Here is an explanation from a walkthrough:
Okay, when you click on the the puzzle, it starts spinning. Here is what you
have to do. You have to remember which squares light up at the end of each
spin, and click on them in the right order. There are a total of three spins and
it gets harder and harder each spin.
Think of the puzzle as a clock. The first square(yellow) is on the 12:00, the
green square is on 3:00, orange is on 6:00, and red square is on 9:00.
There are a total of 16 squares. Below are the listings of each spin.
1st spin: 14,7,6
2nd spin: 6,12,4,1,10
3rd spin: 15,8,12,9,3,1,5,10
Also, if you are playing in Casual mode, there is an option to skip some of the puzzles - you can access the skip option from your Smart Alex. And in Casual mode there are in-game hints as well.