On numerous occasions during the Second World War, members of the German high command had reason to believe that the allies knew the contents of some of their most secret communications. Naturally, the Nazi leadership was most eager to locate and eliminate this dangerous leak. They were convinced that the problem was one of treachery. The one thing they did not suspect was the simple truth: the British were able to systematically decipher their secret codes. These codes were based on a special machine, the “Enigma,” which the German experts were convinced produced coded messages that were entirely secure. In fact, a young English mathematician, Alan Turing, had designed a special machine for the purpose of decoding messages enciphered using the Enigma. This is not the appropriate place to speculate on the extent to which the course of history might have been different without Turing’s ingenious device, but it can hardly be doubted that it played an extremely important role.


Word Problem Computing Procedure Input String Punctuation Mark Post Word 
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Suggestions for Further Reading


  1. Chaitin, Gregory. Randomness and mathematical proof. Scientific American 232 (May 1975) 47–52.CrossRefGoogle Scholar
  2. Davis, Martin and Hersh, Reuben. Hilbert’s 10th problem. Scientific American 229 (November 1973) 84–91.CrossRefGoogle Scholar
  3. Knuth, Donald E. Algorithms. Scientific American 236 (April 1977) 63–80, 148.CrossRefGoogle Scholar
  4. Knuth, Donald E. Mathematics and computer science: coping with finiteness. Science 194 (December 17, 1976 ) 1235–1242.CrossRefMATHMathSciNetGoogle Scholar
  5. Turing, Sara. Alan M. Turing. W. Heifer, Cambridge, 1959.Google Scholar
  6. Wang, Hao. Games, logic and computers. Scientific American 213 (November 1965) 98–106.Google Scholar


  1. Davis, Martin. Computability and Unsolvability. McGraw-Hill, Manchester, 1958.MATHGoogle Scholar
  2. Davis, Martin. Hilbert’s tenth problem is unsolvable. American Mathematical Monthly 80 (March 1973) 233–269.CrossRefMATHMathSciNetGoogle Scholar
  3. Davis, Martin. The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions. Raven Pr, New York, 1965.Google Scholar
  4. Davis, Martin. Unsolvable problems. In Handbook of Mathematical Logic, by Jon Barwise (Ed.). North-Holland, Leyden, 1977.Google Scholar
  5. Minsky, Marvin. Computation: Finite and Infinite Machines. Prentice-Hall, Englewood Cliffs, 1967.MATHGoogle Scholar
  6. Rabin, Michael O. Complexity of computations. Comm. Assoc. Comp. Mach. 20 (1977) 625–633.MATHMathSciNetGoogle Scholar
  7. Trakhtenbrot, B.A. Algorithms and Automatic Computing Machines. D.C. Heath, Lexington, 1963.Google Scholar

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© Conference Board of the Mathematical Sciences 1978

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  • Martin Davis

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