Tiling Problem

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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.


Input n = 3
Output: 3
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
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


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.

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source and credits.
source and credits: http://www.geeksforgeeks.org
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