How to Solve Polyfit: Fill the Footprint Without Gaps

Polyfit gives you a bank of colorful pieces and an amber footprint. Drag every piece onto amber cells, tap a bank piece to rotate it 90 degrees, and fill the footprint with no gaps or overlaps. Placed pieces can be picked up and repositioned. The controls rotate; they do not reflect pieces, so a mirror image is not automatically available.

Packing puzzles punish vague experimentation. A better approach is to rank pieces by how constrained they are, then protect the rare spaces that only those pieces can fill.

First pass: inventory the bank

Find the long pieces, the pieces with narrow hooks, and any asymmetric shapes. Rotate them in the bank and count their distinct orientations. A symmetric piece may have only two meaningful rotations, while an awkward branch may have four.

The fewer usable orientations a piece has inside the footprint, the earlier you should consider placing it.

Second pass: inspect the footprint's rare spaces

Concave corners, one-cell notches, and narrow corridors cannot be filled by every piece. Match those locations to the stubs and turns in the constrained pieces. Flat outer edges are usually easier to fill later with flexible shapes.

If a notch has only one possible visitor, that relationship is more useful than any guess in the open center.

Place anchors, then examine the cavities

After seating one constrained piece, stop. Look at each remaining empty region as its own packing problem. Is its area compatible with the sizes of the pieces left in the bank? Does it contain a one-cell corner that none of them can reach?

A placement can fit perfectly by itself and still be wrong because it leaves an impossible cavity.

Prefer clean boundaries

Good placements tend to leave rectangular blocks, straight corridors, or cavities that visibly match a remaining piece. Jagged one-cell fjords are suspicious unless a specific piece is waiting to fill them.

This is not an absolute rule, but it is an excellent tie-breaker when the same flexible piece fits in several places.

Rotation discipline

Rotate with a target orientation in mind. If you need the long arm horizontal and the short arm pointing down, get that shape in the bank before dragging. Randomly rotating after every failed drop makes it hard to remember which arrangements you have actually tested.

Since reflection is not a control, distinguish a rotation from a mirror image. Some apparent matches will never line up no matter how many times you tap.

How to backtrack without starting over

• Keep the most constrained anchor unless its cavity was already suspicious.
• Pick up the most recently placed flexible piece.
• Try that piece in the next clean cavity rather than moving every piece at once.
• Recount the area of each leftover region after the change.

Use area as a fast impossibility test

Count the cells in a cavity and compare that total with the sizes of the unused pieces. If a seven-cell cavity remains but your available pieces contain four and four cells, no arrangement can fill it. You have learned something decisive without trying a single orientation.

Area is necessary but not sufficient. An eight-cell cavity and two four-cell pieces have the right total, yet the cavity may include a narrow hook neither piece can reach. After the count passes, compare the cavity's corners and corridors with the actual shapes.

Which piece should move first?

When a packing fails, rank the placed pieces by flexibility. A straight piece along a long flat edge may have several alternative homes, while a branched piece filling a notch may have only one. Move the flexible piece first. This preserves the deductions that gave the board its structure.

Polyfit is less about finding a magical first move than about preserving feasible leftovers. Place one piece, inspect what remains, and make every cavity earn its place on the board.

A new Polyfit puzzle is available every day. For more repetitions with the same rules, open Practice and choose Polyfit.