1.1.2 Sudoku

Activity symbolActivity: Do You Do Sudoku?

[ These puzzles are closely linked to a branch of mathematics known as Graph Theory. ] In 2005, a puzzle craze known as sudoku swept across the world, starting in newspapers. Sudoku involves putting numbers on a square grid, and its creators claimed that solving it needed just a logical mind—no mathematics required. Most sudoku puzzles are nine squares long and nine squares wide, which is called a nine-by-nine (9×9) layout, but we are going to look at a smaller example shown below.

Our large square is composed of four blocks, each of which has four smaller squares within. The blocks are framed by thicker lines. There are also four rows across and four columns down, so this setup is a four-by-four (4×4) puzzle. Draw this puzzle in your notebook.

The idea is to arrange the numbers 1, 2, 3, and 4 in each block, so that each row and each column contains only one of the numbers 1, 2, 3, and 4. For example, the bottom left-hand block already has a 4 in it, so you’ll need to put 1, 2, and 3 in the remaining cells in that block. Make sure that you do not end up with two numbers that are the same in any row or in any column, though. Try it! Write your ideas directly on the copy of the diagram you drew. When you have either solved the puzzle or spent about ten minutes on it, read on. You can reveal the hint at any time that you feel you want more guidance. Just click on “Reveal discussion.”

Hint symbol

Discussion

If you have never seen these puzzles before, you may find this one quite tricky. There are many different ways that you can tackle this problem. The first step is to try to sort out exactly what you are being asked to do and to make sure you understand the problem. You may find it helpful to use a highlighter pen to mark rows, columns, or blocks as you work.

Then you might like to get a feel for the problem by putting in a few numbers just by guesswork. Unless you have been lucky, you will probably realize fairly soon that this does not work very well, but it will have given you a better idea of what is involved. Starting to get a feel for what might be involved is a very important first stage for any problem.

The next tactic to try is to make the problem simpler by breaking it down into steps concentrating on just one kind of number. You can choose the number that occurs most often and therefore is the number which you have most information about—in this case, 2. Begin by focusing on the number you chose to start with.

Now, you need to find the number that makes most sense to fill in that does not violate any of the rules given. Remember: Each number 1, 2, 3, and 4 appears only once in each block, in each row, and in each column. Keep the time limit of ten minutes in mind for the active part of this puzzle. You can spend twice this time if you wish, but don’t go beyond that.

Try a solution now. If you need more guidance, there is another hint to reveal below.

Hint symbol

Discussion

In this case, there are two 2s on the grid already, so you only need to add two more, one in each of the bottom blocks. There is already a 2 in the first column, so no more 2s can go in that column and so the only place for the 2 in the bottom left-hand block is in the square under the 4.

If you find this difficult to follow, draw a pencil line through the column and row that the given 2 is in to show that the 2s in the other blocks cannot be placed in this row or column. (Here’s a helpful tip: Write your numbers in pen and make the lines in pencil, so you can erase the pencil lines after each thought.)

Figure 8

Similarly, there is already a 2 in the third column, so no more 2s can go in the third column. So the 2 in the bottom right-hand block must go above the 3.

Now, let’s move on. You can see that the bottom row already has a 3 in it, so the 3 in the bottom left-hand block must go next to the 4.

Now look at the third row. Which number is missing?

As more numbers are added, the puzzle gets easier. If you have not finished the puzzle for yourself, try to do so now.

Solution symbol

Answer

Your final grid should look like this:

Although there is only one correct answer to each sudoku puzzle, the approach described in the hint is not the only way to get started on this puzzle. If, for example, you focus on the number 4 in your first move, then you can reason that the block in the lower right-hand corner must have a 4 placed beside the given 3. This would lead to a different start as well as different steps to reason through, but it leads to the same answer.

Each sudoku puzzle has only one possible arrangement of the numbers in the answer. If you don’t reason your way through, reaching the correct answer is almost impossible.

1.1.3 Reflection on Sudoku