May 2007
Ponder This Solution:
The answer is 3*sqrt(2)/4 = 1.06066+.
Suppose the unit cube has vertices (x,y,z) where x,y,z are in {0,1}. Then an example of a maximum size embedded square has corners (0,0,.75), (1,.25,1), (1,1,.25) and (0,.75,0).
We sketch a proof that this is largest possible embedded square. Let (r,s,t) be the center of the square. Let (r+a,s+b,t+c) and (r-a,s-b,t-c) be one pair of opposite corners of the square and let (r+x,s+y,t+z) and (r-x,s-y,t-z) be the other pair of opposite corners. Since these corners must lie within the unit cube it is easy to see that |a|,|b|,|c|,|x|,|y|,|z| <= .5 and that we may assume that r=s=t=.5. Furthermore the diagonals of a square meet at right angles so a*x+b*y+c*z=0. Also the diagonals of a square have equal length so a*a+b*b+c*c=x*x+y*y+z*z=d*d/4 (d is the length of the diagonal). We want the maximum value of d so that we can find a,b,c,x,y and z satisfying the above. If we let d=1.5 then setting (a,b,c) = (.5,.5,-.25) and (x,y,z) = (.5,-.25,.5) satisfies these constraints and produces the above example. It is easy although a bit tedious to verify that we can't do better.
I came up with this problem and solution myself but it seems to be fairly well known as "Prince Rupert's Problem".
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