There are two players 1 and 2, and the game begins with player 1 selecting one of the boxes marked 1 to 16. Following such a selection, the selected box, as well as all boxes in the square of which the selected box constitutes the leftmost and lowest corner, will be deleted. For example, if he selects box 7, then all the boxes, 3, 4, 7 and 8 are deleted. Similarly, if he selects box 9, then all boxes 1 to 12 are deleted. Next it is player 2’s turn to select a box from the remaining boxes. The same deletion rule applies in this case. It is then player 1’s turn again, and so on. Whoever deletes the last box loses the game? What is a winning strategy for player 1 in this game?
I can't even begin to form a payoff matrix or anything for this question, do we have to consider ALL the possible alternatives?
This is the game of Chomp. It is a first player win for any board size, but the strategy is not known in general. The proof is "stategy-stealing". Assume player 2 can win. Then player 1 can take box 4. Any move that the second player can play now, the first player could have played on the first turn, so the first player plays the second player strategy and can win. We have a contradiction, so we know the first player can win. We don't know whether player 1 should take 4 or something else. It varies with board size.
For a square board like yours, the first player has an easy win by taking the square diagonally next to the poison square, in this case $10$. The first player can then maintain equal lengths on the vertical and horizontal legs, which forces the second player to take $13$.