Let dn denote the number of ways to cover the squares of a 2-by-n board using plain dominos. then it is easy to see that d1 = 1, d2= 2, and d3 = 3. compute a few more values of dn and guess an expression for the value of dn and use induction to prove you are right.

See explanation below.

Step-by-step explanation:

The area of a plain domino equal the area of a 2-by-1 board, so it is easy to see that D4=4, D5=5, and so on.

Let's prove by induction than Dn=n for every natural number n.

1) Obviously, D1 is true.

2) Assume now that Dn is true for a given natural n, that is to say, a 2-by-n board can be covered with n dominoes.

we must prove that this implies D(n+1) is also true.

But the area of a 2-by-(n+1) board equals the area of a 2-by-n board plus the area of a 2-by-1 board.  

By 2), we can cover the 2-by-n board with n dominoes and by 1) we can cover the 2-by-1 board with 1 domino, so we just need n+1 dominoes to cover the 2-by-(n+1) board, which is what we wanted to prove.