No easy way around this. Here are some tips:

• Do not use hints, as you will lose a significant chunk off your scoring bar at the bottom
• Speed counts. If you have a fast eye and fast fingers, you will score big.
• Set yourself up. If you can make hyper gems and power gems and refrain from using them until you are stuck, do so.
• You cannot run out of moves. The game will drop combos into the level if you are low on moves so check near the top for quick moves.
• Once you hit level 5, make as many single moves as you can but ALWAYS keep your eye out for combos. Single moves will barely do anything to the scoring bar.

You are going to need a lot of time for this. You cannot run out of moves because the game will automatically start throwing "possible" moves in on the top when you start running low. The best part is, you can save your game, so it doesnt have to be in one sitting. If you're stuck don't bother using the hint button, for this will make your score bar go down, just wait the 15 seconds and it gives you a tip for free.

Alternative method by Zef...

A very time consuming achievement, taking around 15,000 minutes (250 hours or 11 days) to complete. You WILL NOT have to play in one go, the game allows you to save your progress when you exit the game. So you can load up your game and chip away at the levels over a longer period.

You cannot 'lose' in this game mode as the computer will always leave you at least one move to keep the game going, so if you are struggling to find a move look at the top of the grid as that is where you will usually find them. You can also wait for the game to tell you where a moveable piece is, just stay idle for a few seconds and you get a free hint. Useful if you don't feel like looking yourself.

Besides having a lot of free time to play you should try and make combinations to maximise your score, making power gems and hyper cubes gives you a bonus and destroying them will also give you more points. Take your time to make these pieces as you are not playing against the clock, you might as well spend a little time to get those extra points as well as work towards two other achievements.

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!!
click here

Meet the requirements to unlock the 4 mystery modes. 1 mystery mode is unlocked with each of the classic mode scenarios.
Finity Mode -- Complete 280 levels of Endless Mode
Hyper Mode -- Complete 8 levels of Action Mode
Twilight Mode -- Complete 17 levels of Classic Mode
Cognito Mode -- Complete Puzzle Mode