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    <p>So you start with a <code>X * Y</code> rectangle. Say the optimal solution involves making a vertical (or horizontal) cut, then you have two new rectangles with dimensions <code>X * Y1</code> and <code>X * Y2</code> with <code>Y1 + Y2 = Y</code>. Since you want to maximize your profit, you need to maximize the profit on these new rectangles (<em>optimal substructure</em>). So your initial recursion goes as follows: <code>f(X, Y) = max(f(X, Y1) + f(X, Y2), f(X1, Y) + f(X2, Y))</code> for all posible values of <code>X1, X2</code> (horizontal cut) and <code>Y1, Y2</code> (vertical cut).</p> <p>Now the question is <em>when do I actually decide to make a product ?</em> You can decide to make a product when one of its dimensions equals one of the dimensions of your current rectangle (why ? Because if this doesn't hold, and the optimal solution includes making this product, then sooner or later you will need to make a vertical (or horizontal) cut and this case is already handled in the initial recursion), so you make the appropriate cut and you have a new rectangle <code>X * Y1</code> (or <code>X1 * Y</code>), depending on the cut you made to obtain the product), in this case the recursion becomes <code>f(X, Y) = cost of product + f(X1, Y)</code>.</p> <p>The solution of the original problem is <code>f(X, Y)</code>. The running time of this dp solution would be <code>O(X * Y * (X + Y + number of available products))</code>: you have <code>X * Y</code> possible rectangles, for each of these you try every possible cut (<code>X + Y</code>) and you try to make one of the available products out of this rectangle.</p> <p>Also, check out this similar problem: <em>Sharing Chocolate</em> from the 2010 ICPC World Finals.</p>
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    1. COThank you for this response. If you don't mind, it would be very helpful for me if you could include some psuedo-code to illustrate this algorithm. For some reason I'm just having a very hard time visualizing this.
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    2. COSpecifically, how would I go about checking for maximum profit on the rectangles?
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    3. COI'll include it when I get home :)
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