20200227, 21:44  #1 
Dec 2019
Kansas
2^{4} Posts 
March 2020

20200424, 05:46  #2 
Jan 2020
1B_{16} Posts 
What about tiling an infinite board?
Asymptotically, leave a fraction r of empty squares, and place each symbol in 1/3 of the remaining squares; ensure winning chance for no player.
Can you do so for some explicit fraction r>0? Can you state (and eventually reach) some upper bound on r? Let's make "asymptotically" more precise. Weak version: choose some square as the origin, consider a (2L1)x(2L1) board centered around it and find the fraction r(L) of empty squares; take the limit as L grows to infinity. Strong version: for each square, consider the four 1xL boards with a corner on it (along the directions +x,x,+y,y); as L grows to infinity, the four limits must be equal, and such value must not change for different choices of the starting square. Last fiddled with by 0scar on 20200424 at 05:56 
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