White to play and checkmate in two moves.
[title "E.Umnov's chess problem, 1944"] [FEN "k7/5QK1/3n1pb1/1R1p1r2/5p2/8/1p6/4R3 w KQkq - 0 1"]
1.Re4, threatening 2.Ra4#. Black has many possible moves, but all of them allow either Ra4#, Qa2# Qb7#, or Rxe8# (did I miss any?).
(In the spirit of "teaching a man to fish", if you can't solve a checkmate problem and decide to give up, you can always feed it to a chess engine and get an instant answer without having to wait for someone to reply to an online post. For example, try https://lichess.org/analysis ).
Very nice - thanks for posting. The solution is:
1. Re4! threat 2. Ra4#
1. ... Nc8/Nb7/Nxe4 2. Q(x)b7#
1. ... Ne8+/Nxb5 2. R(x)e8#
1. ... Nc4 2. Qb7, Re8#
1. ... dxe4/d4 2. Qa2#
Note that after 1. ... Nc4, there are two possible mates. This is not generally regarded as a defect because each of them can be found by itself in other lines.
I am really not an expert in these kinds of problems, but I guess the idea is that White begins with two units en prise, and must paradoxically offer up the third for sacrifice as well.
General purpose chess engines can be used to tackle direct mates like this, but they may not pick up all the artistic points (e.g. tries, set play, duals, etc) and will not be applicable to helpmates, selfmates, retros, fairy problems, etc. Specialist problem engines exist, see this Wikipedia page.
EDIT: Of course at one level, all the engines are pointless. The challenge is for the individual to solve the problem by themselves, without tech support. Direct mates in 2 are fairly simple to solve in this way, but the difficulty grows rapidly as the number of moves increases.
One key difference between solving a problem and playing a game of chess is that in the former, one knows that there is a solution, and that probably it's quite artistic. So part of the approach is to hunt for such cool effects. There are also deliberate red herrings which are part of the fun. E.g. a try is a false answer which nearly works, but there is just one black move which defeats it. E.g. in the Umnov problem, how should Black respond to 1. Rxb2? The right move is often quiet and paradoxical - apparently giving something to the opponent, but in reality compelling the opponent to create a fatal weakness in their defence.
On the other hand, one beauty of over-the-board chess is that there's no certainty of a winning move existing at all, certainly not within a specific time span. If it's a published position from a game then there is probably a unique move, but it's possibly quite flashy maybe involving a sacrifice to break through a defensive position.