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 A Square Problem (Posted on 2004-12-26)
Given a unit cube, what is the largest square that can be placed completely inside the cube?

 No Solution Yet Submitted by SilverKnight Rating: 4.0000 (5 votes)

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 Probable solution | Comment 3 of 17 |

The best I could find was 3/4*sqrt(2) or about 1.06066

Call the top face ABCD and the bottom EFGH where the other four edges are AE, BF, CG, and DH.

Mark points at distances of x from A to B, from C to B, from E to H, and from G to H.  These four points will always form a rectangle.  To make then a square form set the lengths of the sides equal.  And solve for x.

sqrt((1-x)^2 + (1-x)^2) = sqrt(x^2 + 1 + x^2)

2 - 4x - 2x^2 = 2x^2 + 1

x = 1/4

Substitute this into either side length to get 3/4*sqrt(2)

I tried many other configuartions for the square, but this is the best I could find.

-Jer

 Posted by Jer on 2004-12-26 17:48:38

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