Time for a little exercise in game theory! Yes, there is a branch of mathematics called game theory. If you ever wondered what mathematicians do for a living, well, some of them research game theory. This game, however,is simple enough that anyone can analyze it without training.
We have two players. Each player has a marker in one of the colored regions in the picture above. The second player chooses the initial positions of the markers. The players take turns, the first player going first. During each player's turn, the player may either move his marker into an adjacent region or choose to do nothing. If the first player ever moves his marker into the same region as the second player's marker, then the first player wins. If the first player doesn't win after twenty of his turns, the second player wins.
Which of the two players can always win? I do not require a proof, but you get extra points for showing reasoning.
Now, if I may digress a bit, have you ever heard of the four color theorem? The four color theorem states that if I have any group of regions like in the picture above, and I want to color each region such that no two adjacent regions are the same color, then I only need four different colors. Interesting, huh? I am told that the proof is highly nontrivial.
ETA: my thoughts on the solution here