NOUN (2)

1. any very difficult problem; insoluble in its own terms;

2. an intricate knot tied by Gordius, the king of Phrygia, and cut by the sword of Alexander the Great after he heard that whoever undid it would become ruler of Asia;

The Collaborative International Dictionary of English v.0.48:

Gordian \Gor"di*an\, a.
   1. Pertaining to Gordius, king of Phrygia, or to a knot tied
      by him; hence, intricate; complicated; inextricable.
      [1913 Webster]

   Gordian knot, an intricate knot tied by Gordius in the
      thong which connected the pole of the chariot with the
      yoke. An oracle having declared that he who should untie
      it should be master of Asia, Alexander the Great averted
      the ill omen of his inability to loosen it by cutting it
      with his sword. Hence, a Gordian knot is an inextricable
      difficulty; and to cut the Gordian knot is to remove a
      difficulty by bold and energetic measures.
      [1913 Webster]

   2. (Zool.) Pertaining to the Gordiacea.
      [1913 Webster]

WordNet (r) 3.0 (2006):

Gordian knot
    n 1: any very difficult problem; insoluble in its own terms
    2: an intricate knot tied by Gordius, the king of Phrygia, and
       cut by the sword of Alexander the Great after he heard that
       whoever undid it would become ruler of Asia

Moby Thesaurus II by Grady Ward, 1.0:

37 Moby Thesaurus words for "Gordian knot":
   Chinese puzzle, Rube Goldberg contraption, can of worms, complex,
   crux, dilemma, enigma, jungle, knot, knotty point, labyrinth, maze,
   meander, mesh, mess, node, nodus, nonplus, oxymoron, paradox,
   perplex, perplexity, pons asinorum, poser, puzzle, quandary, ravel,
   snafu, snake pit, snarl, tangle, tangled skein, teaser,
   vexed question, webwork, wheels within wheels, wilderness