A polyomino fence is a placement of nonoverlapping polyominoes on the square grid whose union is connected by shared cell edges. Free cells are regarded as connected when they share an edge or a corner. The fence area is the number of cells in its largest bounded connected region of free cells, or zero if none exists (Langlois-Rémillard et al. 2026). This convention excludes diagonal leaks and counts only the largest region when several regions are enclosed.
The 
-omino
fence problem asks for the largest fence area using every free 
-polyomino
exactly once, with rotations and reflections
allowed. For the twelve pentominoes, the maximum is
128 cells (Gardner 1973, Langlois-Rémillard et al. 2026). Langlois-Rémillard
et al. (2026) enumerated 5760 optimal pentomino fences up to translation,
rotation, and reflection,
with 1440 optimal cyclic piece orders. Among all cyclic orders, the smallest achievable
optimal area is 111, attained by 313 orders.
For all 35 free hexominoes, Langlois-Rémillard et al. (2026) proved that the maximum is 1597. Their lower and upper bounds for all free heptominoes are 19733 and 19830, and those for octominoes are 279134 and 280389. These latter bounds do not determine exact optima. The authors report that Claude contributed proof drafts and identified omitted enumeration cases.
