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Numberlink
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A simple example of a Numberlink puzzle
Solution to the Numberlink puzzle

Numberlink is a type of logic puzzle involving finding paths to connect numbers in a grid.

Rules

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The player has to pair up all the matching numbers on the grid with single continuous lines (or paths). The lines cannot branch off or cross over each other, and the numbers have to fall at the end of each line (i.e., not in the middle).

It is considered that a problem is well-designed only if it has a unique solution[1] and all the cells in the grid are filled, although some Numberlink designers do not stipulate this.[2]

Another rule that is included in some versions of the puzzle is that a path cannot have any U-turns, as these would allow it to be shortened without changing the other paths.[2]

History

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In 1897, a slightly different form of the puzzle was printed in the Brooklyn Daily Eagle, in a column by Sam Loyd.[3] Another early, printed version of Number Link can be found in Henry Ernest Dudeney's book Amusements in mathematics (1917) as a puzzle for motorists (puzzle no. 252).[4] This puzzle type was popularized in Japan by Nikoli as Arukone (アルコネ, Alphabet Connection) and Nanbarinku (ナンバーリンク, Number Link). The only difference between Arukone and Nanbarinku is that in Arukone the clues are letter pairs (as in Dudeney's puzzle), while in Nanbarinku the clues are number pairs.

Versions of this known as Wire Storm, Flow Free and Alphabet Connection have been released as apps for iOS, Android, Web and Windows Phone.[5][6][7][8][9][10][11]

Computational complexity

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As a computational problem, finding a solution to a given Numberlink puzzle is NP-complete, for the versions in which the problem is only to connect all of the pairs of numbers,[12][13] for paths without U-turns that must cover all squares of the grid,[14] and for the "zig-zag" version in which all squares must be covered but U-turns are allowed.[2]

These hardness results require the number of pairs of numbers to grow with the size of the puzzle. For fixed numbers of pairs, even on arbitrarily large grids, connecting all pairs (without necessarily filling the grid) can be solved in polynomial time as an instance of the vertex-disjoint paths problem on undirected graphs.[15]

See also

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References

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[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Numberlink

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from Grokipedia
Numberlink is a in which players connect pairs of identical numbers placed on a rectangular grid by drawing non-intersecting paths that run horizontally or vertically through the centers of empty cells, ensuring no cell is used more than once and paths do not branch or pass through numbered cells. The objective is to link all pairs while adhering to these constraints, often resulting in a unique solution that fills portions of the grid without overlaps. The puzzle traces its origins to 1897, when American puzzle designer Sam Loyd published an early variant known as "The Puzzled Neighbors" in the Brooklyn Daily Eagle, challenging readers to connect houses to resources like water and gas without crossing pipes. This concept evolved and gained widespread popularity in through the publisher Nikoli, which formalized and commercialized Numberlink (known as Arukone in Japanese) starting in 1989, incorporating it into puzzle books and magazines. Nikoli's version emphasizes intuitive solving through deduction, with grids typically ranging from 5×5 to larger sizes, and it has since inspired digital adaptations and variants worldwide. Numberlink is renowned for its deceptive simplicity, as the basic rules belie its ; the problem of solving it is NP-complete, meaning that finding a solution can be computationally intensive for large instances, which has led to its use in algorithmic research and studies on neurological function. Popular modern iterations, such as the mobile game Flow Free, extend the core mechanic by requiring paths to cover the entire grid, amassing over 300 million downloads as of 2025 and introducing it to broader audiences. Despite its age, Numberlink remains a staple in puzzle communities for honing and skills.

Gameplay

Rules and Objective

Numberlink is a logic puzzle consisting of a rectangular grid of cells, where certain cells are pre-filled with pairs of identical numbers, such as two 1's, two 2's, up to a set of k distinct numbers, each appearing exactly twice. The rules require connecting each pair of identical numbers with a single continuous path that traverses the centers of adjacent cells, moving only horizontally or vertically—no diagonal moves are permitted. Paths may change direction at any cell but cannot branch, cross other paths, or pass through any cells containing numbers; additionally, all paths must occupy distinct cells without overlap or revisiting. While paths connect the numbered endpoints without forming closed loops, they may wind circuitously as long as they adhere to these constraints. The objective is to connect all pairs of identical numbers with such paths. Well-designed puzzles have a unique solution in which the paths together cover every empty cell without gaps or overlaps. Puzzles are typically designed with 2 to 10 pairs on rectangular grids ranging from 2×2 to 20×20 cells, though common examples use 5×5 to 10×10 grids with 4 to 8 pairs.

Example Puzzles

In Numberlink puzzles, numbers are placed in the grid cells, often positioned in corners or along edges to constrain possible paths and maximize difficulty, forcing solvers to consider limited routing options early. A medium-difficulty 5x5 example features five pairs and fifteen empty cells, demonstrating more intricate weaving, as in this unsolved grid (numbers indicated, empty cells as dots):

1 . . . 1 2 . . . . 3 4 . 4 . . . . . . 3 2 5 . 5

1 . . . 1 2 . . . . 3 4 . 4 . . . . . . 3 2 5 . 5

This puzzle has a unique solution that connects the pairs without crossing and fills the entire grid, as verified computationally.

History

Origins and Early Publications

Numberlink originated as a type of connecting puzzle in the late , with its earliest known variant appearing in American print media. The puzzle's first documented instance was "The Puzzled Neighbors," created by renowned puzzler Sam Loyd and published in the Brooklyn Daily Eagle on February 28, 1897. In this challenge, eight neighboring houses on a street needed to be connected to their respective front gates using paths that did not cross, presented on a simple grid-like diagram of the block. Loyd's version focused on non-intersecting paths between specific points, laying foundational mechanics for later iterations without requiring full grid occupancy. The puzzle gained further traction through the works of British puzzle designer Henry Ernest Dudeney, who adapted and refined connecting concepts in his publications. In his 1917 book Amusements in Mathematics, Dudeney presented puzzle number 252, "A Puzzle for Motorists," which required linking eight houses to eight churches via non-crossing roads on a grid. This example emphasized filling the entire grid with paths, marking a closer alignment to modern Numberlink rules and attributing the design directly to Dudeney. Dudeney revisited the theme in his 1932 posthumous collection Puzzles and Curious Problems, where puzzle number 270, "Planning Tours," featured matching (using letters as proxies) for closed-loop paths on a road map that players connected without overlaps. This iteration enhanced the puzzle's structure by requiring non-intersecting routes, providing examples attributed to Dudeney. Throughout the early , similar "number connecting" puzzles appeared in American newspapers, often as standalone brainteasers inspired by Loyd's original framework. These variants typically involved pairing numbered dots or points on a grid, promoting logical without the full-grid constraint seen in Dudeney's later works. Such connecting puzzles were independently developed multiple times in various forms before formal standardization.

Popularization in Japan and Worldwide

In the 1980s, Nikoli, 's leading puzzle publisher, introduced Numberlink to its audience through two variants: Arukone, which uses pairs of letters, and Nanbarinku (Number Link), which employs numerical pairs. The puzzle first appeared in volume 17 of Puzzle Communication Nikoli in 1987, authored by Kazuki Nogi, marking its formal entry into Japanese puzzle culture. Nikoli played a pivotal role in standardizing Numberlink by enforcing a unique-solution rule for all its puzzles, ensuring logical solvability without ambiguity and elevating the genre's design quality. This approach contributed to its growing popularity within , further amplified through inclusions in the World Puzzle Championships organized by the World Puzzle Federation, where Arukone variants have appeared as competitive events since the . Additionally, Kazuki Nogi's contributions extended to puzzle books that helped disseminate Numberlink, solidifying its place in Nikoli's portfolio alongside icons like Sudoku. The puzzle's global spread accelerated in the digital era, particularly post-2010 with the rise of mobile gaming. The 2012 app Flow Free by Big Duck Games reimagined Numberlink using colored dots and pipes, achieving over 100 million downloads by 2025 and introducing the mechanic to a broad international audience. Other implementations, such as the 2013 game Wire Storm by Bit Hax, which themed paths as circuit wires, and various Connection apps echoing Arukone's letter-based format, further embedded Numberlink in mobile platforms. This digital adaptation fostered a cultural shift, transforming the puzzle from niche print media to a staple of casual gaming worldwide, with millions engaging daily in its intuitive, non-crossing path challenges.

Variants and Adaptations

Rule Variations

One notable variation of Numberlink, known as Zig-Zag Numberlink, prohibits paths from making immediate 180-degree turns, or U-turns, which forces more serpentine routes and increases the puzzle's complexity, particularly on grids requiring full coverage of empty cells. In partial filling variants, solvers are not required to occupy every empty cell with a path segment, permitting isolated empty regions as long as all numbered pairs are correctly connected without overlaps or crossings. Themed adaptations replace numerical labels with alternative symbols to enhance visual appeal or fit specific motifs; for instance, the Arukone variant published by Nikoli often uses letters from A to Z for pairing, maintaining the core connection rules but shifting the theme to alphabetic matching. Similarly, in the Flow Free series, pairs are indicated by colored dots rather than numbers, with paths connecting same-color endpoints while adhering to non-crossing and grid-filling constraints where specified. Irregular grid shapes introduce further modifications by departing from rectangular layouts; hexagonal grids, for example, adapt the orthogonal path rules to six-directional adjacency, creating puzzles with altered connectivity patterns and heightened spatial challenges.

Digital Implementations

Numberlink has been widely adapted into mobile applications, enhancing accessibility through intuitive touch interfaces and expansive puzzle collections. One of the most popular implementations is Flow Free, developed by Big Duck Games and released in June 2012 for and Android platforms. In this app, players connect pairs of colored dots with pipe paths on grids ranging from 5x5 to 14x14 cells, filling the entire board without overlaps, with levels escalating in complexity across multiple packs. The game includes over 2,500 free puzzles in free-play mode, alongside daily challenges that introduce fresh content to maintain engagement. Another notable mobile adaptation was Number Link by Tapps Games, released around 2012 for Android and other platforms. This app provided over 800 handmade puzzles across varying difficulty levels and board sizes, emphasizing logical path connections between numbered pairs. Key features in these digital versions include touch-based drawing for creating paths, integrated hint systems to assist stalled progress, and built-in tracking of completed levels and achievements, allowing players to resume sessions seamlessly. More recent apps, such as "Number link. " by Leaflock (updated October 2024), continue to offer Numberlink-style puzzles on Android. Web-based implementations further extend Numberlink's reach, offering browser-playable puzzles and tools without requiring downloads. Sites like Puzzle Baron host over 50,000 unique online Numberlink puzzles, enabling instant play with adjustable difficulties and printable options for offline solving. Additionally, dedicated web solvers, such as the one at noq.solutions, allow users to input custom grids and generate solutions algorithmically, while generators on platforms like facilitate creating new puzzles for personal or educational use. These digital formats have made Numberlink more approachable, supporting casual play on smartphones, tablets, and computers while preserving the puzzle's core logic.

Solving Techniques

Basic Strategies

One effective starting point for solving Numberlink puzzles is to identify and connect isolated pairs of numbers that have obvious, short paths available, particularly those with minimal obstacles or direct adjacency. These connections often serve to block off sections of the grid early, simplifying the remaining layout and preventing future conflicts with other paths. For instance, if two identical numbers are positioned adjacent to each other or separated by a single empty cell, linking them immediately can delineate boundaries for adjacent pairs. In standard Numberlink puzzles, all empty cells must be filled by the paths. Another fundamental technique involves applying parity considerations to the empty cells within defined regions of . Since each path connects a pair and fills an number of cells, the total count of unoccupied cells in a bounded area must align with the requirements of the remaining paths to avoid dead ends—specifically, ensuring that no isolated single cell remains unfillable, as paths cannot occupy just one cell. This approach helps eliminate impossible path routings by verifying that the parity of available spaces matches the needs of unpaired . Prioritizing the edges and borders of the grid is a practical , as cells along the perimeter typically offer fewer options and can constrain the puzzle if left until later. By connecting border-placed numbers first—such as those in corners or along sides with clear edge-adjacent paths—solvers can secure these limited avenues early, reducing the risk of blocking essential routes for interior pairs. For example, a number in a corner cell must connect outward along one of only two possible directions, making it advisable to resolve such positions promptly to maintain grid flow. To maintain solvability, solvers must practice loop avoidance by ensuring that partial paths do not enclose or trap empty cells in ways that prevent complete filling of the grid. This involves monitoring for premature closures that might isolate regions with an incompatible number of cells for the remaining pairs, such as leaving a single empty cell unconnectable or forcing overlapping paths. By extending paths judiciously and checking for potential enclosures at each step, one can uphold the rule that all paths remain non-intersecting and the entire grid is utilized without isolated voids.

Advanced Methods

Backtracking serves as a systematic trial-and-error approach for solving medium-sized grids, where solvers tentatively draw paths between numbered pairs and undo invalid choices that lead to dead ends or violations of non-crossing rules. This method begins by selecting an unsolved pair and exploring possible routes, branches that exceed grid boundaries, intersect existing paths, or leave insufficient space for remaining connections, thereby reducing the search space through constraint propagation. For human solvers, is enhanced by prioritizing pairs with limited options, such as those near edges or blocked by partial paths, to minimize revisions. In graph-theoretic modeling, the Numberlink grid is represented as an undirected graph where each cell is a vertex, adjacent cells are connected by edges, and numbered cells act as endpoints for vertex-disjoint paths that must connect matching pairs without sharing vertices. This formulation transforms the puzzle into finding a of disjoint paths in the grid graph, allowing solvers to apply connectivity theorems to identify impossible configurations. Human application involves mentally partitioning the grid into subgraphs around numbered pairs, ensuring each path respects degree constraints (at most two per vertex) and global coverage. Heuristics extend these methods to variants; a key technique across variants is identifying forced moves, where surrounding constraints—such as enclosed regions or parity imbalances in adjacent cells—dictate a unique path for a pair, eliminating ambiguity without trial. These forced connections, often arising from nodes with only one viable neighbor configuration, propagate to unlock further deductions, bridging basic parity checks with deeper .

Computational Complexity

Complexity Results

The for Numberlink—determining whether there exists a valid filling of a given grid that connects all numbered pairs with non-intersecting paths—is NP-complete. This holds for the standard formulation where paths must connect the pairs without sharing cells, as shown by reduction from the in grid graphs. Membership in NP follows from verifying a proposed solution in time by checking path connectivity and non-intersection for each pair. For the variant requiring all empty cells to be covered by the paths (often called the "filling" or "cover all" version), NP-completeness was established by Kotsuma and Takenaga (2010). An earlier proof for the non-covering version, without the full coverage requirement, demonstrated NP-hardness via reduction from the Hamiltonian path problem in grid graphs (Lynch 1975). The problem remains NP-complete for the U-turn-free variant (Zig-Zag Numberlink), where paths cannot make 180-degree turns, proven by a 3-SAT reduction adapted with parity-preserving gadgets to maintain coverage and disjointness. Additionally, the partial filling problem—deciding whether a partially completed grid can be extended to a valid solution—is NP-complete, as it inherits the hardness of the empty-grid case through gadget extensions that force completion of pre-filled segments. When the number of pairs kk is fixed, the problem becomes solvable in time using algorithms for finding kk vertex-disjoint paths in planar graphs, such as dynamic programming over graph separators that run in time O(nc)O(n^c) for constant cc depending on kk.

Practical Solving Algorithms

Practical solving algorithms for Numberlink puzzles typically rely on with to systematically explore possible path configurations. The algorithm begins by selecting an unconnected pair and attempting to extend a path from one endpoint to the other, choosing adjacent empty cells while ensuring no cycles or isolations occur. Optimization is achieved through mechanisms, such as connectivity checks that verify whether the remaining unconnected pairs can still reach each other given the current partial solution; if not, the branch is abandoned to avoid futile exploration. This approach efficiently handles typical puzzles with 5 to 10 pairs on grids up to 10x10, as demonstrated in open-source implementations. Another class of algorithms models Numberlink as the problem of finding vertex-disjoint paths between fixed pairs in the underlying grid graph. To enforce vertex disjointness, the graph is transformed into a by splitting each vertex into an incoming and outgoing node connected by an edge of capacity 1, with original edges directed from outgoing to incoming nodes of neighbors also at capacity 1. For a fixed number of pairs, multi-commodity maximum flow algorithms can then determine if flows of value 1 exist for each pair without exceeding capacities, solvable in time relative to the grid size. These methods are particularly useful for verifying solvability or generating puzzles with guaranteed unique solutions, though their implementation grows complex with increasing pairs. For larger or more constrained instances, heuristic solvers combine greedy path extension—prioritizing connections that resolve the most immediate constraints, such as pairs with few possible moves—with local search techniques to iteratively refine suboptimal paths by swapping segments or rerouting around blockages. These s approximate optimal solutions quickly, often succeeding where exhaustive search would be prohibitive. Open-source Python implementations employing such optimized and s solve 10x10 grids in seconds, making them suitable for puzzle generation and validation tools. Despite the of Numberlink for arbitrary instances, these practical algorithms bridge theoretical limits with real-world efficiency for standard puzzle dimensions.

References

  1. https://www.nikoli.co.jp/en/puzzles/numberlink/
  2. https://www.mathematica-journal.com/2007/08/06/number-link/
  3. https://arxiv.org/pdf/2202.04113
  4. https://www.researchgate.net/publication/262144434_Finding_All_Solutions_and_Instances_of_Numberlink_and_Slitherlink_by_ZDDs
  5. https://www.bigduckgames.com/
  6. https://arxiv.org/pdf/2002.01143
  7. https://mellowmelon.wordpress.com/2010/07/24/numberlink-primer/
  8. https://www.puzzler.com/puzzles-a-z/numberlink
  9. https://numberlinks.puzzlebaron.com/
  10. https://www.gmpuzzles.com/blog/2016/10/numberlink-serkan-yurekli-2/
  11. https://stackoverflow.com/questions/35043600/solving-the-numberlink-puzzle-with-prolog
  12. https://www.gutenberg.org/files/16713/16713-h/16713-h.htm
  13. https://archive.org/details/in.ernet.dli.2015.200332
  14. https://pmc.ncbi.nlm.nih.gov/articles/PMC7188385/
  15. https://wpcunofficial.miraheze.org/wiki/Arukone
  16. https://play.google.com/store/apps/details?id=com.bigduckgames.flow
  17. https://www.bigduckgames.com/flowfree
  18. https://doug-osborne.com/the-level-generator/
  19. https://apps.apple.com/us/app/flow-free/id526641427
  20. https://www.amazon.com/Tapps-Top-Apps-and-Games/dp/B0081TQLL4
  21. https://play.google.com/store/apps/details?id=com.leaflock.connectnumbers
  22. https://www.noq.solutions/numberlink
  23. https://github.com/thomasahle/numberlink
  24. https://www.snappywords.com/games/puzzles/arukone
  25. https://puzzling.stackexchange.com/questions/49687/numberlink-strategy
  26. https://arxiv.org/pdf/2210.02888
  27. https://www.mdpi.com/1999-4893/5/2/176
  28. https://dl.acm.org/doi/10.1145/800146.804097
  29. https://www.ieice.org/publications/ER/ER2010/er2010_16.html
  30. https://arxiv.org/abs/1410.5845
  31. https://www.101computing.net/connect-flow-backtracking-algorithm/
  32. https://www.iajit.org/downloadfile/4773
  33. https://courses.grainger.illinois.edu/cs473/sp2010/notes/17-maxflowapps.pdf
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