Okie Dokie can be solved using linear algebra or with the
aid of the following chart. To use the chart, draw a 5x6 chart and look up each
lit 'okie' in the following table. Then place a mark in the squares indicated.
Do this for all of the okies.

Then, simply hit the okie on each square which contains an odd number of marks.
Once this is done, the puzzle will be solved.

CODE

For example, if the center two okies are set and no other ones are, one's chart
would look like:

CODE

(with numbers indicating the number of marks in each square). Thus, one would click all of the squares that show an odd number ("1" in this case, though if there were more lights lit in the original puzzle, there might be 3's, 5's, etc. as well). As another example, suppose that the left center square and the three squares next to it were lit. This would yield the following chart:

CODE

Thus, one notices that all of the marks cancel out to even numbers except the
left-center square itself (which ends up with an odd number, "3").

Addendum: To solve any position reasonably quickly, it is sufficient to remember three patterns (and be able to figure out their reflections). Clearing all but the bottom row is trivial--just start at the top and click below each lit 'okie'. To clear the bottom row, you just need to remember the patterns to clear each of the left three okies and be able to apply them reflected to the right okies.