Problem Statement
Domino and Tromino Tiling
You have two types of tiles: a 2 × 1 domino shape and a tromino shape. You may rotate these shapes.
Given an integer n, return the number of ways to tile an 2 × n board. Since the answer may be very large, return it modulo 109 + 7.
In a tiling, every square must be covered by a tile. Two tilings are different if and only if there are two 4-directionally adjacent cells on the board such that exactly one of the tilings has both squares occupied by a tile.
Examples
Example 1:
Input: n = 3
Output: 5
Explanation: The five different ways are shown below.
[Image showing 5 different tilings of a 2x3 board]
Example 2:
Input: n = 1
Output: 1
Explanation: There is only one way to tile a 2x1 board.
Constraints
- 1 <= n <= 1000
Problem Breakdown
To solve this problem, we need to:
- This problem can be solved using dynamic programming.
- We need to define multiple states to represent different configurations of the board.
- The recurrence relation involves considering all possible ways to place dominos and trominos.
- The modulo operation needs to be applied at each step to avoid integer overflow.
String Problems
Learning Objective
Apply string manipulation concepts to solve a real-world problem.
Domino and Tromino Tiling
You have two types of tiles: a 2 × 1 domino shape and a tromino shape. You may rotate these shapes.
Given an integer n, return the number of ways to tile an 2 × n board. Since the answer may be very large, return it modulo 109 + 7.
In a tiling, every square must be covered by a tile. Two tilings are different if and only if there are two 4-directionally adjacent cells on the board such that exactly one of the tilings has both squares occupied by a tile.
Examples
Example 1:
The five different ways are shown below. [Image showing 5 different tilings of a 2x3 board]
Example 2:
There is only one way to tile a 2x1 board.
Problem Breakdown
Constraints:
- 1 <= n <= 1000
Key Insights:
This problem can be solved using dynamic programming.
We need to define multiple states to represent different configurations of the board.
The recurrence relation involves considering all possible ways to place dominos and trominos.
The modulo operation needs to be applied at each step to avoid integer overflow.
Real-World Application
This problem has several practical applications:
Combinatorial Problems
Solving problems that involve counting the number of ways to arrange objects under certain constraints.
Tiling and Packing
Optimizing the arrangement of tiles or objects in a confined space, relevant in manufacturing and logistics.
Game Design
Creating puzzles and game mechanics that involve fitting shapes into a grid, as seen in games like Tetris.