You Just Found a Fascinating Number Grid
You’re looking at a puzzle that looks like a Sudoku board crossed with a math equation. The grid has numbers in some squares, and little boxes with mathematical symbols in the corners. This is Kendoku, also known as KenKen or Calcudoku. It’s a brilliant logic puzzle that challenges your arithmetic and reasoning skills in a uniquely satisfying way.
If you love Sudoku but want something that engages a different part of your brain, Kendoku is your next obsession. The good news is, learning the rules is straightforward. Mastering the strategy is the deeply rewarding part. This guide will walk you through everything, from the basic rules to the advanced tactics that will have you solving the hardest puzzles.
Understanding the Kendoku Battlefield
Before we dive into solving, let’s break down what you’re looking at. A classic Kendoku grid is a square, typically 4×4, 5×5, or 6×6 for beginners, going all the way up to 9×9 for experts. Like Sudoku, your goal is to fill the grid with numbers, following one core rule: you cannot repeat a number in any row or column.
The twist is the “cages.” These are the thick-bordered groups of cells. In the top-left corner of each cage, you’ll see a target number and a mathematical operation: addition (+), subtraction (-), multiplication (×), or division (÷).
Your second rule is that the numbers within a single cage must combine, using the given operation, to produce that target number. For example, a cage marked “6+” with two cells means the two numbers you place in those cells must add up to 6. A cage marked “2÷” with two cells means one number divided by the other must equal 2.
The Non-Repeating Number Rule
This is your anchor. In a 4×4 puzzle, you will use the numbers 1 through 4. In a 5×5, you use 1 through 5, and so on. Every single row and every single column must contain each number exactly once. This rule alone creates powerful constraints that you will use constantly.
It means if you already have a 3 in a row, you cannot place another 3 anywhere else in that same row. This simple logic eliminates possibilities and is the first tool in your solving kit.
Deciphering the Cage Operations
The cage system is where the arithmetic comes in. Each operation has its own logic for finding possible number combinations.
For addition (+) and multiplication (×) cages, the order of the numbers doesn’t matter. 3+4 is the same as 4+3. For subtraction (-) and division (÷) cages, order is critical. The operation is always performed as (larger number) – (smaller number) or (larger number) ÷ (smaller number) to get a positive, whole-number result.
A “1-” cage with two cells can only be [2,1] or [3,2] or [4,3], etc., depending on the puzzle size. A “2÷” cage can only be [2,1], [4,2], or [6,3], and so on.
Your First Puzzle: A Step-by-Step Walkthrough
Let’s solve a simple 4×4 puzzle together. We’ll use a grid where the only possible numbers are 1, 2, 3, and 4. Remember the two rules: no repeats in any row/column, and cages must hit their target.
Look for the easiest entry point. This is often a single-cell cage. A cage that is just one square with a target like “3” is already solved—just write a 3 in that cell. Also, look for small cages with unique solutions.
Starting with a Multiplication Cage
Imagine a two-cell cage in the top row marked “6x”. In a 4×4 puzzle, what two different numbers between 1 and 4 multiply to give 6? The only possibility is 2 and 3 (since 1×6 is invalid as 6 isn’t allowed). So that cage must contain a 2 and a 3.
You don’t know which order yet, but you’ve narrowed it down massively. Now, look at the rows and columns those cells are in. If one of the cells shares a row with another cell that already has a 2, then that cell cannot be 2, so it must be 3, solving the other cell by default.
Leveraging the Row and Column Constraints
As you place numbers, immediately use the “no repeat” rule. Cross out possibilities in your mind (or with pencil marks) for the rest of that row and column. This will quickly reveal what numbers are left for other cages in that line.
For a cage marked “5+” with two cells, the possibilities in a 4×4 are [1,4] or [2,3]. If the row one of those cells is in already has a 1 and a 4, then the pair cannot be [1,4], so it must be [2,3]. This process of elimination is the core of Kendoku logic.
Advanced Strategies for Tougher Grids
Once you’re comfortable with the basics, harder puzzles require more sophisticated deduction. You’ll need to think about the grid as a whole, not just individual cages.
Looking for “Hidden” Single Numbers
Sometimes, a row or column will be almost complete except for one cage that spans multiple cells. You can sum the entire row. In a 5×5 puzzle, a row must contain 1+2+3+4+5 = 15. If you can calculate the sum of the numbers you already have in that row, and the sum of the target numbers for the complete cages, you can deduce the sum for the remaining partial cage. This can reveal its possible combinations.
For example, if a row’s total must be 15, and you have two solved cells summing to 7, and a complete cage with a target of 8, then the remaining two-cell cage must have numbers that sum to 15 – 7 – 8 = 0, which is impossible. This tells you your earlier assumption was wrong, helping you backtrack.
The Power of Factor Lists for Multiplication
For large multiplication cages, write out all possible factor combinations. A three-cell “60x” cage in a 6×6 puzzle (numbers 1-6) is daunting. But list the factors: 60 can be made from 5 x 4 x 3. Could it be 6 x 5 x 2? That’s also 60. Systematically listing these, while respecting the “no repeat in cage” rule (cages can have repeats unless restricted by row/column), helps you see the options clearly.
Navigating Subtraction and Division Cages
These cages are often the key to unlocking a puzzle because they have very few possible number pairs. A “1-” cage is incredibly restrictive—it’s always two consecutive numbers. A “3÷” cage in a 6×6 can only be [6,2] or [3,1].
Use these highly constrained cages as your anchors. Place them in the grid conceptually, even if you don’t know the order. Then, see how they interact with the rows and columns. Often, the row/column constraint will force the order for you.
Common Mistakes and How to Avoid Them
The most frequent error is forgetting the order of operations in subtraction and division. Remember, it’s always (big number) op (small number). Writing down the possible pairs next to the cage can prevent this.
Another pitfall is focusing too narrowly on one cage and missing a row/column conflict. Always double-check the entire row and column after placing a tentative number. Pencil marks are your best friend for harder puzzles—lightly writing small possible numbers in the corners of cells helps visualize constraints.
Where to Practice and Find Puzzles
Now that you know how to play, you need puzzles. Kendoku was popularized by the Japanese educator Tetsuya Miyamoto. You can find daily puzzles of varying difficulties on many puzzle websites dedicated to KenKen. Reputable newspapers often feature them in their puzzle sections as well.
Start with 4×4 and 5×5 puzzles to build confidence. Don’t be discouraged if a 6×6 takes longer; the logic chains become more complex. The satisfaction of finally cracking a difficult puzzle is worth the effort.
There are also excellent mobile apps that offer endless puzzles with instant feedback. These are fantastic for practice because they can highlight rule violations, helping you learn from mistakes in real-time.
Taking Your Skills to the Next Level
Once you’re breezing through standard puzzles, look for variants. Some puzzles use only addition and subtraction, which changes the strategic feel. Others might use larger grids or even irregular, non-square “cages” for a supreme challenge.
The ultimate test is creating your own Kendoku puzzle. Designing a valid, solvable grid with a unique solution requires deep mastery of the interplay between cage constraints and row/column rules. It’s a fantastic way to reverse-engineer your understanding.
The Final Move Is Yours
Kendoku is more than just a pastime. It’s a workout for your logical deduction and mental math skills, wrapped in a deceptively simple package. The journey from staring at a blank grid to placing that final, confident number is a pure exercise in problem-solving.
Grab a pencil, find a beginner’s puzzle, and start with the smallest, most constrained cage. Apply the two golden rules: no repeats, and respect the cage math. Use elimination, look for the unique combinations, and let each solved cell guide you to the next. Before long, you’ll not only know how to play Kendoku—you’ll know how to think like a solver.
The grid is waiting. Your first puzzle is just a few logical steps away.
