OFFICIAL GUIDE EXPLANATIONThe left figure below shows a cube with edge length 4, where P is the lower left vertex on its front face and Q is the upper right vertex on its back face. It is clear that the shortest path on the surface between P and Q consists of a path on the front face joined to a path on the top face, or a path on the front face joined to a path on the right face. In fact, each of these two approaches can be used in essentially the same way to give a path from P to Q, whose length is the surface distance between P and Q. To simplify the exposition, we will consider the case where the surface distance is the length of a certain path PF on the front face plus the length of a certain path PT on the top face.
The middle figure below shows the top face of the cube lifted about 45 degrees, and the right figure below shows only the front and top faces of the cube after the top face has been lifted 90 degrees. Since rotations of the top face about this “hinge” do not change the length of any path on the front face or on the top face, this rotation by 90 degrees will not change the length of PF or the length of PT, and hence this rotation will not change the sum of the lengths of PF and PT.
In the right figure below, the shortest path from P to Q is the dashed segment shown in the figure. Moreover, the length of this dashed segment is the surface distance between P and Q because if PF and PT did not correspond to portions of this dashed segment on the front and top faces, respectively, then the sum of the lengths of PF and PT would be greater than the length of the dashed segment, since the shortest distance between P and Q in the right figure below is the length of the line segment with endpoints P and Q.
An illustration shows 3 cubes with its upper face being lifted on one side, progressively. The edges of the cube are 4 units in length. The lower left vertex on the cube's front face is labeled P, and the upper right vertex of the back face is labeled Q. The first illustration shows a closed cube. The second illustration shows the upper face being lifted while hinged at the line connecting the upper face and the front face. The third illustration shows the upper face fully lifted and a dotted line connecting point P and point Q.
From the discussion above, it follows that the surface distance between P and Q is the length of the dashed segment in the right figure, which is easily found by using the Pythagorean theorem:
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4
2
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2
=
√
4
2
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=
4
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