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Shikaku Rectangle Puzzles: Complete Guide to Area Division Logic

Master the Nikoli puzzle where numbered cells define rectangular regions that cover the entire grid.

1. What Is Shikaku?

Shikaku (四角に切れ, literally "cut into quadrilaterals") is a rectangle division logic puzzle created by the Japanese puzzle company Nikoli. It's also known as "Boxes" or "Rectangles" in English.

The puzzle presents a rectangular grid with some cells containing numbers. Your objective is to divide the entire grid into rectangular regions such that each region contains exactly one numbered cell, and that number equals the total area (number of cells) of the region.

Shikaku's unique appeal lies in the interplay between area constraints (each number defines region size) and geometric constraints (regions must be rectangles that tile the grid perfectly). This combination creates elegant deduction chains that are satisfying to unravel.

<div class="tip-box">
  <strong>💡 Why Two Names?</strong><br>
  The Japanese name "Shikaku" (四角) means "square" or "rectangle." The full phrase "Shikaku ni kire" means "cut into rectangles." In English, it's commonly called "Boxes" or simply "Rectangles." All refer to the same puzzle.
</div>

2. The Three Rules of Shikaku

Every Shikaku puzzle, regardless of grid size or difficulty, follows exactly three rules. Master these and you can solve any puzzle.

<div class="rule-box">
  <strong>Rule 1 — Rectangle Shape:</strong> Every region must be a rectangle (or square). No L-shapes, no irregular polygons. Each region's boundaries must align with the grid lines.
</div>

<div class="rule-box">
  <strong>Rule 2 — Single Number per Region:</strong> Each rectangular region contains exactly one numbered cell. The number in that cell equals the total area (number of cells) of the entire region.
</div>

<div class="rule-box">
  <strong>Rule 3 — Complete Coverage:</strong> Every cell in the grid must belong to exactly one region. No cells can be left uncovered, and regions cannot overlap.
</div>

<p>These three rules interact in powerful ways. Rule 2 provides area constraints, Rule 1 forces rectangular geometry, and Rule 3 ensures complete tiling. The art of solving Shikaku lies in using numbered cells as anchor points and then reasoning about where rectangular boundaries must fall to satisfy all three rules simultaneously.</p>

3. Anatomy of a Shikaku Grid

Understanding the structure of a Shikaku grid is essential before applying solving techniques. Let's dissect a sample grid:

<svg viewBox="0 0 500 400" role="img" aria-label="Shikaku grid anatomy showing numbered cells and rectangular regions" title="Shikaku Grid Anatomy">

  <rect x="50" y="50" width="400" height="300" fill="#fff" stroke="#2a3545" stroke-width="2"></rect>

  <g stroke="#ccc" stroke-width="0.5">

    <line x1="100" y1="50" x2="100" y2="350"></line>
    <line x1="150" y1="50" x2="150" y2="350"></line>
    <line x1="200" y1="50" x2="200" y2="350"></line>
    <line x1="250" y1="50" x2="250" y2="350"></line>
    <line x1="300" y1="50" x2="300" y2="350"></line>
    <line x1="350" y1="50" x2="350" y2="350"></line>
    <line x1="400" y1="50" x2="400" y2="350"></line>

    <line x1="50" y1="100" x2="450" y2="100"></line>
    <line x1="50" y1="150" x2="450" y2="150"></line>
    <line x1="50" y1="200" x2="450" y2="200"></line>
    <line x1="50" y1="250" x2="450" y2="250"></line>
    <line x1="50" y1="300" x2="450" y2="300"></line>
  </g>

  <rect x="50" y="50" width="150" height="100" fill="rgba(79,195,247,0.2)" stroke="#4fc3f7" stroke-width="2"></rect>
  <rect x="200" y="50" width="250" height="100" fill="rgba(129,199,132,0.2)" stroke="#81c784" stroke-width="2"></rect>
  <rect x="50" y="150" width="100" height="200" fill="rgba(255,183,77,0.2)" stroke="#ffb74d" stroke-width="2"></rect>
  <rect x="150" y="150" width="200" height="100" fill="rgba(239,83,80,0.2)" stroke="#ef5350" stroke-width="2"></rect>
  <rect x="350" y="150" width="100" height="200" fill="rgba(171,71,188,0.2)" stroke="#ab47bc" stroke-width="2"></rect>
  <rect x="150" y="250" width="200" height="100" fill="rgba(255,241,118,0.3)" stroke="#fff176" stroke-width="2"></rect>

  <circle cx="125" cy="100" r="18" fill="#fff" stroke="#333" stroke-width="2"></circle>
  <text x="125" y="106" text-anchor="middle" fill="#333" font-size="16" font-weight="bold">6</text>

  <circle cx="325" cy="100" r="18" fill="#fff" stroke="#333" stroke-width="2"></circle>
  <text x="325" y="106" text-anchor="middle" fill="#333" font-size="16" font-weight="bold">10</text>

  <circle cx="100" cy="250" r="18" fill="#fff" stroke="#333" stroke-width="2"></circle>
  <text x="100" y="256" text-anchor="middle" fill="#333" font-size="16" font-weight="bold">8</text>

  <circle cx="250" cy="200" r="18" fill="#fff" stroke="#333" stroke-width="2"></circle>
  <text x="250" y="206" text-anchor="middle" fill="#333" font-size="16" font-weight="bold">8</text>

  <circle cx="400" cy="250" r="18" fill="#fff" stroke="#333" stroke-width="2"></circle>
  <text x="400" y="256" text-anchor="middle" fill="#333" font-size="16" font-weight="bold">8</text>

  <circle cx="250" cy="300" r="18" fill="#fff" stroke="#333" stroke-width="2"></circle>
  <text x="250" y="306" text-anchor="middle" fill="#333" font-size="16" font-weight="bold">8</text>

  <text x="125" y="75" text-anchor="middle" fill="#4fc3f7" font-size="11">Region: 6 cells</text>
  <text x="325" y="75" text-anchor="middle" fill="#81c784" font-size="11">Region: 10 cells</text>
  <text x="75" y="275" text-anchor="middle" fill="#ffb74d" font-size="11">8 cells</text>
  <text x="250" y="175" text-anchor="middle" fill="#ef5350" font-size="11">8 cells</text>
  <text x="425" y="275" text-anchor="middle" fill="#ab47bc" font-size="11">8 cells</text>
  <text x="250" y="325" text-anchor="middle" fill="#fff176" font-size="11">8 cells</text>
</svg>

<p>Key elements of a Shikaku grid:</p>
<ul>
  <li><strong>Numbered cells:</strong> Cells containing numbers (1–16+ depending on grid size). Each number equals the area of its region.</li>
  <li><strong>Rectangular regions:</strong> Each numbered cell is the center of a rectangle whose area equals the number. Regions are typically shown with colored backgrounds to distinguish them.</li>
  <li><strong>Grid boundaries:</strong> The outer edges of the grid form natural boundaries for some regions.</li>
  <li><strong>Region borders:</strong> Thick lines separate different rectangular regions. Every cell belongs to exactly one region.</li>
</ul>

<table>
  <thead><tr><th>Grid Size</th><th>Numbers (typical)</th><th>Difficulty</th><th>Typical Solve Time</th></tr></thead>
  <tbody>
    <tr><td>7×7</td><td>6–10</td><td>Easy</td><td>2–5 minutes</td></tr>
    <tr><td>10×10</td><td>10–15</td><td>Medium</td><td>5–12 minutes</td></tr>
    <tr><td>12×12</td><td>15–20</td><td>Hard</td><td>12–25 minutes</td></tr>
    <tr><td>15×15</td><td>20–30</td><td>Expert</td><td>25+ minutes</td></tr>
  </tbody>
</table>

4. Six Solving Techniques

These techniques progress from beginner to advanced. Master the first three to solve easy puzzles; all six are needed for hard grids.

<h3>Technique 1: Start with Large Numbers</h3>
<p>Large numbers (8, 10, 12, etc.) have fewer possible rectangular configurations. For example:</p>
<ul>
  <li>A "12" can only be: 1×12, 2×6, 3×4, 4×3, 6×2, or 12×1 (but must fit within grid boundaries).</li>
  <li>A "10" can only be: 1×10, 2×5, 5×2, or 10×1.</li>
  <li>An "8" can only be: 1×8, 2×4, 4×2, or 8×1.</li>
</ul>
<p>Scan the grid for large numbers and enumerate their possible rectangle orientations. Often, grid boundaries or neighboring numbers force a specific orientation.</p>

<div class="tip-box">
  <strong>💡 Pro Tip:</strong> A number equal to a prime (2, 3, 5, 7, 11, 13) can only form rectangles of 1×N or N×1. This severely limits placement options and often forces immediate deductions.
</div>

<h3>Technique 2: Corner and Edge Constraints</h3>
<p>Numbers near corners or edges have restricted rectangle orientations. A "6" in a corner can only extend in 2 directions (not 4), and grid boundaries limit how far it can stretch.</p>
<p>Similarly, numbers along edges can't extend beyond the grid, which eliminates some rectangle orientations. Use these geometric constraints to narrow down possibilities.</p>

<svg viewBox="0 0 500 200" role="img" aria-label="Diagram showing corner and edge constraints in Shikaku" title="Corner and Edge Constraints">
  <text x="250" y="20" text-anchor="middle" fill="#4fc3f7" font-size="14" font-weight="bold">Corner and Edge Constraints</text>

  <text x="80" y="45" text-anchor="middle" fill="#8899aa" font-size="12">Corner "4"</text>
  <g transform="translate(20, 55)">
    <rect x="0" y="0" width="120" height="120" fill="#fff" stroke="#2a3545" stroke-width="1"></rect>
    <g stroke="#ccc" stroke-width="0.5">
      <line x1="40" y1="0" x2="40" y2="120"></line>
      <line x1="80" y1="0" x2="80" y2="120"></line>
      <line x1="0" y1="40" x2="120" y2="40"></line>
      <line x1="0" y1="80" x2="120" y2="80"></line>
    </g>

    <rect x="0" y="0" width="80" height="40" fill="rgba(129,199,132,0.3)" stroke="#81c784" stroke-width="2"></rect>
    <rect x="0" y="0" width="40" height="80" fill="rgba(79,195,247,0.3)" stroke="#4fc3f7" stroke-width="2"></rect>
    <circle cx="20" cy="20" r="15" fill="#fff" stroke="#333" stroke-width="2"></circle>
    <text x="20" y="26" text-anchor="middle" fill="#333" font-size="14" font-weight="bold">4</text>
  </g>

  <text x="180" y="115" text-anchor="middle" fill="#ffb74d" font-size="24">→</text>

  <text x="320" y="45" text-anchor="middle" fill="#8899aa" font-size="12">Edge "6"</text>
  <g transform="translate(220, 55)">
    <rect x="0" y="0" width="200" height="120" fill="#fff" stroke="#2a3545" stroke-width="1"></rect>
    <g stroke="#ccc" stroke-width="0.5">
      <line x1="40" y1="0" x2="40" y2="120"></line>
      <line x1="80" y1="0" x2="80" y2="120"></line>
      <line x1="120" y1="0" x2="120" y2="120"></line>
      <line x1="160" y1="0" x2="160" y2="120"></line>
      <line x1="0" y1="40" x2="200" y2="40"></line>
      <line x1="0" y1="80" x2="200" y2="80"></line>
    </g>

    <rect x="60" y="0" width="120" height="40" fill="rgba(129,199,132,0.3)" stroke="#81c784" stroke-width="2"></rect>
    <rect x="60" y="0" width="60" height="80" fill="rgba(79,195,247,0.3)" stroke="#4fc3f7" stroke-width="2"></rect>
    <circle cx="100" cy="20" r="15" fill="#fff" stroke="#333" stroke-width="2"></circle>
    <text x="100" y="26" text-anchor="middle" fill="#333" font-size="14" font-weight="bold">6</text>
  </g>
</svg>

<h3>Technique 3: Area Elimination</h3>
<p>Once you place a rectangle, all cells within it are claimed. This eliminates those cells from consideration by other numbered cells. Use this "area elimination" to narrow down where other rectangles can go.</p>
<p>For example, if a "6" in the top-left corner must be a 2×3 rectangle, those 6 cells are now occupied. Any other numbered cell that could have extended into that area must now use a different orientation.</p>

<h3>Technique 4: Forced Single Orientations</h3>
<p>Some numbers have only one possible rectangle orientation due to grid boundaries and neighboring constraints. For example:</p>
<ul>
  <li>A "5" in a corner can only be 1×5 or 5×1. If one orientation would overlap with a forced rectangle from another number, the other orientation is forced.</li>
  <li>A "3" along an edge with limited space might only fit as 1×3, not 3×1.</li>
</ul>
<p>Scan for these forced single orientations—they cascade quickly once you identify them.</p>

<h3>Technique 5: Region Interaction</h3>
<p>Look at how numbered cells interact with each other. If two numbers are close together, their rectangles might compete for the same cells. Use this competition to deduce boundaries.</p>
<p>For example, if a "4" and a "6" are adjacent, and the "4" must be a 2×2 square, the "6" cannot extend into that 2×2 area. This forces the "6" into a different orientation.</p>

<h3>Technique 6: Complete Tiling Verification</h3>
<p>As you place rectangles, periodically check that every cell is covered by exactly one region. If you notice an uncovered cell that can only belong to one remaining numbered cell, that cell's rectangle is forced.</p>
<p>Similarly, if placing a rectangle would leave an uncovered cell that no remaining number can reach, that rectangle placement is invalid. Use complete tiling as a constraint throughout the solve.</p>

<table>
  <thead><tr><th>Technique</th><th>Difficulty</th><th>When to Use</th><th>Key Insight</th></tr></thead>
  <tbody>
    <tr><td>1. Large Numbers</td><td>Beginner</td><td>Always, first step</td><td>Fewer rectangle orientations</td></tr>
    <tr><td>2. Corner/Edge</td><td>Beginner</td><td>Numbers near boundaries</td><td>Grid limits orientations</td></tr>
    <tr><td>3. Area Elimination</td><td>Easy</td><td>After placing initial rectangles</td><td>Claimed cells eliminate options</td></tr>
    <tr><td>4. Forced Orientations</td><td>Easy</td><td>When only one shape fits</td><td>Boundaries force single option</td></tr>
    <tr><td>5. Region Interaction</td><td>Medium</td><td>Numbers close together</td><td>Competition for cells</td></tr>
    <tr><td>6. Complete Tiling</td><td>Medium</td><td>Near end of solve</td><td>Every cell must be covered</td></tr>
  </tbody>
</table>

5. Step-by-Step Solving Walkthrough

Let's walk through a simplified Shikaku solve to see how the techniques interact in practice.

<div class="tip-box">
  <strong>💡 Strategy:</strong> Always begin with Technique 1 (large numbers) and Technique 2 (corner/edge constraints). These give you the most information with the least effort. Then move to area elimination (Technique 3) and forced orientations (Technique 4).
</div>

<p><strong>Step 1 — Scan for large numbers.</strong> Look for numbers like 8, 10, 12, etc. Enumerate their possible rectangle orientations (e.g., 8 can be 1×8, 2×4, 4×2, or 8×1). Grid boundaries and neighboring numbers often force a specific orientation.</p>
<p><strong>Step 2 — Check corner and edge numbers.</strong> Numbers near corners or edges have restricted orientations. A "6" in a corner can only extend in 2 directions, and grid boundaries limit how far it can stretch. Use these geometric constraints to narrow down possibilities.</p>
<p><strong>Step 3 — Apply area elimination.</strong> Once you place a rectangle, all cells within it are claimed. This eliminates those cells from consideration by other numbered cells. Use this to force other rectangles into different orientations.</p>
<p><strong>Step 4 — Look for forced single orientations.</strong> Some numbers have only one possible rectangle orientation due to grid boundaries and neighboring constraints. A "5" in a corner might only fit as 1×5 if 5×1 would overlap with a forced rectangle.</p>
<p><strong>Step 5 — Analyze region interactions.</strong> Look at how numbered cells interact with each other. If two numbers are close together, their rectangles might compete for the same cells. Use this competition to deduce boundaries.</p>
<p><strong>Step 6 — Verify complete tiling.</strong> Periodically check that every cell is covered by exactly one region. If you notice an uncovered cell that can only belong to one remaining numbered cell, that cell's rectangle is forced. Iterate until the entire grid is tiled.</p>

6. Shikaku vs Other Nikoli Puzzles

Shikaku belongs to Nikoli's family of logic puzzles. Understanding how it relates to similar puzzles helps you choose which to learn next.

<table>
  <thead><tr><th>Feature</th><th>Shikaku</th><th><a href="/guides/nurikabe-island-puzzle-rules/" style="color:var(--accent)">Nurikabe</a></th><th><a href="/guides/kakuro-number-crosswords/" style="color:var(--accent)">Kakuro</a></th><th><a href="/guides/heyewake-room-puzzle-tips/" style="color:var(--accent)">Heyawake</a></th></tr></thead>
  <tbody>
    <tr><td>Grid</td><td>Rectangular with numbered cells</td><td>Rectangular, no rooms</td><td>Cross-shaped, divided into runs</td><td>Rectangular, divided into rooms</td></tr>
    <tr><td>Core constraint</td><td>Rectangle area = number</td><td>Island size + connected sea</td><td>Sum equals clue</td><td>Room count + connectivity</td></tr>
    <tr><td>Unique rule</td><td>All regions must be rectangles</td><td>Sea cannot have 2×2 block</td><td>No repeat digit in a run</td><td>Three-room rule</td></tr>
    <tr><td>Cell states</td><td>Belongs to a region</td><td>Binary (black/white)</td><td>Digits 1–9</td><td>Binary (black/white)</td></tr>
    <tr><td>Difficulty curve</td><td>Gentle → moderate</td><td>Moderate → steep</td><td>Moderate → steep</td><td>Gentle → steep</td></tr>
    <tr><td>Best for</td><td>Spatial reasoning, geometry</td><td>Island/sea topology</td><td>Arithmetic + placement</td><td>Room-based spatial logic</td></tr>
  </tbody>
</table>

<p>If you enjoyed <a href="/guides/hashiwokakero-bridge-puzzles/" class="internal-link">Hashiwokakero</a>, Shikaku is a natural next step—both involve spatial reasoning and geometric constraints, but Shikaku adds area-based rectangle division. If you prefer arithmetic, try <a href="/guides/kakuro-number-crosswords/" class="internal-link">Kakuro</a>.</p>

7. Brain Training Benefits

Regular Shikaku practice exercises multiple cognitive faculties simultaneously:

  • Spatial reasoning: Mentally rotating and comparing rectangle orientations, similar to skills used in Sokoban planning and real-world geometry.
  • Constraint satisfaction: Holding three simultaneous rules in working memory and checking each candidate rectangle against all of them.
  • Cascading deduction: Following chains of "if A then B then C" across the grid—the same skill used in critical thinking puzzles and Minesweeper.
  • Working memory: Tracking which cells are claimed, which numbers are satisfied, and which orientations are eliminated—across multiple interacting constraint systems.
  • Pattern recognition: Recognizing common rectangle configurations for specific numbers, accelerating solve speed over time.
<div class="tip-box">
  <strong>💡 Transfer effect:</strong> Research shows that logic puzzle practice improves performance on unrelated tasks requiring working memory and executive function. Shikaku's unique combination of area constraints and geometric rectangle requirements may provide broader cognitive transfer than puzzles with only one type of constraint.
</div>

<p>Pair Shikaku with other brain training activities on our site for a well-rounded cognitive workout: <a href="/guides/best-brain-training-games/" class="internal-link">best brain training games</a>, <a href="/guides/brain-games-focus/" class="internal-link">focus and concentration games</a>, or try our <a href="/games/sudoku/">Sudoku</a> for complementary number logic.</p>

8. FAQ

<div class="faq-item">
  <h3>What are the rules of Shikaku?</h3>
  <p>Three rules: (1) Every region must be a rectangle (or square) with boundaries aligned to grid lines. (2) Each region contains exactly one numbered cell, and that number equals the region's area. (3) Every cell in the grid must belong to exactly one region—complete coverage with no overlaps.</p>
</div>

<div class="faq-item">
  <h3>What does "Shikaku" mean?</h3>
  <p>Shikaku (四角) is Japanese for "square" or "rectangle." The full name "Shikaku ni kire" (四角に切れ) means "cut into rectangles." In English, it's also called "Boxes" or "Rectangles."</p>
</div>

<div class="faq-item">
  <h3>Can regions be non-rectangular?</h3>
  <p>No. Every region must be a rectangle (including squares). L-shapes, T-shapes, and other irregular polygons are not allowed. This geometric constraint is what makes Shikaku unique among Nikoli puzzles.</p>
</div>

<div class="faq-item">
  <h3>What's the best grid size for beginners?</h3>
  <p>Start with 7×7 grids with 6–10 numbered cells. These let you practice the basic deduction techniques (large numbers, corner constraints) without overwhelming geometric complexity. Move to 10×10 or larger once you're comfortable with area elimination and region interactions.</p>
</div>

<div class="faq-item">
  <h3>How do I know which rectangle orientation to use?</h3>
  <p>Enumerate possible orientations based on the number (e.g., 6 can be 1×6, 2×3, 3×2, or 6×1). Then use grid boundaries, neighboring numbers, and area elimination to narrow down which orientations are possible. Often, only one orientation fits without conflicts.</p>
</div>

<div class="faq-item">
  <h3>Can a region extend outside the grid?</h3>
  <p>No. All regions must fit entirely within the grid boundaries. Grid edges form natural boundaries that limit how far rectangles can extend. Use these boundaries as constraints when determining possible orientations.</p>
</div>

<div class="faq-item">
  <h3>How is Shikaku different from Nurikabe?</h3>
  <p>Nurikabe requires you to shade cells so that white "islands" have specific sizes and the black "sea" is connected. Shikaku requires you to divide the entire grid into rectangles where each rectangle's area equals its number. Nurikabe is about island/sea topology; Shikaku is about geometric rectangle division.</p>
</div>

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