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<p>
  Wonderful post!  I am preparing a video on Langford pairings and
  your proof was incredibly helpful and written in great detail.  Many
  articles come to the necessary condition and just assume sufficiency
  which is wrong.  Yours is the only article that proves both the
  necessity and the sufficiency of the condition.  The construction of
  the sequence in your post is wonderful.
</p>
<p>
  By the way, I be