A unit square of paper is to be cut into six rectangles and reassembled to form the sides of a cuboid. (Wasted paper is allowed.)
What is the maximum volume, , of the cuboid thus formed?
Let be the least integer such that . Enter the value of .
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Following Jeremy's dissection of the unit square, suppose instead the two smallest faces have dimensions a × b , while the other two pairs have dimensions 2 1 × a and 2 1 × b , so that a + b + a = 1 where a < b . The volume of the cuboid is then
V = 2 1 a ( 1 − 2 a )
so that the maximum occurs at a = 4 1 . Any departure from this (resulting in leftover paper) will mean less volume.