To play the frog puzzle, you just need a tame frog. In the frog's information screen, choose Games from the upper-right corner, then choose Puzzle.
Objective of the Game
The objective of the game is to line up at least three frogs of the same color in a row so that they're all facing the same direction. The trick is that in order to complete the puzzle, you have to make all of the frogs in the entire puzzle be lined up so that there aren't any frogs that aren't in a line.
The frog puzzle can be solved with trial and error, but this could take a while and be pretty frustrating. Instead, here's what I recommend.
What you should do is try to line up a color of frog along a side of the puzzle so that you don't have to worry about that color anymore. For example, if there is a color of frog that has exactly six frogs, you should line them up along a side of the puzzle so that all six are lined up along the edge. This way, those frogs are taken care of and don't have to be dealt with anymore.
Once you line up a group of frogs, look once more for a group of frogs that can all be lined up along a side of the puzzle. For example, if you already lined up a group of six, then the remaining puzzle space is 6x5 rather than 6x6 (since the line of six frogs takes up a whole row or column). This means that if there is a group of exactly five frogs, you can line them up on one of the sides of the puzzle that has room for five.
Sometimes this method isn't perfect and you have to shuffle the frogs around a bit in order to solve the puzzle. However, this is a simple way to help you solve the frog puzzle more quickly.