Patches puzzle strategy: think in factors, not guesses

A good Patches puzzle strategy starts from a simple observation: every clue can only become a handful of rectangles. A 6 is 1×6, 2×3, 3×2 or 6×1, and nothing else. On a 4×4 board you can try those in your head almost without noticing. On a 7×7 or 9×9 board, with a dozen or more clues fighting for space, drawing shapes and undoing them quickly becomes tiring.

The method below replaces drawing with counting. For each clue you ask three questions: which rectangle sizes multiply to this number, which of them the clue's shape allows, and which of them actually fit on the board without swallowing another clue. Most clues fail at least one of those tests for most shapes, and that is where your progress comes from.

The number on a clue is the area of its rectangle, so the possible shapes are the factor pairs of that number. It is worth knowing the common ones by heart, because you will use them on every board.

• 2: 1×2 or 2×1.
• 3, 5 and 7: prime, so only a straight line one cell thick.
• 4: 1×4, 2×2 or 4×1.
• 6: 1×6, 2×3, 3×2 or 6×1.
• 8: 1×8, 2×4, 4×2 or 8×1.
• 9: 1×9, 3×3 or 9×1.

Then cut the list down by board size. A line of 8 cannot exist on a 7×7 board, so an 8 there must be 2×4 or 4×2. A 6 on a 5×5 board cannot be a 1×6 line. Big prime clues are the most valuable early: a 7 on a 7×7 board is a full row or a full column through the clue, and the cells around it immediately tell you which.

Patches marks clues as square, tall, wide or freeform, and the shape filter is stricter than it first looks. Tall means taller than it is wide, and wide means wider than it is tall. A square shape is neither, so a tall or wide clue can never be a square.

• Tall 4: the 2×2 square is ruled out, so it is a 1×4 line standing up.
• Wide 4: the same logic leaves only a 4×1 line lying down.
• Tall 9: the 3×3 square is ruled out, so it is a vertical line of nine cells, which only fits on a 9×9 board.
• Wide 8: 4 wide and 2 tall, or 8 wide and 1 tall.
• Square clues: only perfect squares, so a square 4 is always 2×2 and a square 9 is always 3×3.

After this filter, many shaped clues have one or two shapes left. That still leaves the question of position, which is the next step.

A rectangle of a known shape can usually sit in several positions that still cover its clue. Any cell that every position covers belongs to that clue for certain, even before you know where the rectangle ends.

Take a tall 3 in the middle of a column, with free cells above and below it. The line can sit with the clue at its top, middle or bottom. Whatever happens, the clue cell itself is covered, but nothing else is shared yet. Now suppose another clue sits two cells above it. The positions that would reach that clue are illegal, because a rectangle may never hold two clues. Only the positions covering the clue and the two cells below it, or the cell above and the cell below, remain, and the cell directly below the clue is in both. You now own a cell without having drawn the whole shape.

Clues tell you what a rectangle needs, but empty cells tell you what the board needs. Every cell without a clue must be covered by some rectangle, and that rectangle must contain a clue. So pick an awkward empty cell, often one tucked into a corner or wedged between placed shapes, and list every clue whose remaining options could still cover it.

• If only one clue can reach it, that clue's rectangle must include the cell. Remove every option that does not.
• If two clues can reach it, check whether one of them would have to stretch past another clue to get there. Often it would, and you are back to one.
• If no clue can reach it, an earlier rectangle is wrong. Use Undo and look again at your last few placements.

Reach checks are especially useful on larger boards, where many cells are far from any number. A lonely corner cell four steps from the nearest clue can only be collected by a rectangle long enough to get there, and that usually means one specific clue with one specific shape.

The clue numbers always add up to the number of cells on the board. The same accounting works for any region you have sealed off with placed rectangles and the board edge. If a pocket of 10 empty cells contains a 4 and a 6, and no other clue can reach in, those two clues must fill the pocket exactly between them.

Imagine the top-left corner of a 6×6 board. The corner cell holds a wide 3. Two cells below it, in the first column, sits a freeform 4. No other clue sits in the first three rows.

1. Factor list: 3 is prime, so the wide 3 is a 3×1 line lying along the top row. It is in the corner, so it can only run to the right: it covers the first three cells of row one.
2. The first column now has an empty cell in row two, between the 3 and the 4. The 3 cannot reach down, because it is a flat line. So the 4 must cover that cell.
3. A freeform 4 is 1×4, 2×2 or 4×1. A horizontal 4×1 line through the clue would stay in row three and miss row two, so it is gone. That leaves a vertical line or a square, and both must include rows two and three of the first column.
4. Those two cells are now certain, even though the exact shape is not. Fill in the rest of the board around them and come back; a neighboring clue will soon decide between the line and the square.

Notice that nothing here required drawing a shape and hoping. Each step crossed options off a list, and the board did the rest.

• Placing a large freeform clue first. A freeform 8 has many shapes and positions; let the small, shaped clues around it box it in.
• Forgetting that tall and wide exclude squares. A tall 4 that you draw as 2×2 will look fine until the column runs out of room.
• Committing to a shape when you only know its core. Mark the certain cells in your head and move on, rather than drawing the whole rectangle and undoing it later.

Patches starts on 4×4 boards and grows to 9×9 by level 251, and as the boards grow fewer clues show their shape, so counting and reach checks matter more. That makes it a good place to practise this routine one step at a time: factor lists on the early boards, shared cores and reach checks once you reach the 7×7 and 8×8 range.

Undo is free, so testing a deduction costs nothing, and a Hint draws one correct rectangle when you want to check your reasoning. Free hints refill over time. Boards are generated on your device and play offline, so you can work through a few on a commute and see the method get faster with every board.