Zero-Knowledge Bingo
This seems entirely straight forward. Each player publishes a commitment, essentially a hash of their board. Possibly signed. 

If the boards are not high enough entropy themselves we will tack on some random data to increase their entropy to make the result unguessable. 

When a player declares victory he will provide a Zero Knowledge proof that he knows of a winning board matching his commitment. 

Note Zero Knowledge proofs are very powerful stuff, but we need to open the black box of the functions we use (such as hashing) in order to use them. For any efficiently computeable predicate function $f$ we can prove we have $x$ so that $f(x)=true$ without revealing anything about $x$ beyond the statement. 

In our case we would have $f(board)$ be the predicate which checks the board is a winning board and matches our commitment. Obviously trivial to compute.