Problem Statement
Knight Dialer
The chess knight has a unique movement, it may move two squares vertically and one square horizontally, or two squares horizontally and one square vertically (with both forming the shape of an L). The possible movements of chess knight are shown in this diagram:
A chess knight can move as indicated in the chess diagram below:
We have a chess knight and a phone pad as shown above, the knight can only stand on a numeric cell (i.e. blue cell).
Given an integer n, return how many distinct phone numbers of length n we can dial.
You are allowed to place the knight on any numeric cell initially and then you should perform n - 1 jumps to dial a number of length n. All jumps should be valid knight jumps.
As the answer may be very large, return the answer modulo 10^9 + 7.
Examples
Example 1:
Input: n = 1
Output: 10
Explanation: We need to dial a number of length 1, so placing the knight over any numeric cell of the 10 cells is sufficient.
Example 2:
Input: n = 2
Output: 20
Explanation: All possible valid knight moves from any numeric cell to dial a number of length 2 are:
Knight at cell 1 can jump to cell 6 and 8
Knight at cell 2 can jump to cell 7 and 9
Knight at cell 3 can jump to cell 4 and 8
Knight at cell 4 can jump to cell 3 and 9 and 0
Knight at cell 5 cannot jump to any cell
Knight at cell 6 can jump to cell 1 and 7 and 0
Knight at cell 7 can jump to cell 2 and 6
Knight at cell 8 can jump to cell 1 and 3
Knight at cell 9 can jump to cell 2 and 4
Knight at cell 0 can jump to cell 4 and 6
Thus the total number of distinct phone numbers we can dial is 20.
Example 3:
Input: n = 3131
Output: 136006598
Explanation: Please take the answer modulo 10^9 + 7.
Constraints
- 1 <= n <= 5000
Problem Breakdown
To solve this problem, we need to:
- This problem can be solved using dynamic programming.
- We need to keep track of the number of ways to reach each digit after a certain number of jumps.
- The state of the DP array is defined by the current digit and the number of jumps made so far.
- The recurrence relation involves considering all possible previous digits that can lead to the current digit.
- 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.
Knight Dialer
The chess knight has a unique movement, it may move two squares vertically and one square horizontally, or two squares horizontally and one square vertically (with both forming the shape of an L). The possible movements of chess knight are shown in this diagram:
A chess knight can move as indicated in the chess diagram below:
We have a chess knight and a phone pad as shown above, the knight can only stand on a numeric cell (i.e. blue cell).
Given an integer n, return how many distinct phone numbers of length n we can dial.
You are allowed to place the knight on any numeric cell initially and then you should perform n - 1 jumps to dial a number of length n. All jumps should be valid knight jumps.
As the answer may be very large, return the answer modulo 10^9 + 7.
Examples
Example 1:
We need to dial a number of length 1, so placing the knight over any numeric cell of the 10 cells is sufficient.
Example 2:
All possible valid knight moves from any numeric cell to dial a number of length 2 are: Knight at cell 1 can jump to cell 6 and 8 Knight at cell 2 can jump to cell 7 and 9 Knight at cell 3 can jump to cell 4 and 8 Knight at cell 4 can jump to cell 3 and 9 and 0 Knight at cell 5 cannot jump to any cell Knight at cell 6 can jump to cell 1 and 7 and 0 Knight at cell 7 can jump to cell 2 and 6 Knight at cell 8 can jump to cell 1 and 3 Knight at cell 9 can jump to cell 2 and 4 Knight at cell 0 can jump to cell 4 and 6 Thus the total number of distinct phone numbers we can dial is 20.
Example 3:
Please take the answer modulo 10^9 + 7.
Problem Breakdown
Constraints:
- 1 <= n <= 5000
Key Insights:
This problem can be solved using dynamic programming.
We need to keep track of the number of ways to reach each digit after a certain number of jumps.
The state of the DP array is defined by the current digit and the number of jumps made so far.
The recurrence relation involves considering all possible previous digits that can lead to the current digit.
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 or select objects under certain constraints.
Game Design
Designing games with complex movement patterns and calculating the number of possible game states.
Telecommunications
Analyzing patterns in phone number dialing and designing secure phone authentication systems.