HOW RECTANGLE CONSTRAINTS FORCE PLACEMENTS
Every numbered cell must sit inside a rectangle whose area equals that number. This one constraint does most of the solving work — but it interacts with the grid boundaries and other rectangles in ways that aren't always obvious at first.
The key insight: a number constrains both the shape and the position of its rectangle. A cell containing "4" must be inside a 1×4, 4×1, or 2×2 rectangle. A cell containing "6" must be inside a 1×6, 6×1, 2×3, or 3×2 rectangle. As grids get larger, some shapes become geometrically impossible given the cell's position — and ruling out impossible shapes directly tells you what the valid shapes are.
TECHNIQUE 1: CORNER AND EDGE CONSTRAINTS
Cells near corners or edges have fewer valid rectangle orientations. A numbered cell in the corner of the grid can only extend in two directions; one near an edge can only extend in three.
Start with corner cells. A number in a corner with value N can only form certain shapes — and if the grid is small enough relative to N, only one shape fits. A 1 in a corner is a single cell. A 4 in a corner on a 5-wide grid can be 1×4, 4×1, or 2×2 — but if the corner is at the very edge, the 4×1 pointing left or the 1×4 pointing up might run off the grid. Eliminate impossible orientations and you narrow the options quickly.
Edge cells (not corners) are next. The number of valid shapes is still restricted by which directions the rectangle can extend. A 6 on the top edge of a grid can be 1×6 (horizontal), 2×3, or 3×2 — but NOT 6×1 (would extend above the grid). Ruling out impossible orientations is free information.
TECHNIQUE 2: LARGE NUMBERS FIRST
Large numbers — cells where the area is close to the grid's total size — have very few valid rectangle shapes and are usually the most constrained placements. On a 7×7 grid (49 cells), a numbered cell with value 12 or higher covers a significant fraction of the grid, and only a few configurations fit without overlapping other cells.
Solve large numbers before small ones. A large rectangle, once placed, subdivides the remaining space and creates new edge/boundary constraints for the smaller rectangles nearby. Small numbers are flexible — they can fit in many positions — so solving them first gives you fewer forced constraints.
This is the opposite of the instinct most beginners have. The temptation is to solve the "easy" small cells (1s, 2s) first, but their flexibility means many valid placements exist and each choice is a guess that needs to be undone later. Large cells do the heavy lifting.
TECHNIQUE 3: OVERLAP ANALYSIS
For a given numbered cell, list all valid rectangle placements. Find the cells that appear in EVERY valid placement. Those cells are guaranteed to be part of that rectangle — you can mark them as belonging to it regardless of which valid shape is ultimately correct.
Example: a cell containing "4" at position (row 3, col 4) in an 8×8 grid has several valid placements: 1×4 horizontal (cols 1–4, or 2–5, or 3–6, or 4–7, or 5–8), 4×1 vertical (rows 1–4, or 2–5, or 3–6, or 4–7, or 5–8), 2×2 (four possible positions). If all horizontal placements include col 4, then col 4 in the relevant rows is guaranteed to be in this rectangle — mark it. If, further, a nearby rectangle has already claimed some of those positions, the remaining valid placements are reduced and the overlap might become total (forcing).
Overlap analysis is the core technique for intermediate Shikaku. It does not require guessing — it derives certainty from the intersection of possibilities.
TECHNIQUE 4: BOUNDARY FORCING
Once some rectangles are placed, their boundaries become new constraints. A cell that is adjacent to a placed rectangle boundary cannot extend into it — it must form a rectangle that stays on the other side. This is boundary forcing.
Check every numbered cell adjacent to a placed rectangle boundary after each placement. The new boundary may rule out orientations that were previously valid, creating new forced placements or narrowing overlap analysis.
Boundary forcing cascades naturally as the grid fills in — later placements are often trivially forced by the accumulated boundaries of earlier ones. This is why solving the most-constrained cells first (corners, edges, large numbers) is so effective: each one creates more boundaries that force the next.
TECHNIQUE 5: PARITY AND AREA ACCOUNTING
The total area of all rectangles must equal the total number of cells in the grid. On a 7×7 grid, the numbered cells' values must sum to 49. This is a useful sanity check, but it also creates a forward-inference tool.
If a region of the grid is bounded on all sides by placed rectangles and grid edges, and the remaining area doesn't match the numbered cells inside it, you have a contradiction — backtrack. If the remaining area does match and only one configuration of the remaining cells' rectangles fills the region exactly, that configuration is forced.
Area accounting is particularly useful near the end of a solve, when large portions of the grid are already claimed and only small regions remain. The available shapes are tightly constrained by both the remaining area and the boundaries of what's already placed.
THE SOLVING ORDER THAT WORKS
- Step 1: Identify corner and edge cells — list their valid rectangle orientations.
- Step 2: Solve large numbers first — they have the fewest valid shapes and create the most constraints for their neighbours.
- Step 3: After each placement, run boundary forcing on adjacent cells.
- Step 4: For cells not yet forced, run overlap analysis — mark cells in every valid placement as guaranteed members.
- Step 5: Use area accounting on bounded regions to verify or force endgame placements.
Most beginner and intermediate Shikaku grids resolve entirely in steps 1–3. Overlap analysis is mainly needed on intermediate and hard grids; area accounting is a tool for the endgame. The key is trusting the spatial reasoning rather than guessing orientations — every forced placement comes from a constraint, and chasing those constraints is how Shikaku is meant to be solved.