Hopefully interesting challenge :P

[Kevin]

^
Yeah, I forgot to mention that reactions only works for 0, 3, 6 and 9 starting circles in each boxes, and actually having some empty boxes in the middle isn't a problem :P . Wow, I don't think I will be able to solve it without seeing the hints if I were not the creator xD .

!! (open only if you have finished the challenge because it spoils the answer ^^ )

Spoiler (Click to View)

Actually I found the pattern accidentally when playing with my calculator during some boring or too easy studies at my school. Firstly, I found out that
3x4 = 12
33x4 = 132
333x4 = 1332
3333x4 = 13332
Later I found out it works for 6 and 9 too,
66x4 = 264
666x4 = 2664
6666x4 = 26664

It doesn't work with any other number. It's funny because it seems like the last digit gives 1/3 of it's value to make a new digit on the left.
I made my first quiz and most of my friends can figure it out:
f(3) = 12
f(33) = 132
f(333) = 1332
f(3333) = 13332
What the heck is f(x)?

Soon I noticed that it happens because 3x4 = 12, so
33333x4 = 120000 + 12000 + 1200 + 120 + 12, the number one carries through all the way, converting all number '2' back into 3.

Then somehow I got the idea to combine these numbers.
36x4 = 144
336x4 = 1344
3336x4 = 13344
First, I thought it fails to work, as well as when using other numbers. Soon, though, I realized that each digit just give 1/3 of its value to the previous digit.
The first funny ruleset that I found is very interesting for me. There is a bag, which walks from the right to the left.
It always wants to carry 1/3 of the value of the digit it meets.

For example, 3669693 x 4 = ...?
The bag first comes to the last digit, i.e. number 3, so it takes 1 of it. So the answer's last digit must be 2 (because 1 is taken away by the bag).
Next, when it meets the number 9, the bag wants to have 3. Since it already have 1, it only take 2, thus the number 9 decreases to 7.
When it meets the next number, 6, the bag only want to have 2 while it already have 3, so it gives 1 to the number 6 and it becomes 7.

This works everytime, if the number consists of only the numbers 0, 3, 6 and 9.

I guess, when I reverse it, people may realize the more complex patterns, but not that it is actually as simple as a multiplication of four.