diff --git a/hw1/LinZhu/9.6 b/hw1/LinZhu/9.6
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+code is in the .py file with explaination
diff --git a/hw1/LinZhu/knapsack 9.6.py b/hw1/LinZhu/knapsack 9.6.py
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+++ b/hw1/LinZhu/knapsack 9.6.py	
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+def knapsack_value_dp(weights, values, capacity):
+    l = len(weights)
+    Vmax = max(values)
+    total_value = sum(values)
+
+    # Initialize DP table with infinity
+    INF = float('inf')
+    dp = [[INF] * (total_value + 1) for _ in range(l + 2)]
+    
+    # Base case: 0 value requires 0 weight
+    dp[l + 1][0] = 0
+
+    # Build table from item l down to 1
+    for j in range(l, 0, -1):
+        for v in range(total_value + 1):
+            # Option 1: exclude item j-1
+            dp[j][v] = dp[j + 1][v]
+            # Option 2: include item j-1, if value allows
+            if v >= values[j - 1]:
+                dp[j][v] = min(dp[j][v], weights[j - 1] + dp[j + 1][v - values[j - 1]])
+
+    # Find the maximum value v such that dp[1][v] <= capacity
+    for v in range(total_value, -1, -1):
+        if dp[1][v] <= capacity:
+            return v  # maximum value achievable within capacity
+
+    return 0  # no items can be included
+
+# Example usage
+weights = [2, 3, 4, 5]
+values = [3, 4, 5, 6]
+capacity = 7
+
+result = knapsack_value_dp(weights, values, capacity)
+print("Maximum value within capacity:", result)
+
+
+#If P = NP, the NP problem will be solved in polynomial time,
+#O(l^2 * v_max) where v_max is a constant, so bits required for v_max is log2(v_max), which is not a polynomial
+#P != NP
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