Here are the solutions to last week's problems:
Benko H#6-
1 e4 e5 2 Qh5 Nc6 3 g4 d6 4 g5 Kd7 5 Bh3+ f5 6 gxf6 e.p. #
Some transposition of moves is possible.
H#4-
1 f3 e6 2 Kf2 Qf6 3 Kg3 Qxf3+! 4 Kh4 Be7#
"This problem is almost impossibly difficult, because Qxf3+ is such a horrible move by normal chess standards; it is hard for a competent player to consciously consider it. We no longer know who composed this problem. There can be some disagreement as to whether it's flawed, because Black's first move can be either e6 or e5. In general, "cooked" problems are flawed, but here I find it natural to say that the problem has just one solution, with two trivially different variants. Qxf3+ is the real point of the composition."
The above quote, and the problems, come from the chapter, The Reviled Art by Stuart Rachels, from the book, Philosophy Looks At Chess. The problem by Benko is also in his book, Pal Benko: My Life, Games and Compositions by Pal Benko and Jeremy Silman. Stuart Rachels, when he was eleven, became the youngest chess master in American history, and was on the cover of Chess Life. There is an excellent section, in his chapter, on Chess Compositions, and I will just quote one relevant paragraph. "I cannot exaggerate how marvelous chess problems are. There are
thousands upon thousands of gloriously beautiful problems. They are awesome, fantastic, stunning. Yet, if chess players are ignored and shunned by American society, problemists are even more isolated and unappreciated--not only in America, but around the world. Just as American culture is indifferent or hostile to chess, chess players themselves are indifferent or hostile to chess compositions. Why is that?"
In the notes, there is a quote from Sam Loyd, "My theory of a key move was always to make it just the reverse of what a player in 999 cases out of 1000 would look for." Lasker's Chess Magazine, December 1904.
David Dana-Bashian mentioned a problem similar to one above, a helpmate in 4, in which you construct a game in which Black's fourth move is a discovered checkmate. One solution is 1 f3 e5 2 Kf2 h5 3 Kg3 h4+ 4 Kg4 d5#.

Coincidentally, the June issue of Chess Life had an article by GM Andy Soltis, Problematic Problems, which had some similar unorthodox problems.