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 2020-02-27, 21:44 #1 what   Dec 2019 Kansas 24 Posts March 2020
 2020-04-24, 05:46 #2 0scar   Jan 2020 11002 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 (2L-1)x(2L-1) 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 2020-04-24 at 05:56

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