In the game of Chomp, two players start with a chocolate bar, which is scored into an $a$ by $b$ array of squares (where $a\cdot b>1$). The square in the lower left is poisoned.
The players alternate turns. On their turn, a player chooses a square, then eats it, along with all of the squares which are either above it, to its right, or both. This continues until someone eats the poisoned square, wherein the non-poisoned player wins.
Here is an example of how the first two turns might go, where the "x" indicates the square that player chose:
Part 1: Show that the first player can win if she plays perfectly.
What if you have an $a\times\infty$ chocolate bar? This means that there are $a$ rows of squares, where each row continues infinitely to the right, and the left ends of the rows are all aligned. This grid still has a lower left corner, which is still poisoned. Some moves will require players to eat infinitely many squares, which we assume is possible.
Part 2: For which values of $a$ does the first player win on an $a\times\infty$ bar? Why?