Sudoku!
By Michael Naylor

Sudoku is a national craze, and math teachers everywhere couldn't be happier! Not only are these puzzles fun and addictive for teachers and students alike, but they also build logical reasoning, one of the National Council of Teachers of Mathematics process strands for mathematics education.

The puzzle was invented in 1979 in Indiana, and then became popular in Japan in the 1980s where it was renamed "Sudoku," meaning "single digit."

The object is to complete a grid of numbers so that no row, column or subdivided box contains the same number. Usually the grids are 9 x 9 and use the digits 1–9, but smaller and larger grids can be used.

Sudokus make an excellent activity for before or after school, or for those rainy recesses. While fourth graders may be ready for 10 x 10 Sudokus, kids as young as kindergartners can solve 4 x 4 Sudoku puzzles and first graders can successfully puzzle out a 6 x 6 grid. If you've never tried them, work through the following 4 x 4 and 6 x 6 examples – or work with your students and learn together.

4 x 4 Grids (Grades K-3+)
The smallest of the Sudoku, 4 x 4 grids are perfect for young students or for beginners. Give your students a copy of the grid shown in the next column. Tell them that every row, column and sub-box of four numbers must contain the digits 1, 2, 3 and 4. Here are some strategies you can show your students to get them started. They'll develop richer strategies as they play and as their logic expands.

Strategy 1: Missing numbers. Ask your students to first look for rows, columns or boxes that have three numbers already filled in. Fill in the missing numbers where appropriate and use this new information to help find other digits.

Strategy 2: Scanning. An important strategy for more difficult puzzles is called "scanning." Tell your students to choose a number and scan along its row or column. That number cannot appear in any of these squares. Does that open up possibilities elsewhere? This eliminates possibilities, sometimes leaving only one spot where a number can be placed. An example is shown here:

Making 4 x 4 Sudoku
It's very easy for your students to make their own 4 x 4 Sudokus. They can simply copy the grid with all of its dark and heavy lines, and write the numbers 1, 2, 3 and 4, one time each, one number per box, per row and per column. They've just made a solvable 4 x 4 Sudoku! To change it to an easier puzzle for inexperienced students, use the scanning method to fill in one or two additional numbers before you give it to your beginning solvers.

6 x 6 Sudoku (Grades 1-8)
The small 4 x 4 grids are appropriate for beginners, but even first graders will soon be ready for a bigger challenge. 6 x 6 grids are perfect for most students. Here are three 6 x 6 grids – this time, the sub-boxes are rectangles.

Advanced Sudoku strategies
More difficult puzzles require greater organization and greater logical reasoning. The following strategies are very helpful.

Strategy 3: Scanning multiple rows and columns. In this example, there may be no other 1s in the shaded row and column. That leaves one box, the
middle right, with only one remaining spot, shown here with a star. A 1 must go there. Can you spot another place where there must be a 1?

Strategy 4: Listing possibilities. In each square, jot down (write small!) all of the possible digits that could be placed in that square. Some squares may have only one digit that can be placed there. In the following example, the empty squares in the upper right have been labeled with all of the possibilities. There is only possibility for one of the square – the 2 must go in the center bottom of the sub-box. Once the 2 is placed, you can cross out or erase the remaining small 2s in the sub-box (and in this case, find another digit!).

Strategy 5: Number pairs. If you mark all of the possibilities and find a pair of matching numbers in two squares in the same row, column or box, you've found a clue. Those two numbers cannot appear elsewhere in that same row, column or box. In the example below, 5 and 6 are the only choices remaining for the two empty squares in the upper-right box. No 5s or 6s may appear elsewhere in the top row of the grid, and therefore the square marked with a star must contain a 5.

9 x 9 Sudoku (Grades 4-8)
Your students will probably need all of the above techniques in order to tackle 9 x 9 Sudoku. Be sure your class has had success with a 6 x 6 Sudoku before they try these – some can be very difficult. Here's an easier 9 x 9 to get you started.

Sudoku math facts
Sudoku are related to Latin squares, which are arrangements of numbers in a square grid with no repeated numbers in each row or column (like a Sudoku without the sub-boxes). Latin squares were studied by Leonhard Euler, one of the top three mathematicians of all time. Euler was a Swiss mathematician who lived during the 18th century.

There are more than six sextillion (6 followed by 21 zeros) possible 9 x 9 Sudoku answer grids. If you could solve two trillion Sudoku puzzles in a second, you might be able to finish all of the possibilities in 100 years.

The fewest number of starting numbers needed in a 9 x 9 Sudoku puzzle is thought to be 17, but no one's been able to prove it yet. There are too many possibilities for a computer to check them all!

Sudoku resources
Here are three of my favorite free Sudoku resources available online:

• A Sudoku generator with some very nice features, like allowing you to print hints in some of the squares to help kids solve more difficult puzzles, can be found at www.abcteach.com Click on the "Fun Activities" link on the homepage to find the Sudoku generator.
• A great online Sudoku game is at www.jigsawdoku.com The browser-based applet allows you to slide tiles around and drop them on a 4 x 4, 6 x 6 or 9 x 9 grid.
• Over 150,000 free Sudoku puzzles are available for download at www.boldts.net/Sudoku They are packaged in convenient bundles of 100 and organized by difficulty.

Michael Naylor is a professor of math education at Western Washington University, Bellingham, WA.

October, 2006, Vol.37, No.2