<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:
    def knightDialer(self, n: int) -> int:
        phone = [[True, True, True], [True, True, True], [True, True, True], [False, True, False]]
        ROWS = 4 
        COLS = 3
        MOD = 10**9 + 7
        cache = {}

        def canMove(i: int, j: int) -> bool: 
            if i < 0 or i >= ROWS or j < 0 or j >= COLS: 
                return False 
            return phone[i][j]
        
        # m: moves, i, j positions
        def search(m: int, i: int, j: int) -> int: 
            if m == n: 
                return 1 

            if (m, i, j) in cache: 
                return cache[(m, i, j)]

            moves = [[-1, -2], [-2, -1], [-1, 2], [2, -1], [1, -2], [-2, 1], [1, 2], [2, 1]]

            counts = 0 

            for move in moves: 
                nI = i + move[0]
                nJ = j + move[1]
                if canMove(nI, nJ): 
                    counts = (counts + search(m + 1, nI, nJ)) % MOD
            
            cache[(m, i, j)] = counts 
            return counts
        
        res = 0 

        for i in range(ROWS): 
            for j in range(COLS):
                if phone[i][j]:
                    res = (res + search(1, i, j)) % MOD
        # print(cache)
        return res
            

33825