The math logic puzzle — satisfy every cage, never repeat a digit
KenKen is a captivating mathematical logic puzzle invented in 2004 by Japanese educator Tetsuya Miyamoto as a classroom tool to build arithmetic fluency and logical reasoning without making students feel like they were doing rote practice. The name combines the Japanese words for cleverness, and the puzzle rapidly spread worldwide after being featured in The New York Times in 2008. Today it is also known as Calcudoku, MathDoku, or Kendoku, and millions of solvers around the world tackle it daily on paper and online. The rules are elegantly simple yet produce puzzles with enormous depth. You are given an NxN grid — our version supports 3×3 through 6×6 — and your goal is to fill every cell with a digit from 1 to N. Like Sudoku, no digit may repeat in any row or column: this is the Latin square constraint. But KenKen adds a second layer of challenge through cages. Each cage is a group of connected cells outlined with a bold border. The top-left corner of every cage displays a small label showing a target number and an arithmetic operation, for example '12+', '3−', '24×', or '2÷'. The digits you place inside the cage must produce that target when the operation is applied. Single-cell cages simply tell you the exact digit to place — no calculation needed. Addition cages can be any size: the digits must sum to the target. Multiplication cages also work with any number of cells: their product must equal the target. Subtraction and division cages are always exactly two cells, because these operations are not associative across more cells: for subtraction the absolute difference of the two digits must equal the target, and for division the larger digit divided by the smaller must equal the target, in either order. Our online KenKen game brings all the essential features of the best web competitors together in a clean, mobile-responsive interface with a dark-mode-friendly design. You can choose your grid size (3×3 for beginners, up to 6×6 for a real challenge), select Easy, Medium, or Hard difficulty (which affects cage sizes and complexity), and optionally restrict operations — for example 'Addition Only' for younger players or 'All Operations' for experienced solvers. The game generates a new puzzle every time using a genuine backtracking Latin-square generator and a randomized cage-partitioning algorithm, so every puzzle is unique and fresh. Quality-of-life features make this the best free KenKen experience available. Pencil marks (notes mode) let you record candidate digits as small numbers inside cells — toggle with the Notes button or press P. Conflict highlighting instantly shows in red any cell whose digit clashes with another in the same row or column, so you spot mistakes as you make them. When a cage is fully and correctly filled its cells turn subtly green. The Undo button reverses your last move, and you can keep undoing all the way back to the start. The Hint button fills in one random empty cell when you are truly stuck. The Check button highlights any incorrect cells in red after your explicit request. A live timer tracks your solve time, with a Pause button for interruptions. Your in-progress game is automatically saved to local storage so you can close the tab and resume exactly where you left off. When you complete a puzzle a congratulations modal shows your solve time and grid size. Hit New Game to instantly start another — configure your preferred settings in the panel that slides open at the top.
Understanding KenKen
What Is KenKen?
KenKen is a number-placement puzzle combining the uniqueness constraint of Sudoku (no digit repeats in any row or column) with arithmetic cage targets. Invented by Tetsuya Miyamoto in 2004, it is sometimes called Calcudoku, MathDoku, or Kendoku. Each cage — a boldly bordered group of cells — carries a label like '12+' or '3−' that tells you what arithmetic result the cage's digits must produce. Smaller grids (3×3, 4×4) are beginner-friendly while larger ones (6×6, 7×7, 8×8) challenge experienced puzzlers. Single-cell cages act as given hints: just place the stated digit. Unlike Sudoku, which uses only logic of elimination, KenKen also requires arithmetic reasoning, making it an excellent mental workout for both math and logic skills.
How Do Cage Rules Work?
Each cage label consists of a target number and an operation symbol. Addition (+): all digits in the cage must sum to the target — works for cages of any size. Multiplication (×): all digits must multiply to the target — also any size. Subtraction (−): exactly two cells; the absolute difference of the two digits equals the target, in either order. Division (÷): exactly two cells; the larger digit divided by the smaller equals the target, in either order. A single-cell cage shows only a number with no symbol — simply place that digit. Important: digits can repeat within a cage as long as they do not share a row or column. For example, in a 6×6 grid, a two-cell '2+' cage could hold two 1s if those cells are in different rows and different columns — because the Latin-square constraint only forbids repetition within the same row or column, not within the same cage.
Why Play KenKen?
KenKen was designed specifically as a math education tool, and research confirms its benefits. Regular play strengthens mental arithmetic — addition, subtraction, multiplication, and division fluency all improve. The Latin-square constraint trains systematic logical thinking, the same type of reasoning used in coding, chess, and scientific problem-solving. Because the puzzle always has a unique, logically-derivable solution, it rewards patient deduction over lucky guessing. Educators use KenKen in classrooms from elementary through high school to build number sense without the tedium of worksheets. For adults, it serves as an engaging daily brain exercise. Compared to Sudoku, which uses only elimination logic, KenKen adds arithmetic reasoning, engaging a broader range of cognitive skills in every solve session.
Tips & Strategy
Start with single-cell cages (the givens) — they are free information. Next look for large-target or large-product cages that are highly constrained: for example a '6×' cage in a 4×4 grid can only be {1,2,3} or {2,3,1} etc. Use pencil marks liberally — recording candidates prevents errors and reveals forced cells through elimination. The Latin-square constraint is your friend: if a digit appears in three cells of a row with only one empty cell remaining, the last cell is determined. For subtraction and division cages, list all valid digit pairs: in a 4×4 grid '3−' can only be {1,4} or {4,1}. Work cage constraints and row/column constraints together rather than separately for maximum speed. Hard puzzles often require trial-and-error only because the solver has not yet found the right logical chain — revisit constraint propagation before guessing.
How to Play KenKen
Choose Your Settings
Click 'New Game' to open the settings panel. Select a grid size (3×3 is the gentlest introduction; 6×6 is a satisfying challenge). Choose Easy, Medium, or Hard difficulty — this affects how large and complex the cages are. Pick an operation mode: 'Add Only' is great for beginners or younger players; 'All (+−×÷)' is the classic full experience. Then click 'Start New Game'.
Read the Cage Labels
Each cage has a small label in its top-left cell showing a target number and operation (e.g., '12+', '3−', '6×', '2÷'). Single-cell cages show only a number — place that digit there immediately. For multi-cell cages, figure out which digit combinations satisfy the arithmetic target while also obeying the no-repeat-in-row/column rule.
Fill the Grid
Click a cell to select it, then press a number key (1–N) or tap a number on the on-screen pad. Red highlighting appears automatically if you place a digit that conflicts with another in the same row or column — fix it immediately. Use Notes mode (press P or click the Notes button) to write candidate digits as small pencil marks inside cells. When all cells in a cage are correctly filled, the cage turns lightly highlighted to confirm it.
Use the Tools When Stuck
Press Ctrl+Z or click Undo to reverse your last move (unlimited undo history). Click Hint to auto-fill one correct cell when you are completely stuck. Click Check to reveal any incorrect cells in red. Use Reset to clear all your entries and start the same puzzle from scratch. Your progress saves automatically — close the tab and come back anytime to resume exactly where you left off.
Frequently Asked Questions
What is the difference between KenKen, Calcudoku, and MathDoku?
They are essentially the same puzzle under different brand names. KenKen is the original trademark name registered by Tetsuya Miyamoto and popularized by Nextoy LLC in the West. Calcudoku and MathDoku are generic names used by independent puzzle sites to offer the same type of puzzle without trademark issues. Kendoku is another variant name. The rules are identical across all versions: fill an NxN grid so no digit repeats in any row or column, and every cage's digits produce the stated arithmetic result. Some sites introduce minor rule variations — for example offering 'no-op' mode where the operation symbol is hidden — but the fundamental puzzle is the same invention.
Can digits repeat inside a cage?
Yes — digits CAN repeat within a cage, as long as they do not share a row or column. This is a common point of confusion for players coming from Sudoku, where no repetition of any kind is allowed within a box. In KenKen, the only hard rule is that no digit may appear twice in the same row or twice in the same column across the entire grid. So two cells in the same cage can both hold the digit 2 if those cells are in different rows AND different columns. For example, a '4+' cage in a 4×4 grid could contain {2,2} if the two 2s occupy different rows and different columns.
Why are subtraction and division limited to two-cell cages?
Subtraction and division are not associative operations: the result of (a − b) − c depends on the order of operations and is not the same as a − (b + c). For three or more cells there would be many different valid interpretations of how to apply the operation, making the puzzle ambiguous. By restricting subtraction and division to exactly two-cell cages, each cage has a single unambiguous arithmetic relationship: |a − b| = target for subtraction, and max(a,b) ÷ min(a,b) = target for division, regardless of which order the digits appear in the cells.
How do I use pencil marks effectively?
Toggle Notes mode by pressing P or clicking the Notes button — the button turns highlighted when active. In notes mode, pressing a digit adds it as a small candidate mark inside the selected cell rather than placing it as the definitive answer. Build up the set of possible digits for each cell based on cage constraints and which digits already appear in the cell's row and column. When only one candidate remains in a cell, that is the answer — press the digit again in normal mode to place it. Our game automatically removes pencil marks from the same row and column when you place a confirmed digit, saving you maintenance work.
What does the difficulty setting change?
Difficulty primarily affects cage size and structure. Easy puzzles have more single-cell cages (free given digits) and smaller multi-cell cages (2–3 cells), which are more constrained and easier to deduce. Medium puzzles have fewer givens and allow cages up to 4 cells, increasing the number of candidate combinations you need to consider. Hard puzzles have larger cages (up to 5 cells), fewer single-cell hints, and more reliance on multiplication and division in modes that include those operations, requiring more advanced constraint propagation. Grid size also contributes to difficulty: a 6×6 Hard puzzle is significantly more challenging than a 3×3 Easy.
Is my game saved if I close the browser?
Yes. The game automatically saves your current board state, timer, notes, and all your moves to your browser's local storage every time you make a change. When you return to the page it will load your in-progress puzzle exactly as you left it, including your pencil marks and elapsed time. Your save is device-specific and browser-specific — it will not transfer to another device or browser. If you clear your browser data or use private/incognito mode the save will be lost. Completing a puzzle or starting a new game will replace the saved state with the new game.