I have a chess puzzle here:
[FEN "7k/8/8/4KPPP/8/8/8/8 - - - 0 0 "]
Can White force mate without promotion?
And if such exist, where can I find some more chess puzzles like this, which feature particular and restrictive directions?
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As Daniel indicated in a comment, the answer here is that, no, White cannot force checkmate without promoting at least one pawn. Enough toying with the position makes this pretty clear, intuitively. The only way it could possibly be done would be to keep the black king in the corner it's in, and while doing so, to then checkmate by advancing the pawns. The intuitive reason this isn't feasible is because the white king is the only option for keeping the black king from running to the other side of the board, but while he's doing that, there's not enough support behind the pawns to advance them for checkmate.
But I want to point out that if one finds that intuitive explanation unconvincing or unsatisfying, there is a more rigorous way to demonstrate that White cannot force checkmate from your position without promoting, and that is to do a little bit of retrograde analysis. (Incidentally, the retrograde analysis subfield of chess problems provides some of what you ask for at the end of your post, namely problems with a very particular sort of solution.) Basically, all I mean is that we should work backwards: with the given material on the board, there's only a small handful of positions that could possibly be our final checkmate position. Thus, if we can show for each of those positions that there couldn't be a forced win that ends that way, then we know it can't be done.
For instance, here's a non-promotion checkmate position with the given material:
[fen "5k2/5PP1/4K2P/8/8/8/8/8 b - - 0 1"]
But I claim we couldn't possibly force Black into this position: the final move would have to have been
100.g7# (where I just picked 100 arbitrarily), the move before that would have to have been
99...Kf8 (moving from either e8 or from g8), and the move before that had to be
99.f7+. OK, so we know that White must have played this
99.f7+ with the black king either on e8 or on g8. But if the black king was on e8, he could have instead played
99...Kd8 in reply, and if the black king was on g8 then he could have instead played
99...Kh8 in reply. The point is that, either way, White couldn't actually have forced Black into that final checkmate position.
Alright, so that's one possibility down. There are others of course, but a similar retrograde analysis for each of those will confirm that none of them can actually be forced, and that's all you need for an airtight proof that White can't force checkmate from your initial position without promoting a pawn.