And the second solution is better! After 2 Qxe6 Kf8, 3 Nf6 leads to mate in one! There are 2 mate threats (Qf7# and Nh7#) and black can only stop one of them. Or am I missing something?
Yes, you are missing uritbon's comment on the problem, 3... Bh6+.
What Xavier said. Unfortunately this problem is one of those "find the quickest" mates when there are plenty of quick mates available.
I don't think 3 ... Bh6+ can stop white. After 4 Rxh6 white still has way too many overwhelming threats (chiefly Qf7# and Qg8#). After 4 Rxh6...
4 ... Kg7 5 Rh7+ Kany 6 Qf7#
4... Ne5 5 Qg8#
4 ... exf6 5 Qf7#
4 ... Nxf6 5 Qf7#
So I was off by a couple of moves in terms of forced mate, but still... 3 Nf6 leads to a very quick mate. So the problem is still not well constructed. Not to mention 3 Nf4, which also leads to a very quick mate.
After 1... fxe6 it's a mate in 4 puzzle, how is it not well constructed? Because you prefer to deliver mate in 5 moves instead?
Yes I do. The only reason my move is a mate in 5 is because of 2 pointless delaying spite moves. Any person OTB would resign after my move. If one solution is superior to another because of one spite check, I believe that is a poorly designed problem.
Well, first of all 3 Nf4 mates just as quickly as 3 Qg8, so the problem is truly cooked.
Secondly, the Tactics Trainer is supposed to simulate real game conditions. In a real game, a forced 5-move mate would be rewarded equally well as a forced 4-move mate. Therefore I would expect Tactics Trainer to reward them equally as well.
Perhaps OTB you will lose on time before your last move is properly executed... though this would be a very unlikely, dramatic situation.
However, we are not addressing "real game conditions" here, but a chess problem, a puzzle. You feel you're good players even if you didn't find the quickest mate, fine!
But there's nothing wrong with the problem's solution. As shown below, 3. Nf3 does not mate as quickly as 3. Qg8+..., unless I missed something.
I looked at the Analysis & Source section and, as a matter of fact, there is an alternate solution (4. Nxe7+ Kf8 5. Nxg6#)! This is why the puzzle stops at 3. Qg8+, proving, as far as I'm concerned, that the problem is well designed.
Let's take a look at the cook, 3 Nf4!
And there is no other defense to the dual threat of Qf7# and Nxg6#. So 3 Nf4! ends in mate no later than move 5, just like the problem composer's solution. Therefore the problem is cooked by any definition, yours or mine. :)
As for chess problem vs. game simulation, it's part of a site feature called "Tactics Trainer," implying that users learn to become stronger players in real games--not better problem composers. So the game simulation aspect ought to be important, IMO.
Of course, there are others who prefer pure problems to real games, and that's a different way to enjoy chess.
In this form, for the second move, the problem asks white to find a mate in 3 when there are some other decisive continuations. Thus, there is room for improvement:
As a note, there are some few quizzes in TT which admit multiple solutions. However, the problems are improved continuously and those with multiple solutions are either corrected or deleted. The best thing you can do if you encounter such a situation is to leave a comment in the dedicated section of the problem and it's very likely that a moderator will see it and take action.
miturr_binesdurtee15 has shown why 3. Nf4 takes one extra move to deliver checkmate (4... d5).