Link to the full guide on wiki
I highly recommend you read the wiki guide for dots and boxes. This will explain to you how to use advanced tactics in order to ‘trick’ your opponent and win the game!
For the puzzle completion you only need to win! That means claiming more than 50% of the board, this is generally pretty easy.
You are on a timer tho, so play quickly! The dots going from green to red at the top of the puzzle is the timer counting down.
There is a separate cheevo called ‘Thinking out of the box’, for completing the board while your opponent has no more than 2 squares .
I am going to attempt to explain how to do that!
In rift, your opponent Omi has 2 basic rules..
RULE 1: if there is a square on the board with 3 sides then she will complete the square
RULE 2: she will only place lines in ‘safe spots’ if possible, meaning she wont make a 3rd line on a square if there are other spots available that wont make a third line.
Knowing that Omi will follow these rules puts you at an advantage and will allow you to control the board.
During the beginning of the game when the board is mostly empty it will all be very random, however as the board fills up you can start to create long chains of nearly complete boxes, towards the end of the game your best situation would be 2 long chains on the board. This wont always happen, it may take a few tries before you get that perfect board, however when it does happen, it is then pretty easy to convince Omi to only take 2 squares! The strategy is explained in the above linked wiki page.. but to put it as simply as possible, you take the majority of the first chain and then sacrifice the last 2 squares so that Omi takes those and then gives you the whole of the second chain. Obviously if you are playing against a real intelligent human then this strategy, although useful, would not always work. but because Omi follows the set rules, she can be tricked!
Here i have made a couple of videos of the puzzle being completed with Omi taking just 2 boxes