To find the number of ways of arranging the numbers 1, 2, 3, 4, 5, 6 in a row such that the product of any two adjacent numbers is even, we need to consider the properties of even and odd numbers.

For the product of two numbers to be even, at least one of the numbers must be even.

Now let's look at the arrangement:

1. If we start with an even number (2, 4, or 6), then the number adjacent to it can be either an even or an odd number.

2. If we start with an odd number (1, 3, or 5), then the number adjacent to it must be even.

Let's analyze these cases:

Case 1: Starting with an even number (2, 4, or 6):

- There are 3 options for the first number (2, 4, or 6).

- For each of these choices, there are 5 options for the second number (either an even number or an odd number).

- For each of the second numbers, there are 4 remaining numbers for the third position, and so on.

Therefore, the total number of arrangements in this case is \( 3 \times 5! \).

Case 2: Starting with an odd number (1, 3, or 5):

- There are 3 options for the first number (1, 3, or 5).

- For each of these choices, there are 3 options for the second number (only even numbers are allowed).

- For each of the second numbers, there are 4 remaining numbers for the third position, and so on.

Therefore, the total number of arrangements in this case is \( 3 \times 3 \times 4! \).

Adding the arrangements from both cases:

\[ Total \, arrangements = (3 \times 5!) + (3 \times 3 \times 4!) \]

Let's calculate the total number of arrangements.

To calculate the total number of arrangements:

\[ Total \, arrangements = (3 \times 5!) + (3 \times 3 \times 4!) \]

\[ Total \, arrangements = (3 \times 120) + (3 \times 3 \times 24) \]

\[ Total \, arrangements = 360 + 216 \]

\[ Total \, arrangements = 576 \]

So, there are 576 ways to arrange the numbers 1, 2, 3, 4, 5, 6 in a row such that the product of any two adjacent numbers is even.