Instead of constructing an equation at random and then converting it into a puzzle, one can also just construct a random puzzle (but don't let yourself use more than 10 different letters). Here are your odds of success with that route: 

Digits       Number of puzzles
  1                     31,200 with unique solutions
                        37,102 with no solutions
                       406,250 with 4..32 solutions
                       474,551 total puzzles

  2                 10,795,200 with unique solutions
                 1,990,652,352 with no solutions
                 6,339,278,400 with 3..476 solutions
                 8,340,725,952 total puzzles

  3             33,433,717,226 with unique solutions
             6,746,392,175,276 with no solutions
           139,816,773,439,850 with 2..1200 solutions
           146,596,599,332,352 total puzzles


  4      1,974,825,396,897,600 with unique solutions
       240,014,957,704,387,2?? with no solutions
     1,147,855,057,502,310,0?? with 2..3840 solutions
     1,389,844,840,603,594,8?? total puzzles

     2,576,581,829,865,418,742 total random letter choices
     1,186,736,989,261,824,0?? with over 10 different letters

Digs   Uniq     None     Dupl    Max
  1    6.57%    7.82%   85.61%    32
  2    0.13%   23.87%   76.00%   476
  3    0.02%    4.60%   95.38%  1200
  4    0.14%   17.27%   82.59%  3840

[download]

And I suspect the odds just get worse after this. (And note that the numbers have gotten too complicated for both Perl and me toward the end there so don't use these for life-and-death calculations.)

In reply to Re^4: Golfing cryptosums by Anonymous Monk
in thread Golfing cryptosums by ELISHEVA

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