# Solving Tips

## Solving Tips

Now matter what your KENKEN skill level is, it never hurts to have a few extra strategies and shortcuts in your back pocket. (OK, you don’t have to actually keep them in your pocket.) Here are “rules of thumb” for different scenarios you’ll encounter in the puzzles.

This tutorial, by David Levy, KENKEN’s technology expert, appears in KENKEN: *Books 1 - 4: The New Brain-Training Puzzle Phenomenon*, which HarperCollins in Britain published.

Overall Strategies

• Write each candidate in every unsolved square. Then look at the unsolved squares one by one, and see if you can eliminate any of the candidates by using the rule that no digit may appear more than once in any row or column, and every digit must appear exactly once in every row and column. When you have done this for every unsolved square on the grid, you may to be able to fill in at least one of them. If so, repeat the process, as your newly filled-ins square(s) should lead to even more.

• If any two squares in the same row or column have the same two candidates, then neither of those candidates can be in any other square in the same row or column.
The same rule applies to three squares with exactly three shared candidates, four squares with four shared candidates, and so on.

*Note: This rule applies for any number of squares.

Example: You’ve determined that the only candidates for the middle two boxes are 2 and 4. Therefore, the far-left box cannot be 2 or 4, as those numbers must be shared between the middle two boxes. Since 3 is already used, the far-left box must be 1.

Unique Situations

Any Grid Size

In a 2x cage with two squares, the candidates must be 1, 2.

In a 3+ cage with two squares, the candidates must be 1, 2.

In a 3x cage with two squares, the candidates must be 1, 3.

An L-shaped 3x cage with three squares must have the numbers arranged like this:

In a 4+ cage with two squares, the candidates must be 1, 3.

An L-shaped 4+ or 2x cage with three squares must have the numbers arranged like this:

In a 5x or 5÷ cage with two squares, the candidates must be 1, 5.

An L-shaped 5x cage with three squares must have the numbers arranged like this:

In a 6+ cage with 3 squares in a line the candidates must be 1,2,3

In a 6+ cage with three squares in the same row or column, the candidates must be 1, 2, 3.

In a 6x cage with three squares in the same row or column, the candidates must be 1, 2, 3.

In a 7+ cage with three squares in the same row or column, the candidates must be 1, 2, 4.

In a 10x cage with three squares in the same row or column, the candidates must be 1, 2, 5.

In a 15x cage with two squares, the candidates must be 3, 5
.

In a 15x cage with three squares in the same row or column, the candidates must be 1, 3, 5.

In a 20x cage with two squares, the candidates must be 4, 5.

In a 20x cage with three squares in the same row or column, the candidates must be 1, 4, 5.

4x4 or Larger

In a 6+ cage with three squares in the same row or column, the candidates must be 1, 2, 3.

In a 6x cage with three squares in the same row or column, the candidates must be 1, 2, 3.

In an 8x cage with three squares in the same row or column, the candidates must be 1, 2, 4.

3x3

An L-shaped 4x cage with three squares must have the numbers arranged like this:

4x4

In a 3– cage with two squares, the candidates must be 1, 4.

In a 6+ cage with two squares, the candidates must be 2, 4.

In a 7+ cage with two squares, the candidates must be 3, 4.

In a 4x or 4÷ cage with two squares, the candidates must be 1, 4.

In a 6x cage with two squares, the candidates must be 2, 3.

In an 8x cage with two squares, the candidates must be 2, 4.

5x5

In a 4– cage with two squares, the candidates must be 1, 5.

In a 6x cage with two squares, the candidates must be 2, 3.

In a 9+ cage with two squares, the candidates must be 4, 5.

In an 8+ cage with two squares, the candidates must be 3, 5.

In an 8x cage with two squares, the candidates must be 2, 4.

In a 12x cage with two squares, the candidates must be 3, 4.

In a 12x cage with three squares in the same row or column, the candidates must be 1, 3, 4.

In a 60x cage with three squares in the same row or column, the candidates must be 3, 4, 5.

6x6

In a 5– cage with two squares, the candidates must be 1, 6.

In an 11+ cage with two squares, the candidates must be 5, 6.

In a 10+ cage with two squares, the candidates must be 4, 6.

In a 15+ cage with three squares in the same row or column, the candidates must be 4, 5, 6.

In an 18x cage with two squares, the candidates must be 3, 6.

In an 18x cage with three squares in the same row or column, the candidates must be 1, 3, 6.

In a 14+ cage with three squares in the same row or column, the candidates must be 3, 5, 6.

In a 120x cage with three squares in the same row or column, the candidates must be 4, 5, 6.

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## Welcome to **KENKEN!**

No matter what you're looking for — learning or fun — KenKen.com has it! Free online puzzles. Cool math games. Number games. (More addictive than Sudoku or Kakuro? You decide!) Educational games for kids. Visit our For Teachers section for math teacher resources, our free KenKen Classroom program, and a message from Tetsuya Miyamoto, the Japanese educator behind KenKen. Miyamoto, chess master David Levy, and the rest of Team KenKen have crafted these math puzzles into great games for learning and brain training. Need even more KenKen? Check out our Will Shortz Presents books. Try our mobile app on iPhone and iPad or our Kindle version. Play on the New York Times puzzle page and NCTM website. Regardless of why, how, or where you play, KenKen are the math puzzles that make you smarter!

*Sincerely, Team KenKen*