Solution to self-referential enumeration problem.  

There are exactly four A's, three C's, two D's,
twenty nine E's, seven F's,
five H's, eleven I's, three L's,
sixteen N's, six O's, nine
R's, twenty seven S's, fourteen T's,
five U's, six V's, four W's, six X's, and
four Y's in this sentence.


(This problem is posed on the index page of Daniel W. VanArsdale.)

I sent this problem (without the drawing) and solution to the American Mathematical Monthly problem editor on 10 October 1972. It was not accepted for publication. Probably the first published example appeared in the "Metamathematical Themas" column in the January 1982 Scientific American. It was constructed by Lee Sallows, who named such a self-enumerating sentence an "autogram".
Douglas R. Hofstadter reproduced this column in his 1985 book Mathematical Themas (Chapters 1 and 2). Wikipedia has an entry for "autogram" that gives an example similar to the above containing 167 letters including "one z." The above contains 161 letters,

The method used to find the above solution was simply to check many tries with a computer program, changing one guess after each unsuccessful try and avoiding getting hung up in a cycle. The above was the only solution found in about 70,000 tries. There likely are other solutions. If you find another with fewer letters, if you wish I will place it here and give you credit.

Daniel W. VanArsdale

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