The second solution by Emmanuel:

 

 

My Sierpinski Carpet (1=red,3=yellow,5=green,4=white):

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