To get the achivement for Cognito Mode first complete Puzzle mode completely. Then play Cognito mode - if you get stuck do not use the hint button because it will not give you any points for that level. Instead, quit the game (it will save your score and your level) and go back into puzzle mode. Press start and go the the level select screen and find the puzzle that looks like the one you are stuck on in Cognito (All the puzzles for Cognito and Puzzle are the same, they are just in a different order). Go into the right puzzle and use the hint button as many times as you need until you memorize the right order. Then go back to your game in Cognito and complete the level.
Or you can refer to this guide at gamefaqs -- IT HELPS!!