How Do You Build a Circuit Using a 4:1 Multiplexer?

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Hmm, that's a challenging one. I think you've done the correct job of making the K-map, and now you need to play with different combinations of what to make the don't-cares, and what you can do with a 4:1 multiplexer. Your 4:1 MUX actually has 6 inputs and 1 output, right? There's the 4 inputs to the MUX, plus the 2 selector lines. So look at some variations of the truth table with different configurations of the X values (don't cares), and see if the simplest truth table variations are amenable to using a 4:1 MUX to form the outputs of the truth table. I haven't solved the problem myself yet, but that's how I would approach it.