I just did it (without reading the above solution). The key is to think ahead at each step. There are usually only two moves you can make at each point (since there’s only one open space). The wrong moves will get you stuck within two steps by trapping all the frogs on one side – e.g., if you have ggbb on the right side, your browns can never get out (since the greens can’t move backwards.) So just make sure to keep picking the move that won’t get you stuck.

I think the three minute time limit is to trick you into rushing (to try to prove how smart you are.) If you ignore the three minute rule and just check your moves at every step, you should end up finishing it in less than three minutes.

For those who want the solution (with the thought process behind it), here it is:

[spoiler]

First move: It doesn’t matter if I start with a brown or a green, since the problem is symmetric. I’ll start with a green. If I click the second green from the left, my browns are trapped. So I’ll click the rightmost green.

Second move: If I click the second green from the left, my browns are trapped. So I’ll go with my only other choice: the leftmost brown.

Third move: If I click the rightmost green, I can follow with either the second green from the left (trapping two browns on the right), or I can follow with the second brown from the right (trapping two greens on the left.) Since both of those are losing outcomes, I won’t click the rightmost green this turn. Instead, I’ll click the second brown from the right.

Fourth move: Clicking the rightmost brown gets me stuck, So I’ve got to click the rightmost green.

Fifth move: Clicking the middle brown traps my greens. So I’ll click the second green from the left.

Sixth move: If I advance the leftmost brown, I can either follow with my leftmost green (trapping my right two browns), or follow with my middle brown (trapping my leftmost green). So that’s a bad move. Instead, advance the leftmost green.

You can keep proceeding in the same way, but from here it’s pretty obvious you just advance all three browns, then advance all three greens, then advance the last two browns, then advance the last green.[/spoiler]

Incidentally, I think the general proceedure for solving this problem with N frogs on each side is:

Move 1 green

Move 2 browns

move 3 greens

move 4 browns

…

until you get to moving N of one color.

Then move N of the other color.

Then move N of the first color

Then move N-1 of the second color

Then move N-2 of the first color.

…

Until you move the last one.

With each step, you should be moving the frontmost frog first.

So, for four on each side, it’s

gbbgggbbbbggggbbbbgggbbg