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Sock Sorting Space (Posted on 2023-09-30) Difficulty: 3 of 5
Imagine sorting a basket of laundry consisting of n pairs of socks, each of a different color. Randomly pulling out socks, one at a time, you will either put each on the sorting surface or match it up with its mate to make a pair and put them away in the sock drawer.

The maximum number of socks in the sorting space can then be any number from 1 to n. For example with 3 pairs, pulling them in the order AABBCC will only require one space, ABACCB requires two, and ABCABC will require three spaces.

Find the expected number of sorting spaces required in terms of n.

No Solution Yet Submitted by Jer    
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Some Thoughts random trials of one million each n Comment 2 of 2 |
for n=2:17
  alpha='aabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzz';
  socks=alpha(1:2*n);
  totspace=0;
  for trial=1:1000000
    srce=socks; space=''; mx=0;
    while ~isempty(srce)
      r=randi(length(srce));
      c=srce(r);
      srce(r)='';
      sf=strfind(space,c);
      if ~isempty(sf)
        space(sf)='';
      else
        space=[space c];
      end
      if length(space)>mx
        mx=length(space);
      end
    end
    totspace=totspace+mx;
  end
  fprintf('%4d %10d %10d %10.8f\n',n,totspace,trial,totspace/trial)
end

finds for n= 2 to 17

       total of             expected
   n    needed     trials     value
        spaces
   2    1666101    1000000  1.666101
   3    2332919    1000000  2.332919
   4    2961024    1000000  2.961024
   5    3574332    1000000  3.574332
   6    4174017    1000000  4.174017
   7    4762736    1000000  4.762736
   8    5343320    1000000  5.343320
   9    5919300    1000000  5.919300
  10    6488262    1000000  6.488262
  11    7054210    1000000  7.054210
  12    7620058    1000000  7.620058
  13    8177731    1000000  8.177731
  14    8732739    1000000  8.732739
  15    9289301    1000000  9.289301
  16    9836566    1000000  9.836566
  17   10387656    1000000 10.387656

  Posted by Charlie on 2023-09-30 12:59:31
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