This achievement pops once you've beaten all of the Standard and Hard puzzles. The Standard ones are typical situational checkmate scenarios, teaching you how to mate quickly. As such, the standard ones are harder to post solutions for since they change based on your moves.
The best way? Get a computer to beat the scenario for you!
Go here: http://www.chessvideos.tv/endgame-training/crafty-posit....php
Put the colors in backwards: you're going to be playing the white and the computer will play black, trying to checkmate you. Since white always goes first, you'll have to set up the scenario one move ahead so that you move into the position you're trying to beat. Then just move your king where the king in the game moves, and the computer will tell you where to move.
The generator is pretty good. Every scenario I plugged in worked the first time.
As for the Hard puzzles, TSG Dwaggie posted a great solution on the PC page.
I'll copy it for you here, but it's his solution, not mine.
The Normal puzzles are just normal-like gameplay, and I can't advise you on that because the computer's responses will change every time.
However, for the hard puzzles, the solutions are all "Checkmate in 2 moves" except for Hard Puzzle #1, which is "Checkmate in 3 moves". The solutions for the Hard Puzzles are below:
Hard Puzzle #1
Short Form: F7 to D6, D3 to C5, C2 to E4.
Long Form: From the start, move your Knight from F7 to D6, causing Black King to be in Check. The computer will respond by taking your Knight with his Castle, by moving his Castle D8 to D6. You will then want to move your other Knight from D3 to C5, once again causing Black King to be in Check. The computer will then move his King from E4 to F4. You can then move your Queen from C2 to E4 for Checkmate, and puzzle completion.
Hard Puzzle #2
"Transvaal Leader", 1906
Short Form: G4 to C8, H1 to A8.
Long Form: You will need to sacrifice your bishop by moving it from G4 to C8. This will force Black King to take it, by moving it from B8 to C8. This will then leave the line open for corner=to=corner movement from your Queen, from H1 to A8 for Checkmate, and puzzle completion.
Hard Puzzle #3
"Chess Player's Review", 1861
Short Form: F6 to H8, H7 to F5
Long Form: This is a hard one, if you don't know what you're doing. First, you need to block the 'H' squares with your Queen, by moving it from F6 to H8. This puts the 'H' squares under your command. The Black King will move from G4 to H4. Now, to Checkmate, you need to block his diagonal escape from the Queen, and to do that, move your Bishop from H7 to F5 for Checkmate, and puzzle completion.
Hard Puzzle #4
"Lead's Mercury", 1880
Short Form: A2 to B1, C2 to F5
Long Form: You need to have your Queen protected for it's movement in move #2. To do this, move Bishop from A2 to B1, to which the computer's King will move from E5 to F6. You will the need to move your Queen from C2 to F5 for Checkmate, and puzzle completion.
Hard Puzzle #5
Short Form: H4 to A4, A8 to C6
Long Form: What you need to do is get the Black King across the line of White Queen's diagonal line of threat. To do this, move your castle from H4 to A4, which will force the Black King to take it. At this point, you can now Checkmate the Black King because Squares A3, B3, B4, and A5 are covered by White Queen & White King. Move Bishop from A8 to C6 for Checkmate, and puzzle completion.
Hard Puzzle #6
"Musical World", 1859
Short Form: D4 to D1, B5 to E2
Long Form: This is an annoying one. First you'll need to move your Castle from D4 to D1, to force the Black King to take it, to leave the line of Checkmate. Once the Black King has moved from C2 to D1, move your Queen from B5 to E2 for Checkmate, and puzzle completion.
Hard Puzzle #7
"Birmingham Post", 1890
Short Form: G4 to H3, H3 to F1
Long Form: To be honest, the Black King is pretty much Checkmated even at the start of this puzzle. The Pawn and the Bishop are protecting each other, so what you need to do is move the Queen twice to win. Move your Queen from G4 to H3. The Black King will try and get away from the Pawn and Bishop, by moving from B5 to A6. You can then move your Queen from H3 to F1 for Checkmate, and puzzle completion.
Hard Puzzle #8
"Magyar Sakkvilag", 1911
Short Form: G3 to G4, D7 to E7
Long Form: Move your lowest-most pawn forward one space, from G3 to G4. This will cause the computer's King to move from G5 to F6. You can then do a Close-range Checkmate by moving your Queen from D7 to E7 for Checkmate, and puzzle completion.
Hard Puzzle #9
"Grand Rapid Herald", 1932
Short Form: F1 to H3, H3 to D7
Long Form: The stronger pieces are a distraction. All you need to do is move your bishop. F1 to H3 with your bishop will make the King move from C6 to B5. You can then move your bishop from H3 to D7 for Checkmate, and puzzle completion.
Hard Puzzle #10
"Deutsche Schachzeitung", 1923
Short Form: H2 to H5, H5 to F3
Long Form: You need to move your Queen twice on this. First, move your Queen from H2 to H5, which will cause the black Pawn to move from G2 to G1, upgrading to a Queen. However, it's too late to save it's King, as with it's desertion to promote itself, it's left the King open. Move your Queen from H5 to F3 for Checkmate, and puzzle completion.
Hard Puzzle #11
Short Form: A6 to A3, A3 to C5
Long Form: Move your Queen from A3 to A6 to put the black King in Check. the black King will take your Bishop, by moving D6 to C6, allowing you to move your Queen from A3 to C5 for Checkmate, and puzzle completion.
Hard Puzzle #12
Short Form: H2 to H6, G6 to G7
Long Form: The King is already cornered. All you have to do is finish the game. Move your Castle from H2 to H6. The computer's pawn will take it, leaving the way forward for you to move your pawn forward from G6 to G7 for Checkmate, puzzle completion, and if you've done the Normal-difficulty puzzles, the 'Sage' achievement :)