Start with a row, say, the first ro

Start with a row, say, the first row. Choose a number for a11. Let a12¼a11þ3
(modulo 9), and a13¼a12þ3 (modulo 9). If the value of a12 or a13 happens to be 0
due to ‘modulo 9’, then use 9 instead of 0. The same is done below for all ‘modulo 9’
cases. Then, let a14 be a number other than the first three numbers, and make
a15¼a14þ3 (modulo 9) and a16¼a15þ3 (modulo 9). Finally, let a17 be a number
other than the first six numbers, and make a18¼a17þ3 (modulo 9) and a19¼a18þ3
(modulo 9). It is easy to see that now the first row contains a permutation of
{1, 2, . . . 9}.