I won't give detailed, step by step instructions for building the robot because it's physical design does not need to be exactly the same as mine. I used LOZ blocks. You can build the robot however you want using whatever blocks you have. It could also be made from Meccano or Lego.
The supplied Arduino code (see step 4) allows you to teach the robot arm where each position on the board is using the TV remote. The positions are stored in the controllers EEPROM memory. This means your robot can be any size or shape as long as the arm can reach all positions.
The program also displays the game on your PC via the serial monitor so you can play against the controller even without the robot. This is ideal for experimenting with the robot's A.I. My code is not very smart so even at the hardest level you can beat it.