Here is an interesting game. You start with an empty 4x4 grid. At each turn you can choose an empty cell and place a value in it. The placed value is given by the following rules:
- If the chosen cell has no neighboring (horizontal or vertical) values then the placed value is 1.
- Otherwise the placed value is the sum of all neighboring (horizontal or vertical) values.
What is the largest value that you can achieve in this game?