Langford's Problem

Arrange copies of the $n$ digits 1, ..., $n$ such that there is one digit between the 1s, two digits between the 2s, etc. For example, the $n=3$ solution is 312132 and the $n=4$ solution is 41312432. Solutions exist only if $n\equiv 0,3\ \left({{\rm mod\ } {4}}\right)$. The number of solutions for $n=3$, 4, 5, ... are 1, 1, 0, 0, 26, 150, 0, 0, 17792, 108144, ... (Sloane's A014552).

References

Gardner, M. Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, pp. 70 and 77-78, 1978.

Sloane, N. J. A. Sequence A014552 in ``The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.