SOLO-028 · Pentomino tiling

The Eight Shapes

Pave the rectangle 5 × 8 completely with the 8 pentominos given, without overlap.

This page contains the solution.

The answer

The paving below is unique for the indicated pieces and clues. A letter identifies an entire pentomino.

The reasoning

1. The silhouettes are defined by the squares bearing their letter. The dots are not part of the room. Build possible orientations: four rotations, then their mirror, by removing the duplicates.
2. Start with the boxes bearing an index. For each orientation of the piece concerned, check that it covers all the indices of its letter and no indication of another letter. Then choose an empty box with the least possible investments.
3. Place I on L1C1, L1C2, L1C3, L1C4, L1C5. Verify that these five cells form a rotation or mirror of its silhouette and do not cover any parts already laid.
4. Place L on L1C7, L1C8, L2C8, L3C8, L4C8. Verify that these five cells form a rotation or mirror of its silhouette and do not cover any rooms already laid.
5. Place T on L3C7, L4C7, L5C6, L5C7, L5C8. Verify that these five boxes form a rotation or mirror of its silhouette and do not cover any rooms already laid.
6. Place X on L1C6, L2C5, L2C6, L2C7, L3C6. Verify that these five boxes form a rotation or mirror of its silhouette and do not cover any parts already laid.
7. Place W on L2C4, L3C4, L3C5, L4C5, L4C6. Verify that these five cells form a rotation or mirror of its silhouette and do not cover any parts already laid.
8. Place P on L4C3, L4C4, L5C3, L5C4, L5C5. Verify that these five cells form a rotation or mirror of its silhouette and do not cover any rooms already laid.
9. Place Z on L3C2, L3C3, L4C2, L5C1, L5C2. Verify that these five cells form a rotation or mirror of its silhouette and do not cover any parts already laid.
10. Place V on L2C1, L2C2, L2C3, L3C1, L4C1. Verify that these five cells form a rotation or mirror of its silhouette and do not cover any parts already laid.
11. The proof of uniqueness explores 980 subpaves and eliminates 634 dead ends where a free box no longer accepts any remaining parts. Only one complete cover remains.
12. To independently check the answer, count five boxes per letter, compare each group to its close rotation/turn silhouette, and then check the clues. To prove uniqueness on paper, take the placement tree by the most constrained box and bar each branch that leaves an isolated box or impossible to cover. The validation script performs this exhaustive exploration; the displayed groups give the winning path.

Common mistakes and false leads

• A mirror is allowed, but not a partial displacement of only one box.
• Two groups of the same range are not necessarily of the same form.
• A cavity of five cells may remain unfilled if its shape does not correspond to any remaining room.

{
  "nodes": 980,
  "dead_ends": 634,
  "max_depth": 8
}

← Return to puzzle SOLO-028