<p>The chess knight has a <strong>unique movement</strong>,&nbsp;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 <strong>L</strong>). The possible movements of chess knight are shown in this diagram:</p>

<p>A chess knight can move as indicated in the chess diagram below:</p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/08/18/chess.jpg" style="width: 402px; height: 402px;" />
<p>We have a chess knight and a phone pad as shown below, the knight <strong>can only stand on a numeric cell</strong>&nbsp;(i.e. blue cell).</p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/08/18/phone.jpg" style="width: 242px; height: 322px;" />
<p>Given an integer <code>n</code>, return how many distinct phone numbers of length <code>n</code> we can dial.</p>

<p>You are allowed to place the knight <strong>on any numeric cell</strong> initially and then you should perform <code>n - 1</code> jumps to dial a number of length <code>n</code>. All jumps should be <strong>valid</strong> knight jumps.</p>

<p>As the answer may be very large, <strong>return the answer modulo</strong> <code>10<sup>9</sup> + 7</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> n = 1
<strong>Output:</strong> 10
<strong>Explanation:</strong> We need to dial a number of length 1, so placing the knight over any numeric cell of the 10 cells is sufficient.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> n = 2
<strong>Output:</strong> 20
<strong>Explanation:</strong> All the valid number we can dial are [04, 06, 16, 18, 27, 29, 34, 38, 40, 43, 49, 60, 61, 67, 72, 76, 81, 83, 92, 94]
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> n = 3131
<strong>Output:</strong> 136006598
<strong>Explanation:</strong> Please take care of the mod.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= n &lt;= 5000</code></li>
</ul>

#include <vector>
#include <unordered_map>
using namespace std;

class Solution {
public:
    int knightDialer(int n) {
        const int MOD = 1e9 + 7;
        vector<pair<int, int>> moves = {{2, 1}, {2, -1}, {-2, 1}, {-2, -1}, {1, 2}, {1, -2}, {-1, 2}, {-1, -2}};
        
        vector<vector<int>> grid = {
            {1, 2, 3},
            {4, 5, 6},
            {7, 8, 9},
            {-1, 0, -1} 
        };
        
        vector<vector<vector<int>>> dp(n + 1, vector<vector<int>>(4, vector<int>(3, -1)));
        
        int result = 0;
        for (int i = 0; i < 4; ++i) {
            for (int j = 0; j < 3; ++j) {
                if (grid[i][j] != -1) { 
                    result = (result + solve(n, i, j, grid, moves, dp)) % MOD;
                }
            }
        }
        return result;
    
    }

private:
    int solve(int n, int i, int j, const vector<vector<int>>& grid, const vector<pair<int, int>>& moves, vector<vector<vector<int>>>& dp) {
        if (n == 1) return 1;
        if (dp[n][i][j] != -1) return dp[n][i][j];
        
        const int MOD = 1e9 + 7;
        int result = 0;
        for (const auto& move : moves) {
            int ni = i + move.first;
            int nj = j + move.second;
            if (ni >= 0 && ni < 4 && nj >= 0 && nj < 3 && grid[ni][nj] != -1) {
                result = (result + solve(n - 1, ni, nj, grid, moves, dp)) % MOD;
            }
        }
        return dp[n][i][j] = result;
    }
};

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