<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>

from collections import defaultdict
class Solution:
    def knightDialer(self, n: int) -> int:
        
        def constructStartToDstMap():
            phonePad = [
                [1, 2, 3],
                [4, 5, 6],
                [7, 8, 9],
                [-1, 0, -1]
            ]

            startPosToDsts = defaultdict(list)

            offsetsForPossibleDsts = [
                (2, -1), (2, 1), #move down
                (1, 2), (-1, 2), #move right
                (-2, 1), (-2, -1), #move up
                (-1, -2), (1, -2) #move left
            ]

            for startRow in range(len(phonePad)):
                for startCol in range(len(phonePad[0])):                    

                    startCellValue = phonePad[startRow][startCol]
                    if startCellValue == -1:
                        continue

                    for rowOffset, colOffset in offsetsForPossibleDsts:
                        dstRow = startRow + rowOffset
                        dstCol = startCol + colOffset
                        if dstRow in range(4) and dstCol in range(3):
                            dstCellValue = phonePad[dstRow][dstCol]
                            if dstCellValue != -1:
                                startPosToDsts[startCellValue].append(dstCellValue)

            return startPosToDsts

        startPosToDsts = constructStartToDstMap()                 

        currentPosAndNumberLenToDstsCount = {}
        def dfs(currentPos, currentNumberLen):

            currPosAndLen = (currentPos, currentNumberLen)
            if currPosAndLen in currentPosAndNumberLenToDstsCount:
                return currentPosAndNumberLenToDstsCount[currPosAndLen]

            if currentNumberLen == n:
                currentPosAndNumberLenToDstsCount[currPosAndLen] = 1
                return 1
            
            totalDestinationsFromCurrentPos = 0
            for dstCellVal in startPosToDsts[currentPos]:
                totalDestinationsFromCurrentPos += dfs(dstCellVal, currentNumberLen + 1)

            currentPosAndNumberLenToDstsCount[currPosAndLen] = totalDestinationsFromCurrentPos
            return totalDestinationsFromCurrentPos

        totalCount = 0
        for startPos in range(10):
            totalCount += dfs(startPos, 1)

        return totalCount % (pow(10, 9) + 7)
66180