Imagine being a thief with taste.

Every item has a weight and a value.

Your bag has a hard limit.

Your mission: maximize value without breaking your back.

That’s the 0/1 Knapsack Problem.

You either take the item (1) or leave it (0).

No cutting gold bars in half to cheat the system.

Dynamic Programming cracks it by:

• Splitting the problem into tiny sub-decisions

• Saving results in a 2D table

• Reusing those results instead of recomputing

Every cell asks a simple but brilliant question:

The formula for the optimal choice:

B[i][j] = max( B[i−1][j], V[i] + B[i−1][ j−W[i] ] )

Take it if it improves your value.

Skip it if it weighs you down.

We fill the table bottom-up, then trace back the best loot.

We reconstruct exactly what the thief walks out with — mathematically optimal crime.

It’s not just for burglars:

• Cargo loading

• Budget optimization

• Portfolio selection

• Memory constraints in software

Knapsack teaches a powerful truth:

Optimization is just saying no to the wrong things.

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