You need to make the balls change positions with each other, but the actual act of guiding individual balls to the right hole shouldn't prove too challenging.
Make your moves carefully and keep track of what you're doing. This problem requires you to shift things about a great deal, so just be warned that hints alone won't get you through this one.
The shortest solution for this problem involves less than 30 moves, but in order to solve it that quickly, you need to manage the movement of multiple balls at once.
If you're having trouble, swallow your pride and try the easier solution where you only need to guide one ball at a time.
For puzzles like this, you best bet is to clump as many of the open spaces together as possible to give the ball you're guiding a larger space to move about.
Just don't get too obsessed with consolidating space or you might actually make things harder on yourself.
Solution
Correct
Excellent!
US Version
This puzzle requires 28 moves to solve. Solving it shouldn't be all that difficult if you stay focused and avoid getting confused.
UK Version
This puzzle requires 28 moves to solve. Solving it isn't all that difficult if you stay focused and avoid getting confused.
Initial Position
Move the right green block up
Move the lower blue block right
Move the Red orb up and right
Move the left green block all of the way down
Move the Blue orb directly above the left green block
Move the upper blue block all the way left
Move the Green orb next to the upper blue block
Move the right green block all the way up
Move the yellow block right
Move the Green orb above the Blue orb
Move the Red orb next to the upper Green block
Move the yellow block left
Move the right green block down
Move the Red orb into the red space
Move the yellow block up
Move the right green block up
Move the Blue orb right and down
Move the lower blue block directly below the yellow one
Move the Yellow orb left and up
Move the Blue orb into the blue space
Move the lower blue block next to the Blue orb
Move the Green orb right and down
Move the Yellow orb all the way left and up
Move the yellow block down
Move the upper blue block right
Move the Yellow orb into the yellow space
Move the left green block all the way up
Move the Green orb into the green space
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Trivia
While the puzzle remains the same in all versions, the initial position of the blocks was changed in the European and Korean versions so that in the European and Korean versions 27 moves are required to solve the puzzle rather than 28 moves. This was changed because the original distribution resembled a swastika. However, the text for the solution wasn't changed and still states that the solution to the puzzle requires 28 moves. This change is also present in the HD re-release.