Here's the solution, in case anyone needs it:
Cerff gur gbc yrsg enoovg, cerff gur obggbz evtug enoovg, naq cerff gur enoovg gung vf gjb ubyrf yrsg bs gur obggbz evtug enoovg.
(You can use a rot13 decoder to read the solution.)
Viewing post in Tale of the Bear and the Beech Tree comments
If you want to figure out the solution for yourself, here's the reasoning process:
When you click on any one of the seven rabbit holes, a certain set of rabbits flips their state. Write out those seven possible patterns. Then consider the top-left rabbit: There are only two ways to get this rabbit to emerge. Likewise, consider the bottom-right rabbit: there are only two ways to get this rabbit to emerge, as well.
If you stare at the seven patterns for a second, you should be able to see the solution. There are only four possibilities to consider, and one of them works.
Here's the complete solution, just for fun.
If you give the seven rabbits these names:
a b c d e f g
Then the seven click patterns are:
1: a b 2: a b c 3: b c e 4: d e f 5: c d e 6: d f g 7: f g
If you make a list of the patterns that activate each rabbit, you get this:
a: 1 2 b: 1 2 3 c: 2 3 5 d: 4 5 6 e: 3 4 5 f: 4 6 7 g: 6 7
You can see that rabbits "a" and "g" are special: The only way to activate rabbit "a" is by using pattern 1 or 2. The only way to activate rabbit "g" is by using pattern 6 or 7. That narrows the search down to just four combinations--which is very doable! You *could* try each of those four combinations in turn, and be done in a minute. OR:
You could notice that both patterns 6 and 7 will activate rabbits "f" and "g": The only difference is that pattern 6 will activate rabbit "d" as well. So, is it useful to activate rabbit "d"? No, not really. Pattern 5 activates the middle rabbits ("c", "d", and "e"). That leaves only the first two rabbits ("a" and "b"), and those two rabbits can be activated by pattern 1.
So you can see pretty much immediately that patterns 7, 5, 1 will solve the puzzle.
That's all you need to do! You don't actually need to try all four combinations--even though you could. (Considering all four combinations could lead you to another solution, or show that there aren't any.)