Given a “2 x n” board and tiles of size “2 x 1”, count the number of ways to tile the given board using the 2 x 1 tiles. A tile can either be placed horizontally i.e., as a 1 x 2 tile or vertically i.e., as 2 x 1 tile.

Examples:

Input n = 3 Output: 3 Explanation: We need 3 tiles to tile the board of size 2 x 3. We can tile the board using following ways 1) Place all 3 tiles vertically. 2) Place first tile vertically and remaining 2 tiles horizontally. 3) Place first 2 tiles horizontally and remaining tiles vertically Input n = 4 Output: 5 Explanation: For a 2 x 4 board, there are 5 ways 1) All 4 vertical 2) All 4 horizontal 3) First 2 vertical, remaining 2 horizontal 4) First 2 horizontal, remaining 2 vertical 5) Corner 2 vertical, middle 2 horizontal

## We strongly recommend that you click here and practice it, before moving on to the solution.

Let “count(n)” be the count of ways to place tiles on a “2 x n” grid, we have following two ways to place first tile.

1) If we place first tile vertically, the problem reduces to “count(n-1)”

2) If we place first tile horizontally, we have to place second tile also horizontally. So the problem reduces to “count(n-2)”

Therefore, count(n) can be written as below.

count(n) = n if n = 1 or n = 2 count(n) = count(n-1) + count(n-2)

The above recurrence is noting but Fibonacci Number expression. We can find n’th Fibonacci number in O(Log n) time, see below for all method to find n’th Fibonacci Number.

Different methods for n’th Fibonacci Number.

This article is contributed by Saurabh Jain. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above