Simplifying Digital Logic Equations: Using Karnaugh Maps and 2x2 Method

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The discussion focuses on simplifying a digital logic equation using Karnaugh maps, specifically comparing the effectiveness of the square method and the 2x1 method. The user seeks clarification on whether both methods can yield valid solutions or if the 2x2 method is superior for simplification. Additionally, there is an inquiry about how to simplify a solution derived from the second case back to the first case without using Karnaugh maps. The conversation emphasizes the importance of understanding the simplification process in digital logic design. Ultimately, the goal is to achieve a more efficient representation of the logic equation.
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1. Homework Statement and 3. The Attempt at a Solution
Simplify the equation in the following picture using karnaugh.

http://i.imgur.com/1T2HKpF.jpg

I divided it into two cases, one I used the square and the other I used the 2x1. Will either method work to get the solution, or is the 2x2 better since it is more simplified?

Let's say I wanted to simplify the case 2 solution to get case 1 without karnaugh, how would I got about doing this?
 
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You can simplify the case 2 solution, using that C+\overline{C}B=C+B

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