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 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
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 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.
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
Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.
- Tiling with Dominoes
- Nuts & Bolts Problem (Lock & Key problem) | Set 2 (Hashmap)
- Secretary Problem (A Optimal Stopping Problem)
- Nuts & Bolts Problem (Lock & Key problem) | Set 1
- Transportation Problem | Set 7 ( Degeneracy in Transportation Problem )
- 21 Matchsticks Problem
- Fibonacci problem (Value of Fib(N)*Fib(N) - Fib(N-1) * Fib(N+1))
- Perfect Sum Problem
- Subset Sum Problem | DP-25
- 0-1 Knapsack Problem | DP-10
- Partition problem | DP-18
- The Celebrity Problem
- Box Stacking Problem | DP-22
- Level Ancestor Problem
- Sequence Alignment problem
- Water Jug Problem using Memoization
- N consecutive ropes problem
- Cake Distribution Problem
- Transportation Problem | Set 5 ( Unbalanced )
- Transportation Problem | Set 1 (Introduction)