Try this: Solve a Sudoku

You will need:

• A pencil
• An eraser
• A sudoku

What you do:

The practice sudoku in this newsletter are made up of 16 small squares, called ‘cells’, arranged in a square. The aim is to put a number from 1 to 4 in each of the cells. In the end, you need one of each of the numbers 1-4 in each row across, each column down, and also one of each of the numbers 1-4 in each of the four squares with a thick black border, which are commonly called ‘boxes’.

To explain how sudokus work, we’ll work through one from start to finish. Download a copy here, and print it out so you can follow along.

1. If you look at the top left box of four cells, there is only one cell that doesn’t have a number in it. The box already has a ‘1’, a ‘2’ and a ‘3’, so the empty cell needs to be a ‘4’.
2. Now look at the second row (rows go across). This row also has one empty cell, so you should be able to work out what to put in it.
3. Now look at the first cell in the third row, the one that’s directly under the ‘2’. This cell is in the same column (up and down) as a ‘1’ and a ‘2’, and each number only appears once in each column, so it can’t be a ‘1’ or a ‘2’. It’s also in the same row as a ‘4’, so it can’t be a ‘4’. If it can’t be a ‘1’, ‘2’ or ‘4’ then it must be a ‘3’.
4. Now that you have that ‘3’, you should be able to work out the other cells in the lower left box. Once you have them you’re almost finished.
5. You should now have three cells with the number ‘1’. There should be four of them in total (one for every row). The fourth ‘1’ can’t go in the first, second or third rows across, because they already have ‘1’s. It also can’t go in the first, second or third columns down, because they also have ‘1’s. The only space for it to go is in the bottom right corner.
6. Now that you’ve got this far, you should be able to fit in all the other numbers. When you’ve finished, check to see if there’s only one of each number in every row, column and box.

Most sudokus are actually made up of 81 cells (nine boxes), and use the numbers from 1-9, but the idea is the same. Many sudokus in newspapers or puzzle books will have stars telling you how hard they are. sudokus with lots of stars are a lot harder than those with only one star.

What’s happening:

Solving sudokus is very mathematical, but it’s not the numbers that make it maths. In any sudoku, if you swap all the ‘1’s with ‘2’s, and all the ‘2’s with ‘1’s, then you’ll still only have one of each number in the rows, columns and boxes. In fact, you don’t need to use numbers at all! Some sudoku puzzles use letters, or even symbols. There are even stained-glass windows that are based on sudokus, with one piece of glass of each colour in each row, column and square.

There are a lot of interesting mathematical questions about sudokus. For example, mathematicians have been trying to work out the minimum number of clues (numbers that are already included at the start) that you need before you can solve the puzzle. You’ll definitely need eight, because there are nine numbers in a regular sudoku, and if you only have seven clues, you can switch the other two numbers to give different solutions. Currently, the record for the least number of clues for a sudoku is 17, and many mathematicians don’t think that any 16 clue sudokus exist.

Applications:

A grid where each number only appears once in each column and each row is called a Latin square, and mathematicians have been investigating these for centuries. They are called Latin squares because the mathematician Leonhard Euler used Latin letters instead of numbers when he was writing them out.

Latin squares are used a lot to design experiments. If you want to try different fertilisers out on some grass, you could divide your lawn into strips and give each strip a different fertiliser. However, the best fertiliser might be growing badly because it’s on a slope or it gets less sun than the others. If you use a Latin square arrangement, you end up with a lot of little squares with different fertilisers, and each type of fertiliser will be tested in lots of different environments. This means you can compare the results a lot more easily, and you get much more accurate results, too.

More information:

Some easy sudoku you can play online

A latin square in stained glass

Sudoku variations (more advanced)

Sudoku, mathematics and statistics (very advanced)

Figure 1: Three of these cells are already filled in, so we can work out the remaining one.

Figure 2: This number can't be the same as any number in its row or column.

Figure 3: There need to be four '1's and the all need to be in different columns and rows.

Figure 4: When you've finished, compare your answer to this one.