<p>You are given an integer array <code>coins</code> representing coins of different denominations and an integer <code>amount</code> representing a total amount of money.</p>

<p>Return <em>the fewest number of coins that you need to make up that amount</em>. If that amount of money cannot be made up by any combination of the coins, return <code>-1</code>.</p>

<p>You may assume that you have an infinite number of each kind of coin.</p>

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

<pre>
<strong>Input:</strong> coins = [1,2,5], amount = 11
<strong>Output:</strong> 3
<strong>Explanation:</strong> 11 = 5 + 5 + 1
</pre>

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

<pre>
<strong>Input:</strong> coins = [2], amount = 3
<strong>Output:</strong> -1
</pre>

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

<pre>
<strong>Input:</strong> coins = [1], amount = 0
<strong>Output:</strong> 0
</pre>

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

<ul>
	<li><code>1 &lt;= coins.length &lt;= 12</code></li>
	<li><code>1 &lt;= coins[i] &lt;= 2<sup>31</sup> - 1</code></li>
	<li><code>0 &lt;= amount &lt;= 10<sup>4</sup></code></li>
</ul>

class Solution:
    def coinChange(self, coins: List[int], amount: int) -> int:
        coins.sort(reverse=True)
        coinSet = set(coins)
        mapping = {}

        evenCoins = {}

        for c in coins:
            for d in coins:
                if c > d and c % d == 0:
                    if c in evenCoins:
                        evenCoins[c].append(d)
                    else:
                        evenCoins[c] = [d]

        if amount == 0:
            return 0

        def recurse(n: int):
            if n in coinSet:
                return 1

            if n in mapping:
                return mapping[n]

            found = None

            coinsToSkip = set()

            for c in coins:
                if c not in coinsToSkip and n > c:
                    temp = recurse(n-c)
                    if temp > 0:
                        if c in evenCoins:
                            for e in evenCoins[c]:
                                coinsToSkip.add(e)

                        if found is None:
                            found = temp + 1
                        else:
                            found = min(found, temp + 1)

            if found is None:
                mapping[n] = -1
                return -1
            else:
                mapping[n] = found
                return found

        return recurse(amount)

            
        
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