March 2020
 

[url]http://www.research.ibm.com/haifa/ponderthis/challenges/March2020.html[/url]

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.