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If a square in the interior of the cube or boundaries, Find the largest square side length of $a$, if the length of the cube as $1$ enter image description here

and this problem is from china enter image description hereenter image description here

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I think you'll find what you need at mathoverflow.net/questions/15498/… –  Gerry Myerson Sep 30 '13 at 10:03
    
Thank you,@GerryMyerson,I find sometimes, I can't find solution with the largest is $\dfrac{9}{8}$,can you help me,Thank you,and I think this is good problem –  user94270 Sep 30 '13 at 11:02
    
Maybe a better link is planetmath.org/geometry and scroll down to the discussion of "square in a cube". Someone called dh2718 gives an answer, then has more posts giving more details. Maybe you could study those and then post a summary as an answer here. –  Gerry Myerson Oct 1 '13 at 12:49

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