Story related. Chapter 6. You’ll play the Ball and Cup trick on the handyman on floor 3 on St. Edmund Hall, and to get this you need to do the trick perfectly. The trick is that the handyman can’t move the ball that has the cup in on his first move. So if he moves 1 and 3, the ball is in cup 2, so follow that cup’s movements and swaps to see where the ball is now. This one is kind of weird; on my first play I got it first time using this method, but during the second it took 3 times before it worked. Thankfully you don't have to get it first time, mistakes are allowed. All that matters is that you do the trick correctly the one time needed to continue the story.
Malkaiah of the forums had this method: "Looking at the three cups there is a black mark on Position 1. Remember this for a little later. The first movement is silent remember so you will never know which cup the ball starts under. However, assume the ball is under cup 1 (with the mark on it), follow the movements as the man speaks and then look at the three cups. If the cup with a mark on it is in the position you've measured the ball to be then pick it and you'll get it. Otherwise you'll need to deduce which of the two remaining cups is the one with the ball under. Which is easily enough done if you read the details in the magician handbook. It had me a little stumped at first but after re-reading the instructions in the game it worked first time."
Regardless of what method you choose and how you get it, the achievement will pop when you select the cup with the ball and come out of the game.