In a Russian dictionary, you'd look up головоломка

Quotes from this article:

Russell Wayne Wagner, the convicted murderer whose ashes were deposited last month in Arlington National Cemetery, probably will not be going anywhere soon. But the controversy surrounding his final resting place might make it harder for others convicted of similar crimes to be buried there.

Wagner, 52, died in February while serving two life sentences for the 1994 murder of Daniel Davis, 84, and Wilda Davis, 80, in Hagerstown, Md. After their son objected last week to his parents' killer being interred in Arlington, the Army and the Senate Veterans Affairs Committee said they will take a deeper look at the issue.
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"He served his country honorably in uniform. What he failed to do was to serve society honorably as a citizen," (Joe Davis, a spokesman for the Veterans of Foreign Wars of the United States) said of Wagner, adding that the two sides posed a problem.

"If you looked up 'conundrum' in the dictionary," he said, "this should be the example for one."

Conundrum
noun
1. A riddle in which a fanciful question is answered by a pun.
2. A paradoxical, insoluble, or difficult problem; a dilemma.