Given a 3 x n board, find the number of ways to fill it with 2 x 1 dominoes.
Following are all the 3 possible ways to fill up a 3 x 2 board.
Here is one possible way of filling a 3 x 8 board. You have to find all the possible ways to do so.
Input : 2 Output : 3 Input : 8 Output : 153 Input : 12 Output : 2131
At any point while filling the board, there are three possible states that the last column can be in:
An = No. of ways to completely fill a 3 x n board. (We need to find this) Bn = No. of ways to fill a 3 x n board with top corner in last column not filled. Cn = No. of ways to fill a 3 x n board with bottom corner in last column not filled.
Note: The following states are impossible to reach:
Note: Even though Bn and Cn are different states, they will be equal for same ‘n’. i.e Bn = Cn
Hence, we only need to calculate one of them.
Final Recursive Relations are:
- Golomb sequence
- Newman–Shanks–Williams prime
- Largest divisible pairs subset
- Painting Fence Algorithm
- Perfect Sum Problem (Print all subsets with given sum)
- Tabulation vs Memoizatation
- Choice of Area
- Tiling Problem
- Assembly Line Scheduling | DP-34
- Subset Sum Problem | DP-25
- Cutting a Rod | DP-13
- 0-1 Knapsack Problem | DP-10
- Coin Change | DP-7
- Longest Common Subsequence | DP-4
- Longest Increasing Subsequence | DP-3
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