# Is there a way to prove a checkmate in 3 problem is not solvable?

I have been working on a checkmate-in-3 problem for a long time. So long that I am becoming convinced it is not solvable.

Is there a way to prove it is not solvable?

I want to avoid putting the problem in an engine, in case it will show me a solution. For the same reason I do not want to share the problem.

• Why not just give the position to a friend, ask him to test it with an engine, and to tell you whether it's solvable or not? Mar 20, 2020 at 21:04
• Just to be sure, have you checked en passant as a possible solution? Or sometimes people don't realize that castle is still possible. Mar 20, 2020 at 21:36
• I used to have a mate in three that I liked to give people. No one I showed it to ever solved it. After that experience I am never convinced that any puzzle is not solvable without using a computer to check it. Mar 22, 2020 at 0:03

Three moves is probably in the realm where you could manually brute-force it, although it may take some effort.

Pick a move, a reply, a second move, and a second reply. Then start looking at whether White has a checkmate from that position. If not, then White's second move was incorrect, so change it. If you eliminate all possible White second moves this way, then go back to White's first move, and change it. If you eliminate all White first moves too, then you have proven that a mate in 3 does not exist.

If you do find a White checkmate from one of these positions, then change Black's previous move. If the solution is valid, White will have a checkmate for all possible Black replies.

• Brute force it? Probably not. Three moves in chess, depending on the number of pieces, can easily run into the millions of possibilities. After just three moves in the opening, there are over 121 million possibilities, and after just four, 319 billion. You might stumble upon the answer as we do when calculating any position as a human, but you will not brute force virtually any position with even a minimum of pieces. Mar 21, 2020 at 0:08
• There may be 121 million possibilities after 3 moves in the opening, but that's six ply, and a mate in 3 problem is only 5 ply, reducing it to under 5 million. You don't have to explore all possibilities, though, and that saves time. Not finding a mate after a particular Black second move means White's second move was incorrect, so you don't have to explore the other possible Black responses to that move. And, of course, if checkmate is imminent, the lines can be quite forcing. Whether it's feasible does depend on the position.
– D M
Mar 21, 2020 at 0:38
• I did not consider that half ply, but still, even at 1,000,000, or less, you are not going to "brute force it". You can maybe find it the way we calculate everything...by pattern recognition, but not true brute force. There are people, who will take that at face value, when it is mostly impossible. Mar 21, 2020 at 0:41
• By choosing Black's moves wisely/luckily, you essentially reduce it to 3 ply, since you'd then only need to go through all possible White moves. In some positions that's tedious but doable; in some positions it's admittedly not feasible. But... without using a computer, how else are you going to prove an arbitrary position is not solvable?
– D M
Mar 21, 2020 at 0:54
• My problem is with that specific term. It renders part of your answer incorrect. I am not saying it is not solvable, just HOW. Mar 21, 2020 at 1:02

The only way to prove it with 100% accuracy is to use a special engine. Since you do not want to see the solution yourself, then you could ask someone you trust to use it for you. Having a friend sit there and attempt to solve it will not really prove anything since he is human, and may not be strong enough to guarantee a correct answer.

You are almost asking an impossible question in that you want to know the answer, but not willing to let anyone see it or to use a computer to solve it yourself. You cannot have it all.

There are special engines that are designed to prove or disprove mate-in-x problems. The ChessBase program Fritz has always had this function. I am not sure if others do though.

Here is a screenshot of the help file.

• I will look into it, once I confirm it still works I will accept your answer. I am on Mac though. Mar 21, 2020 at 8:47
• There may be others, but ChessBase products only run on a PC. They made one attempt a long time ago, and stopped, deeming it not worth it business-wise. Mar 21, 2020 at 10:59
• @OrigamiEye If you would like, if you have a chess.com account (of you can create one for free), you can send me the problem via a message there, and I can run it through Fritz with the special mate-in-x engine and send you as much or as little information about the results as you would like. I could simply confirm that the mate is there all the way to sending you the answer, now or in the future if you should change your mind later. Mar 21, 2020 at 14:09