Reconstruction of domino tilings - Combinatorial and probabilistic questions

"Dominosa" is a puzzle game. Before the game begins, the (n+22) domino bones (0,0),(0,1),⋯,(n,n) are placed on an (n+1)×(n+2) grid (constituting a perfect tiling). The domino bones are then being hidden, leaving only the numbers revealed. The purpose of the game is to recover the original tiling of the domino bones. The game raises many questions, such as how various algorithms perform and how many solutions a random puzzle has. In this paper, we study some of these combinatorial, probabilistic and algorithmic questions.