<h1>Number Link Connect Puzzles: Master Path-Routing Logic</h1><p class="subtitle">Connect every matching pair, fill every cell, and never let paths cross — the complete strategy guide.</p><div class="quick-answer"> <h2>⚡ Quick Answer</h2> <ul> <li><strong>What is Number Link?</strong> A Nikoli logic puzzle where you draw paths connecting matching numbered endpoints on a grid</li> <li><strong>Key rules:</strong> Paths go horizontally/vertically, never cross, and every cell must be filled</li> <li><strong>Core strategy:</strong> Start from corners/edges, detect bottlenecks, use parity reasoning</li> <li><strong>Also known as:</strong> Numberlink, Arukone, Flow Free, Nanbarinku</li> <li><strong>Skill built:</strong> Spatial reasoning, path planning, constraint satisfaction</li> </ul></div><h2>What Is Number Link?</h2><p>Number Link (also called <strong>Numberlink</strong>, <strong>Arukone</strong>, or <strong>Nanbarinku</strong>) is a classic Japanese logic puzzle published by Nikoli since 1993. The premise is deceptively simple: you see a grid with pairs of matching numbers, and your job is to draw continuous paths connecting each pair so that every cell in the grid is filled, no paths cross, and no path runs through another numbered endpoint.</p><p>If you have ever played <strong>Flow Free</strong> on your phone, you already know Number Link — Flow Free is essentially Number Link adapted for mobile. The core mechanic is identical: connect matching endpoints with non-crossing paths that fill the entire board. The difference is that paper-based Number Link puzzles are designed with exactly one solution, while casual app versions may accept any valid completion.</p><p>Number Link belongs to the same Nikoli family as <a href="/guides/kakuro-number-crosswords/" class="internal-link">Kakuro</a>, <a href="/guides/sokoban-box-pushing-puzzles/" class="internal-link">Sokoban</a>, and <a href="/guides/nurikabe-island-puzzle-rules/" class="internal-link">Nurikabe</a>. All of these puzzles share a design philosophy: simple rules, pure logic, unique solutions, and deep strategic complexity emerging from minimal constraints.</p><h2>The Three Rules of Number Link</h2><div class="rules-grid"> <div class="rule-card"> <div class="rule-num">1</div> <h3>Connect Matching Pairs</h3> <p>Each number appears exactly twice. Draw a continuous path from one to the other, moving only horizontally or vertically.</p> </div> <div class="rule-card"> <div class="rule-num">2</div> <h3>No Crossing or Overlap</h3> <p>Paths cannot cross each other, share cells, or branch. Each cell belongs to exactly one path — or is a numbered endpoint.</p> </div> <div class="rule-card"> <div class="rule-num">3</div> <h3>Fill Every Cell</h3> <p>The most overlooked rule. Every cell in the grid must be part of some path. No empty spaces allowed — this is what makes the puzzle challenging.</p> </div></div><p>These three rules are the entire game. Every technique you will learn flows from the tension between them: you must connect specific pairs (Rule 1) without blocking other paths (Rule 2) while covering every cell (Rule 3). Most mistakes come from focusing on connection and forgetting the fill-every-cell constraint.</p><figure class="diagram-box"> <svg viewBox="0 0 400 200" xmlns="http://www.w3.org/2000/svg" role="img" aria-label="A simple 5x5 Number Link example showing two connected pairs with paths filling the grid" style="max-width:100%;height:auto;"> <title>5x5 Number Link example with pairs 1-1 and 2-2</title> <g stroke="#2d3140" stroke-width="1"> <line x1="50" y1="20" x2="50" y2="180"></line> <line x1="98" y1="20" x2="98" y2="180"></line> <line x1="146" y1="20" x2="146" y2="180"></line> <line x1="194" y1="20" x2="194" y2="180"></line> <line x1="242" y1="20" x2="242" y2="180"></line> <line x1="20" y1="20" x2="250" y2="20"></line> <line x1="20" y1="52" x2="250" y2="52"></line> <line x1="20" y1="84" x2="250" y2="84"></line> <line x1="20" y1="116" x2="250" y2="116"></line> <line x1="20" y1="148" x2="250" y2="148"></line> <line x1="20" y1="180" x2="250" y2="180"></line> </g> <path d="M 35 36 L 35 68 L 83 68 L 83 100 L 131 100" stroke="#6c5ce7" stroke-width="6" fill="none" stroke-linecap="round" stroke-linejoin="round" opacity="0.7"></path> <path d="M 179 36 L 179 68 L 227 68 L 227 100 L 227 132 L 179 132 L 179 164" stroke="#00b894" stroke-width="6" fill="none" stroke-linecap="round" stroke-linejoin="round" opacity="0.7"></path> <circle cx="35" cy="36" r="14" fill="#6c5ce7"></circle> <text x="35" y="41" text-anchor="middle" fill="white" font-size="14" font-weight="bold">1</text> <circle cx="131" cy="100" r="14" fill="#6c5ce7"></circle> <text x="131" y="105" text-anchor="middle" fill="white" font-size="14" font-weight="bold">1</text> <circle cx="179" cy="36" r="14" fill="#00b894"></circle> <text x="179" y="41" text-anchor="middle" fill="white" font-size="14" font-weight="bold">2</text> <circle cx="179" cy="164" r="14" fill="#00b894"></circle> <text x="179" y="169" text-anchor="middle" fill="white" font-size="14" font-weight="bold">2</text> <text x="300" y="60" fill="#9ca3bf" font-size="12">Every cell filled</text> <text x="300" y="80" fill="#9ca3bf" font-size="12">No paths cross</text> <text x="300" y="100" fill="#9ca3bf" font-size="12">Pairs connected</text> </svg> <figcaption>A solved 5×5 Number Link — two pairs (1-1, 2-2) connected with non-crossing paths covering every cell</figcaption></figure><h2>Essential Solving Techniques</h2><p>Every Number Link puzzle is solvable by pure logic. Here are the techniques you need, ordered from most basic to most advanced.</p><h3>Technique 1: Start From Corners and Edges</h3><p>Endpoints near corners and edges have fewer possible exit directions. A corner endpoint has only 2 exits, an edge endpoint has 3, while a center endpoint has 4. This geometric constraint means corner/edge paths are determined faster.</p><p>When a numbered pair sits in a corner with limited exits, their path direction is often forced immediately. Always scan corners first.</p><figure class="diagram-box"> <svg viewBox="0 0 400 200" xmlns="http://www.w3.org/2000/svg" role="img" aria-label="Diagram showing corner forcing in Number Link — a pair in the corner with only two possible exit directions" style="max-width:100%;height:auto;"> <title>Corner forcing technique — endpoint in corner has only two exits</title> <g stroke="#2d3140" stroke-width="1"> <rect x="40" y="20" width="48" height="48" fill="none"></rect> <rect x="88" y="20" width="48" height="48" fill="none"></rect> <rect x="136" y="20" width="48" height="48" fill="none"></rect> <rect x="40" y="68" width="48" height="48" fill="none"></rect> <rect x="88" y="68" width="48" height="48" fill="none"></rect> <rect x="136" y="68" width="48" height="48" fill="none"></rect> <rect x="40" y="116" width="48" height="48" fill="none"></rect> <rect x="88" y="116" width="48" height="48" fill="none"></rect> <rect x="136" y="116" width="48" height="48" fill="none"></rect> </g> <circle cx="64" cy="44" r="16" fill="#6c5ce7"></circle> <text x="64" y="49" text-anchor="middle" fill="white" font-size="14" font-weight="bold">3</text> <path d="M 64 60 L 64 88" stroke="#e17055" stroke-width="3" stroke-dasharray="5,3" marker-end="url(#arrow-red)"></path> <path d="M 80 44 L 108 44" stroke="#e17055" stroke-width="3" stroke-dasharray="5,3" marker-end="url(#arrow-red)"></path> <text x="28" y="48" fill="#e17055" font-size="11">✕ wall</text> <text x="60" y="14" fill="#e17055" font-size="11">✕ wall</text> <defs><marker id="arrow-red" markerWidth="8" markerHeight="6" refX="8" refY="3" orient="auto"><path d="M0,0 L8,3 L0,6" fill="#e17055"></path></marker></defs> <text x="210" y="50" fill="#e4e7f0" font-size="13" font-weight="bold">Corner Forcing</text> <text x="210" y="75" fill="#9ca3bf" font-size="12">Corner endpoint → only 2 exits</text> <text x="210" y="95" fill="#9ca3bf" font-size="12">Path direction is immediately</text> <text x="210" y="115" fill="#9ca3bf" font-size="12">constrained. Solve these first.</text> <text x="210" y="145" fill="#00b894" font-size="12">✓ Fewer options = faster solve</text> <text x="210" y="165" fill="#6c5ce7" font-size="12">→ Builds momentum for harder pairs</text> </svg> <figcaption>Corner forcing — a corner endpoint has only 2 exits, making its path direction determined faster than center pairs</figcaption></figure><h3>Technique 2: Forced Corridors</h3><p>When narrow passages form between existing paths or walls, only one path can physically fit through. These <strong>bottlenecks</strong> are the most powerful deduction tool in Number Link. If you identify a corridor that must carry a specific path, you can commit to that routing without uncertainty.</p><p>Forced corridors appear in three forms:</p><ul> <li><strong>Single-cell gap:</strong> Two paths (or walls) leave exactly one cell between them — any path needing to pass through must use that cell</li> <li><strong>Edge corridor:</strong> A row or column of cells along the grid edge with limited entry/exit points</li> <li><strong>Pinch point:</strong> Two converging paths create a narrowing channel that funnels remaining paths</li></ul><h3>Technique 3: Avoid Trapping Cells</h3><p>The fill-every-cell rule means you must watch for <strong>isolated pockets</strong>. If your current path layout creates a region where no endpoint can reach the remaining empty cells, you have made an error. After every path placement, ask: "Does any empty cell become unreachable?"</p><p>A common beginner mistake is connecting a pair with the shortest path, only to discover that the detour needed to fill remaining cells becomes impossible. <strong>Shortest paths are not always correct paths</strong> in Number Link.</p><div class="tip-box"> <strong>💡 Pro Tip — Think Space Before Connection:</strong> Before drawing any path, mentally map which cells need to be covered and which endpoints need to reach them. A path that looks obvious can still be wrong if it divides the grid in a way that blocks another pair from connecting.</div><h3>Technique 4: Parity Reasoning</h3><p>Count the empty cells in a region. If there is an odd number of empty cells, an odd number of paths must pass through that region. This <strong>parity argument</strong> is the most advanced technique in Number Link and can reveal which paths must traverse specific corridors when simpler deductions have stalled.</p><p>For example, if a region bounded by existing paths contains 3 empty cells and 2 endpoints need paths through it, then exactly one of those paths must have a cell inside this region (since 3 is odd and each path uses cells in pairs — entering and exiting). This lets you place cells that are not immediately obvious from direct deduction.</p><h3>Technique 5: Work Backwards From Dead Ends</h3><p>When you are stuck, look for cells that can only be reached from one direction. A cell surrounded on three sides by paths or walls must receive its path from the remaining open side. Trace backwards from these forced cells to discover which path must pass through them, then follow that path to its logical endpoint.</p><figure class="diagram-box"> <svg viewBox="0 0 400 180" xmlns="http://www.w3.org/2000/svg" role="img" aria-label="Diagram showing bottleneck detection technique — narrow passage where only one path fits" style="max-width:100%;height:auto;"> <title>Bottleneck detection — narrow passage where only one path can fit</title> <g stroke="#2d3140" stroke-width="1"> <rect x="40" y="10" width="40" height="40" fill="#1a1d27"></rect> <rect x="80" y="10" width="40" height="40" fill="#1a1d27"></rect> <rect x="120" y="10" width="40" height="40" fill="#1a1d27"></rect> <rect x="160" y="10" width="40" height="40" fill="#1a1d27"></rect> <rect x="200" y="10" width="40" height="40" fill="#1a1d27"></rect> <rect x="40" y="50" width="40" height="40" fill="#1a1d27"></rect> <rect x="80" y="50" width="40" height="40" fill="#6c5ce7" opacity="0.3"></rect> <rect x="120" y="50" width="40" height="40" fill="#e17055" opacity="0.3"></rect> <rect x="160" y="50" width="40" height="40" fill="#1a1d27"></rect> <rect x="200" y="50" width="40" height="40" fill="#1a1d27"></rect> <rect x="40" y="90" width="40" height="40" fill="#1a1d27"></rect> <rect x="80" y="90" width="40" height="40" fill="#1a1d27"></rect> <rect x="120" y="90" width="40" height="40" fill="#e17055" opacity="0.3"></rect> <rect x="160" y="90" width="40" height="40" fill="#1a1d27"></rect> <rect x="200" y="90" width="40" height="40" fill="#1a1d27"></rect> </g> <path d="M 60 30 L 100 30 L 100 70" stroke="#6c5ce7" stroke-width="4" fill="none" stroke-linecap="round"></path> <rect x="120" y="50" width="40" height="40" fill="#00b894" opacity="0.2" stroke="#00b894" stroke-width="2" stroke-dasharray="4,2"></rect> <text x="140" y="75" text-anchor="middle" fill="#00b894" font-size="10" font-weight="bold">forced</text> <text x="260" y="40" fill="#e4e7f0" font-size="13" font-weight="bold">Bottleneck Detection</text> <text x="260" y="65" fill="#9ca3bf" font-size="12">The green cell is the only</text> <text x="260" y="83" fill="#9ca3bf" font-size="12">way the orange path can</text> <text x="260" y="101" fill="#9ca3bf" font-size="12">reach the remaining cells.</text> <text x="260" y="130" fill="#00b894" font-size="12">✓ Commit to this routing</text> </svg> <figcaption>Bottleneck detection — when a narrow passage allows only one path through, commit to that routing immediately</figcaption></figure><h2>Strategy Reference: Solving Order</h2><table class="strategy-table"> <thead> <tr> <th>Priority</th> <th>Technique</th> <th>When to Use</th> </tr> </thead> <tbody> <tr> <td>1st</td> <td>Corner/edge forcing</td> <td>Any endpoint with limited exits (corners, edges, near walls)</td> </tr> <tr> <td>2nd</td> <td>Adjacent pair shortcut</td> <td>Matching numbers next to each other — check if direct connection works</td> </tr> <tr> <td>3rd</td> <td>Bottleneck detection</td> <td>Narrow passages between existing paths where only one route fits</td> </tr> <tr> <td>4th</td> <td>Trap avoidance</td> <td>After every placement — check no empty cell becomes unreachable</td> </tr> <tr> <td>5th</td> <td>Dead-end tracing</td> <td>When stuck — find cells reachable from only one direction</td> </tr> <tr> <td>6th</td> <td>Parity reasoning</td> <td>Advanced — count empty cells in regions to force path assignments</td> </tr> </tbody></table><h2>Grid Sizes and Difficulty Progression</h2><p>Number Link puzzles scale from quick warm-ups to marathon challenges. Understanding the difficulty curve helps you build skill progressively:</p><table class="strategy-table"> <thead> <tr> <th>Grid Size</th> <th>Pairs</th> <th>Difficulty</th> <th>Typical Solve Time</th> </tr> </thead> <tbody> <tr> <td>5×5</td> <td>4–6 pairs</td> <td>Beginner</td> <td>1–3 minutes</td> </tr> <tr> <td>7×7</td> <td>6–10 pairs</td> <td>Intermediate</td> <td>3–8 minutes</td> </tr> <tr> <td>9×9</td> <td>10–15 pairs</td> <td>Advanced</td> <td>8–20 minutes</td> </tr> <tr> <td>11×11+</td> <td>15+ pairs</td> <td>Expert</td> <td>20+ minutes</td> </tr> </tbody></table><p>Start with 5×5 puzzles to internalize the rules and basic techniques. Move to 7×7 once you can consistently solve without backtracking. The 9×9 grid is where parity reasoning becomes essential, and 11×11+ grids demand systematic space management from the first move.</p><h2>Common Mistakes and How to Avoid Them</h2><p>Even experienced players fall into predictable traps. Here are the four most common mistakes:</p><ol> <li><strong>Shortest-path bias:</strong> Connecting pairs with the minimum number of cells. Number Link requires filling every cell — the shortest path is often wrong because it leaves unreachable pockets.</li> <li><strong>Ignoring the fill constraint:</strong> Focusing so much on connecting pairs that you forget every cell must be used. After each path placement, scan for isolated empty regions.</li> <li><strong>Committing too early to long paths:</strong> Drawing a long path through the center of the grid before edge paths are resolved. Keep the center flexible until outer paths are determined.</li> <li><strong>Not using pencil marks:</strong> Trying to solve entirely in your head. Mark cells that cannot contain a path (×) or must contain a path (•). This is especially important on larger grids.</li></ol><h2>Number Link vs Other Puzzles</h2><div class="compare-grid"> <div class="compare-card"> <h4>vs <a href="/guides/maze-puzzles-strategies-shortcuts/" class="internal-link">Maze</a></h4> <p>Mazes find one path through a labyrinth. Number Link draws multiple non-crossing paths that must collectively fill the grid.</p> </div> <div class="compare-card"> <h4>vs <a href="/guides/sokoban-box-pushing-puzzles/" class="internal-link">Sokoban</a></h4> <p>Sokoban pushes boxes in space-constrained rooms. Number Link routes paths in cell-constrained grids. Both reward spatial planning.</p> </div> <div class="compare-card"> <h4>vs <a href="/guides/hashiwokakero-bridge-puzzles/" class="internal-link">Hashiwokakero</a></h4> <p>Hashi connects islands with bridges (edges). Number Link connects numbers with paths through cells. Both use constraint propagation.</p> </div> <div class="compare-card"> <h4>vs <a href="/guides/kakuro-number-crosswords/" class="internal-link">Kakuro</a></h4> <p>Kakuro fills cells with digits (like crosswords with math). Number Link fills cells with paths. Both are Nikoli classics requiring pure logic.</p> </div> <div class="compare-card"> <h4>vs <a href="/guides/nurikabe-island-puzzle-rules/" class="internal-link">Nurikabe</a></h4> <p>Nurikabe paints cells black or white to form islands. Number Link draws paths. Both require global spatial reasoning.</p> </div> <div class="compare-card"> <h4>vs <a href="/guides/yinyang-circular-puzzle-guide/" class="internal-link">Yin-Yang</a></h4> <p>Yin-Yang places black/white circles with connectivity rules. Number Link draws paths. Both use constraint-based deduction.</p> </div></div><p>Number Link's unique challenge comes from the combination of pair-connecting and grid-filling. Most path puzzles only require one or the other. The dual constraint creates a richer solving experience where every decision affects both your ability to connect remaining pairs and your ability to cover remaining cells.</p><h2>Why Number Link Builds Real Skills</h2><p>Number Link develops skills that transfer beyond puzzles:</p><ul> <li><strong>Spatial reasoning:</strong> Mentally routing paths through constrained grids strengthens your ability to visualize and manipulate spatial relationships</li> <li><strong>Constraint management:</strong> Juggling three simultaneous rules (connect, don't cross, fill everything) trains multi-constraint decision making</li> <li><strong>Forward planning:</strong> Every path you draw affects future possibilities. Number Link rewards thinking several moves ahead before committing</li> <li><strong>Pattern recognition:</strong> Bottlenecks, forced corridors, and parity patterns become faster to spot with practice, training your eye for structural features</li></ul><p>These skills apply directly to other logic puzzles on our site — from <a href="/guides/akari-lightbulb-puzzle-strategy/" class="internal-link">Akari</a>'s illumination constraints to <a href="/guides/shikaku-rectangle-puzzles/" class="internal-link">Shikaku</a>'s area division. If you enjoy Number Link, you will likely excel at any Nikoli-style puzzle that combines simple rules with deep logical complexity.</p><h2>Frequently Asked Questions</h2><div class="faq-section"> <div class="faq-item"> <h3>What is the difference between Number Link and Flow Free?</h3> <p>Flow Free is the mobile adaptation of Number Link. They share identical core mechanics — connecting matching pairs with non-crossing paths. The key difference: traditional Number Link puzzles are designed to have exactly one solution, while casual Flow Free boards may accept any valid completion. Paper and web-based Number Link puzzles tend to be more carefully crafted with unique solutions.</p> </div> <div class="faq-item"> <h3>Can Number Link paths go diagonally?</h3> <p>No. All paths must move horizontally or vertically — one cell at a time, sharing edges (not corners). Diagonal movement is not allowed. This constraint is fundamental to the puzzle's logic and creates the grid-based routing challenges that define Number Link.</p> </div> <div class="faq-item"> <h3>What if I get stuck on a Number Link puzzle?</h3> <p>Follow this recovery sequence: (1) Re-scan for corner/edge forcing you missed. (2) Look for single-cell bottlenecks. (3) Check if any empty cell has only one possible path reaching it. (4) Count parity in remaining regions. (5) If all else fails, try a hypothesis — draw a path tentatively and see if it creates contradictions. There is no guessing in a well-designed puzzle, but hypothesis testing can reveal forced deductions you missed.</p> </div> <div class="faq-item"> <h3>Is Number Link the same as Arukone?</h3> <p>Yes. Arukone is the Japanese name for Number Link. Both names refer to the same puzzle type. You may also see it called Nanbarinku (ナンバリンク), which is Nikoli's original name. All three names describe the same game: connecting matching numbered endpoints with non-crossing, grid-filling paths.</p> </div> <div class="faq-item"> <h3>What grid size should I start with?</h3> <p>Start with 5×5 grids (4–6 pairs). These are solvable in 1–3 minutes and let you practice corner forcing and bottleneck detection without overwhelming complexity. Once you can solve 5×5 consistently without backtracking, move to 7×7, then 9×9. Most daily puzzle platforms offer sizes from 5×5 to 11×11.</p> </div> <div class="faq-item"> <h3>Can a Number Link puzzle have more than one solution?</h3> <p>Well-designed Number Link puzzles have exactly one unique solution. However, casual mobile apps like Flow Free may accept any valid non-crossing completion. If you find two different solutions for the same puzzle, either the puzzle was not carefully designed, or one solution violates a rule (usually the fill-every-cell constraint).</p> </div> <div class="faq-item"> <h3>How does Number Link compare to Sudoku in difficulty?</h3> <p>Both are pure-logic grid puzzles, but they exercise different skills. Sudoku uses number placement with row/column/box constraints. Number Link uses spatial path routing with fill and non-crossing constraints. At equivalent grid sizes, Number Link tends to be harder for beginners because the solution space is less structured — there are no fixed regions like Sudoku's 3×3 boxes to anchor deductions.</p> </div></div>
