MATLAB: Distinct sum of 3 variable which ranges between 1-255

algorithmmathematicsMATLABmatrix array

Hi, I am trying to retrieve array of sum of 3 numbers (x,y,z) each has a range of 1-255, in such a way that each sum is different from others. I have coded but I couldn't get 100% distinct sum, below result gives Average = 255, Max = 255 , MIN = (254 but I want 255 every time if possible). So each variable's value can be decoded back form the distinct sum. For example sum = 3 is only possible when we have (1+1+1) but for Sum = 4 we have 3 different possible and then it is not decodable; (1+1+2),(1+2+1),(2+1+1), sum = 4. I would be happy if anyone of you can debug this problem. Thank you.
%Updated. 
%Input and Expected Output%
% Input        Output
% [1,1,2]      sum=>4
% [1,2,1]      sum=>[different than above]
% [2,1,1]      sum=>[distinct from others]
% [3,1,1]      sum=>5
% ...          ...
%Code%
clear;clc;
 for n = 1:1000 %using Iteration to have 1000 different results that are stored in Difference(n,:) which gives Uniqueness values of each result.
    Array = zeros(255,5);
    Array(:,1:3) = randi(255,255,3);
    for k = 1:length(Array);
        Array(k,4) = sum(Array(k,1:3));
    end
    for l = 1:length(Array);
        firstVal   = Array(l,1)*0.7;
        secondVal  = Array(l,2)*0.9+firstVal;
        thirdVal   = Array(l,3)*1.4*secondVal;
        Array(l,5) = sum([firstVal,secondVal,thirdVal]);
    end
%253Avg
% first   = Array(l,1)*0.7;
% second  = Array(l,2)*0.9 + first;
% third   = Array(l,3)*1.4 + second;
    oldunique_Length = length(unique(Array(:,4)));
    newunique_Length = length(unique(Array(:,5)));
    Difference(n,:) = [oldunique_Length,newunique_Length];
end
%average no. of distinct sum = 255/255, Max value is 255 and MIN value is 254. It is still not 100% because I am getting 254 as a MIN. Everytime I should be able to get 255 (all rows must be different from others)
["OLD UniqueIds (AVG,MAX,MIN)",round(mean(Difference(:,1))),max(Difference(:,1)),min(Difference(:,1));"NEW UniqueIds (AVG,MAX,MIN)",round(mean(Difference(:,2))),max(Difference(:,2)),min(Difference(:,2))]

Best Answer

As you recognize yourself, if you simply form the sum of three integers, you cannot then recover the original values from only that sum. Sufficient information is lost to recover the original components. Thus, given
2 + 1 + 1 == 1 + 2 + 1 == 1 + 1 + 2 == 4
if the sum was 4, then it is never possible to identify which of the three options caused that sum. Sorry, but that is simple fact. You cannot escape the laws of mathematics.
However, you CAN do something that allows recovery, if you use a proper encoding. Think of it as a base transformation, working in base 255. Given three integers, each in the range 1 - 255. I'll call them K, N, & M. Form an encoding as:
KNM = 255^2*(K-1) + 255*(N-1) + (M-1)
Thus for any values of K, N, M, we will get a distinct result, treating those numbers as effectively "digits" in base 255, then mapping the result to base 10. Try it out.
KNMencode = @(K,N,M) 255^2*(K-1) + 255*(N-1) + (M-1);
KNMencode(2,1,1)
ans =
       65025
       
KNMencode(1,2,1)
ans =
   255
   
KNMencode(1,1,2)
ans =
     1
So we now get a distinct and unique result for any input combination in that range.
The reverse is true too. Now you can decode any possible integer value in the interval [0,16451834], back into the original set that created the number.
KNMencode(1,1,1)
ans =
     0
     
KNMencode(255,255,255)
ans =
    16451834
Again, this is just a case of a base conversion from base 10 back into base 255. Lets try it out.
KNMdecode = @(E) [mod(floor(E/255^2),255) + 1,mod(floor(E/255),255) + 1,mod(E,255) + 1];
KNMencode(1,2,3)
ans =
   257
KNMdecode(257)
ans =
     1     2     3
So any triple of integers (K,N,M) will result in a unique encoded value as needed, that can then be decoded, recovering the original values.
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