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 = 4
Output: 3
Explanation:
For a 2 x 4 board, there are 3 ways
- All 4 vertical
- All 4 horizontal
- 2 vertical and 2 horizontal
Input: n = 3
Output: 2
Explanation:
We need 2 tiles to tile the board of size 2 x 3.
We can tile the board using following ways
- Place all 3 tiles vertically.
- Place 1 tile vertically and remaining 2 tiles horizontally.
Implementation –
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)
Here’s the code for the above approach:
C++
// C++ program to count the // no. of ways to place 2*1 size // tiles in 2*n size board. #include <iostream> using namespace std; int getNoOfWays( int n) { // Base case if (n == 0) return 0; if (n == 1) return 1; return getNoOfWays(n - 1) + getNoOfWays(n - 2); } // Driver Function int main() { cout << getNoOfWays(4) << endl; cout << getNoOfWays(3); return 0; } |
Output:
3 2
The above recurrence is nothing 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.
https://youtu.be/NyICqRtePVs
https://youtu.be/U9ylW7NsHlI
Different methods for n’th Fibonacci Number.
Count the number of ways to tile the floor of size n x m using 1 x m size tiles
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
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