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Atbash Self Referential Number (Posted on 2024-02-27) Difficulty: 2 of 5
One type of self referential number is one in which each digit represents the number of occurrences of that digit in the number itself. For exmaple 21200 has 2 zeros, 1 one, 2 twos, 0 threes, and 0 fours.

The atbash substitution cipher has the reversed alphabet as the encoding rule: A means Z, B means Y, C means X, etc. For decimal digits, the atbash equivalent is simply 9 minus the digit.

Find a 10 digit atbash self referential number such that the leftmost digit is 9 minus the number of zeros, the next digit is 9 minus the number of ones, etc.

See The Solution Submitted by Larry    
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Solution computer solution | Comment 2 of 4 |
Try all possible counts of digits adding to 10.
Form the descriptive number for each.
Check that number to see if it has the given stats.

clc,clearvars
breaks=combinator(19,9,'c');
for i=1:length(breaks)
  bounds=[0,breaks(i,:),20];
  n=''; counts=[];
  for j=0:9
    counts(j+1)=bounds(j+2)-bounds(j+1)-1;
    n=[n num2str(9-counts(j+1))];
  end
  good=true;
  for j=0:9
    counts2(j+1)=length(strfind(n,num2str(j)));
    if counts2(j+1)~= counts(j+1)
      good=false;
    end
  end
  if good==true
    disp(counts)
    disp(n)
    disp(counts2)
    disp(' ')
  end
  if good
    disp(n)
  end
end

checks every possible set of statistics for 10 digits and finds one set that meets the criterion of self reference:

count of each digit:

digit     0     1     2     3     4     5     6     7     8     9

count     0     0     0     1     0     0     0     1     2     6
     
This would be encoded as 9998999873.

Doing the stats on this number we do get the given stats:

     0     0     0     1     0     0     0     1     2     6
 
The answer is 9998999873.


  Posted by Charlie on 2024-02-27 08:33:36
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