The site uses Answer Set Programming with the Clingo engine to compute the optimal solutions for smaller grids. Maximizing grids like this is probably NP-hard.

Note that traditional SAT and SMT solvers are quite inefficient at computing flood-fills.

The ASP specifications it uses to compute optimal solutions are surprisingly short and readable, and look like:

  #const budget=11.
  horse(4,4).
  cell(0,0).
  boundary(0,0).
  cell(0,1).
  boundary(0,1).
  % ...truncated for brevity...
  cell(3,1).
  water(3,1).
  % ...
  
  % Adjacent cells (4-way connectivity)
  adj(R,C, R+1,C) :- cell(R,C), cell(R+1,C).
  adj(R,C, R-1,C) :- cell(R,C), cell(R-1,C).
  adj(R,C, R,C+1) :- cell(R,C), cell(R,C+1).
  adj(R,C, R,C-1) :- cell(R,C), cell(R,C-1).
  
  % Walkable = not water
  walkable(R,C) :- cell(R,C), not water(R,C).
  
  % Choice: place wall on any walkable cell except horse and cherries
  { wall(R,C) } :- walkable(R,C), not horse(R,C), not cherry(R,C).
  
  % Budget constraint (native counting - no bit-blasting!)
  :- #count { R,C : wall(R,C) } > budget.
  
  % Reachability from horse (z = enclosed/reachable cells)
  z(R,C) :- horse(R,C).
  z(R2,C2) :- z(R1,C1), adj(R1,C1, R2,C2), walkable(R2,C2), not wall(  R2,C2).
  
  % Horse cannot reach boundary (would escape)
  :- z(R,C), boundary(R,C).
  
  % Maximize enclosed area (cherries worth +3 bonus = 4 total)
  #maximize { 4,R,C : z(R,C), cherry(R,C) ; 1,R,C : z(R,C), not cherry(  R,C) }.
  
  % Only output wall positions
  #show wall/2.