Math Detours · Grade 5 · Around the corner · 16

Every row, every column, once

A Latin square puts each symbol once in every row and column. Build them, count the small ones, and meet the idea behind Sudoku.

Name ____________________Date ____________

Tonight's idea

Take a grid with 3 rows and 3 columns, and the numbers 1, 2 and 3. Fill it so that each number appears exactly once in every row and exactly once in every column. That is a Latin square.

1 2 3 2 3 1 3 1 2 a Latin square 1 2 3 3 2 1 2 1 3 not a Latin square
Every row on the right is fine. The middle column is not: it has two 2s. The last column has two 3s as well.

Here is one way to build one. Write 1 2 3 in the top row. For each new row, slide the numbers one place left and move the front number to the back. This works for a grid of any size.

A finished Sudoku is a Latin square with 9 rows and 9 columns, plus one extra rule for its boxes.

To fill a box, ask what is already in its row and what is already in its column. The box must be something else.

Watch one: fill a 4 by 4 square

This square uses 1, 2, 3 and 4. Look at the orange box. Its row already has 1 and 3. Its column already has 4. Only 2 is left, so write 2.

1 3 4 2 2 1 start 1 2 3 4 3 4 1 2 4 3 2 1 2 1 4 3 finished

Now the top row is missing only 4. Carry on like this. Always pick a box where the row and column together leave one choice.

Your turn

  1. Which of these is a Latin square? For the one that is not, say what is wrong.
    2 3 1 1 2 3 3 1 2 P 1 2 3 2 3 2 3 1 1 Q
  2. Finish this Latin square. It uses 1, 2 and 3.
    1 2 3 3
  3. Finish this Latin square. It uses 1, 2, 3 and 4.
    2 3 4 4 2 3
  4. (a) Draw every Latin square with 2 rows and 2 columns that uses 1 and 2. How many are there? (b) Find every 3 by 3 Latin square whose top row is 1 2 3. How many are there? Explain how you know you have them all.

Stretch: count them all

The numbers 1, 2 and 3 can be written in a row in 6 different orders. Each order could be the top row of a Latin square. Use question 4 to count every 3 by 3 Latin square.

Answers

  1. P is a Latin square. Q is not: its second row has two 2s and its third row has two 1s. Every column of Q is fine, but the rows must work too.
  2. Second row: 3 1 2. The middle box cannot be 2, because 2 is above it. Third row: 2 3 1.
  3. Reading across each row: 2 4 1 3 / 1 3 4 2 / 3 1 2 4 / 4 2 3 1. A good first box is the end of the bottom row, which must be 1.
  4. (a) Two: 1 2 over 2 1, and 2 1 over 1 2. (b) Two. The box under the 1 must be 2 or 3. If it is 2, the rows are 1 2 3 / 2 3 1 / 3 1 2. If it is 3, they are 1 2 3 / 3 1 2 / 2 3 1. Once that box is chosen, every other box is forced, so there are no more.

Stretch. Each of the 6 top rows can be finished in 2 ways, just as in question 4. \(6 \times 2 = 12\) Latin squares.

← PreviousNext: Sorting with circles →