While the other solutions here detail the nitty-gritty details about when this achievement would pop, that was never a problem for me. My problem was that I just didn't understand the strategy involved in becoming a good Zuma's Revenge player. As a result, I spent many frustrating hours repeatedly failing levels. Eventually, I figured out that Zuma's Revenge is a game of strategy, and once I mastered that, I was able to beat all of the Challenge levels, many of them Aced.
My solution here is to help you become a good enough player to beat the challenge levels. I found that playing in Challenge Mode was quite different than playing in Adventure Mode, requiring different strategies in order to beat enough of the challenge levels to get the achievement. In short, this game is all about strategically placing, and often prudently swapping, balls in order to build up consecutive combos and grabbing bonuses without breaking your streaks.
My 8 Steps to Success in Zuma's Revenge: Challenge Mode:
1. Set Up Combos: The best strategy is to keep combo streaks going. Combos occur when you shoot a ball of a certain color into a group of two or more balls of the same color (i.e. 3 or more makes a combo). But making combos isn't enough: you need to chain together streaks of combos (i.e. two consecutive shots that create combos is a streak). So getting long streaks of combos garners more points than just getting a bunch of combos, meaning, if you don't currently have a combo streak going, you should take that opportunity to set up future combos. This is especially true at the start of each round.
At the start of each round, use a couple of balls to double up an isolated ball. That is, if you have a red ball in your mouth and there is a solitary red ball on the board, shoot it at it to set it up for a future combo. That way, when you get a red ball later, you can shoot it into the two red balls to create a combo. Do this for the first 3-4 shots, if possible, to make it easier to keep combo streaks alive once you start going for streaks.
2. Go for Streaks: Destroying balls (i.e., matching three or more of the same color) in a streak--i.e. with as many consecutive shots as possible--is crucial to getting a high score and also for triggering reversals where the balls start rolling backwards temporarily to give you much needed breathing room. This is where rotating between the color in your mouth and the color on your back is crucial, which is done by simply tapping the back of the frog. Tapping the frog on the back allows you to swap the ball in his mouth with the ball in his back, and thereby keep some streaks alive.
3. Go for Chain Reactions: As much as possible, try to destroy balls that set off combos (i.e. chain reactions). For instance, if you have 2 or more red balls surrounded by 2 or more green balls on either side, themselves surrounded by 2 or more blue balls--Blue.GreenGreen.RedRed.Green.BlueBlue--shoot a third red ball at the other red balls. Those 3 red balls will explode, collapsing the 3 green balls together, which will then explode thereby collapsing the 3 blue balls together. That is a chain reaction and it nets you massive points.
4. Go for Gap Bonuses: When a chain reaction initiates, there is a small window of opportunity for a gap bonus: this occurs when a gap in a line of balls nearer to you opens up after the balls you just destroyed disappear. If you can shoot a new ball through that gap into a line farther away and have it create a new combo, you get a nice bonus. Be careful, though, as often the gap closes so fast that you can easily misfire, prematurely ending your streak.
5. Go for Bonus Balls: Along the way, always go for the luminescent multiplier balls whenever they pop up (trying as much as possible not to break your streak), followed by other types of bonus balls (particularly the lightning strike and the laser strike).
6. Utilize Power Ups: User your power ups intelligently: if you have the lightning strike or laser strike, blow up bonus balls and fruit, or balls that will set off chain reactions. I'll restate that: use the lasers and lightning strikes to directly destroy bonus balls and fruit. If no bonus balls or fruit are on the board, destroy solitary balls that will then trigger chain reactions.
7. Fruits: Try to get as much fruit as possible, but don't break your streak to do so.
8. Recovering: If you get a series of 'bad balls' and have to break your streak, repeat step 1: use 3-4 in a row to fix the problem of isolated balls, so as to set them up for future combos.
If you make use of those strategies, I've no doubt that you'll be able to maximize your score and beat all of the Challenge levels. Good luck!