Tabletop
Archibald and Cookie Monster take turns placing perfectly circular cookies on a perfectly square table. Once placed, cookies cannot be moved. Cookies cannot overlap with other cookies, and must fit fully on the table. The last player to place a cookie on the table wins.
Note that we cannot apply Zermelo’s Theorem, because the number of possible states of the board is infinite. If Archibald goes first, is there a winning strategy?
Archibald has a winning strategy. We can apply a similar strategy as the previous games of mirroring the opponent.
First, start by placing a cookie in the middle of the table. Then, whatever Cookie Monster does, reflect his move across the center of the board. Due to the perfect symmetry maintained by Archibald's moves, then if Cookie Monster ever has a move to play, there will be an open spot on the opposite side of the board. This is because once a cookie is placed in the center, any two points that are reflected across the center have greater distance than one diameter of a cookie, which makes it impossible for any single cookie to cover two points that are reflected over the center.