<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>

class Solution {
public:
    const int mod = 1e9 + 7;
    int dx[8] = {2, 1, -1, -2, -2, -1, 1, 2};
    int dy[8] = {1, 2, 2, 1, -1, -2, -2, -1};
    int knightDialer(int n) {
        vector<vector<int>> dp(4, vector<int>(3, 0));
        
        // Initialize dp for n = 1
        for (int i = 0; i < 4; i++) {
            for (int j = 0; j < 3; j++) {
                if (i == 3 && (j == 0 || j == 2)) continue;  // 5, 9, 0, and 1 don't have two squares vertically
                dp[i][j] = 1;
            }
        }
        
        // Compute dp for n > 1
        for (int k = 2; k <= n; k++) {
            vector<vector<int>> temp(4, vector<int>(3, 0));
            for (int i = 0; i < 4; i++) {
                for (int j = 0; j < 3; j++) {
                    if (i == 3 && (j == 0 || j == 2)) continue;
                    for (int l = 0; l < 8; l++) {
                        int ni = i + dx[l], nj = j + dy[l];
                        if (ni >= 0 && ni < 4 && nj >= 0 && nj < 3 && !(ni == 3 && (nj == 0 || nj == 2))) {
                            temp[i][j] = (temp[i][j] + dp[ni][nj]) % mod;
                        }
                    }
                }
            }
            dp = temp;
        }
        
        // Sum all possible starting positions
        int res = 0;
        for (int i = 0; i < 4; i++) {
            for (int j = 0; j < 3; j++) {
                if (i == 3 && (j == 0 || j == 2)) continue;
                res = (res + dp[i][j]) % mod;
            }
        }
        
        return res;
    }
};
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