A rectangle can be cut into two connected pieces of equal area so that they can be rearranged to get a new rectangle with different side lengths than the original one. Show that this can be done in countable many ways. Can this be done in uncountable many ways?
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Let n ≥ 1 be an integer. Divide the width of an H-by-W rectangle into n+1 equal parts, and the height into n equal parts. Separate the rectangle into two connected and congruent pieces along a path stepped as necessary from a side divided at the 1/(n+1) part of its length, down a distance of 1/n of its height, and so on alternating until the opposite side of the rectangle is reached. The two parts may then be "slid" across one another to form a rectangle of height (n+1)/n H and width n/(n+1) W.
That gives a countable number of ways to rearrange the rectangle, at most one of which could be congruent to the original (i.e. when W = (n+1)/n H for some n, or conversely).
I believe these are the only such dissections of a rectangle into two pieces, and if so there are not uncountably many ways to do it (though of course the roles of height and width can be exchanged). The locations of the four original corners (right angles) are constrained so that they can only appear on the boundary of the rearranged rectangle, and the rearrangement works both ways. The dissection must thus operate so as to separate the boundary of both rectangles into two parts (because an intact boundary of a rectangle cannot be encompassed by a different rectangle of equal area). A bit of case analysis should then show that the two rectangles share a pair of opposite corners and that the path dividing the pieces must enter and exit at the "new" pair of corners.