KenKen Solver, Step by Step
Tap or drag across its squares, choose the operation, type the target.
How to use the KenKen solver
Choose the size of your grid, then draw the cages one at a time: tap or drag across a cage's squares, press its operation, type its target and press Make cage. A one-square cage needs only its number. Tap a finished cage to select it again if you made a mistake. When every square is in a cage, press Explain the next step: the solver fills in the next number that is certain, highlights the row, column or cage that proves it, and says why. Show the answer fills it all at once and says whether the solution is the only one.
How to solve KenKen, step by step
Start with the cages that have few possibilities. A one-square cage is given. A two-square cage with a small product or a big difference has few ways: in a 6 by 6, a 5− can only be 1 and 6, and an 11+ only 5 and 6. Write those candidates in.
Use the rows and columns. Every row and column holds each number once, so a number fixed in a square comes out of the rest of its row and column, and a number with only one square left in a row has to go there. Then go back to the cages: with fewer candidates, fewer fillings of each cage still fit, and each one that disappears can settle another square. In a 6 by 6, a 12× cage of three squares can be 1, 2, 6 or 1, 3, 4 or 2, 2, 3, but 2, 2, 3 only if the two 2s are not in the same row or column. That is exactly the reasoning the solver writes out.
Pairs. When two squares in a row can only be the same two numbers, those numbers are taken in that row, and can be crossed off everywhere else in it.
How it was checked
The list of ways each cage can be filled was compared against brute force, every combination tried. Then the solver was run on 60 puzzles from 4 by 4 to 9 by 9, made from random grids with cages of two to four squares like the published ones and kept only when they had a single solution. It found every one, and every step it explained was checked against the known answer: it never placed a wrong number or crossed off a right one. 50 of the 60 were explained from start to finish without a guess.
More number puzzles
The sudoku solver explains its moves the same way, and handles killer sudoku, the closest relative of KenKen, with its cage sums. The magic square solver and the cryptarithm solver are more puzzles where the numbers have to fit exactly. For the same pleasure in your hands, our brain teasers are made of wood and metal. There are more solvers on the puzzle tools page.
Frequently Asked Questions
How do you solve a KenKen step by step?
Start with one-square cages and the cages with few possibilities, like a 5− in a 6 by 6, which can only be 1 and 6. Then use the rows and columns, where each number appears once, and go back to the cages: as candidates disappear, fewer fillings of each cage still fit. Repeat until the grid is full.
What do the numbers and signs in KenKen cages mean?
Each cage has a target and an operation. The numbers in the cage must make the target: added for +, multiplied for ×, and for − and ÷, which only ever cover two squares, the bigger number minus or divided by the smaller.
Can numbers repeat in a KenKen cage?
Yes, as long as the repeated numbers are not in the same row or column. A 12× cage bent round a corner can be 2, 2, 3.
What is the difference between KenKen and sudoku?
Both put each number once in every row and column, but KenKen has no boxes and no given numbers apart from one-square cages: the arithmetic in the cages does the work instead. Killer sudoku is the closest relative, with sums only.
Does every KenKen have one solution?
A well-made one does. The solver checks: if your cages allow more than one solution, it says so.