The problem is intractable unless you can satisfy at least one of the following conditions:
- You have a list of words that is guaranteed to contain the mystery word
- The mystery word cannot have duplicate letters (violates assumption #2 above)
- The number of letters in common is the exact number of letters, duplicates included. (violates assumption #3 above)
Either you have a list and you brute force it like jdporter's solution or you have to be able to generate words. To generate those words, you have to know how many of each letter there are and you cannot know that without either conditions #2 or #3 satisfied.
Once you give me one of those, then I can solve your puzzle. :-)
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