Level 15 takes luck. After you've cleared the board the first time, the elements become random. The best strategy is to keep looking ahead for opportunities to clear the board.
You really need to clear the board more than once to get past this level, so any time that you can clear the board, make sure that you do it.
It is worth taking some time between moves to make sure that you don't miss an opportunity to clear the board. When you can't, keep the board set up so that you can best take advantage of upcoming elements. For instance, at level 15, with 2 elements in each color, my board would look like this:
- Code:
Y - Y R - -
G - G R - -
B - B - - -
- - - - - -
- - - - - -
- - - - - -
The two Kwarkites would have been used for their only possible combo. From here, I can build any other combination. The upcoming elements will help me decide which ones to build.
Even with the best strategies and most careful analysis, you won't get by level 15 most games. You still need the elements to come up in a way that's actually possible to do it. But if you play it enough, you will have those games where you do get lucky. Level 16 is much easier.
(The tables worked in preview. Sigh...)