Last Whole Piece

Geometry Level 3

The animation to the right implies that we can eat an infinite amount of chocolate from the same chocolate bar, but it is misleading—after each reassembly of the chocolate bar, the height of the chocolate bar actually decreases slightly.

Suppose you kept on cutting, taking out one whole piece while reassembling the remaining pieces into a solid rectangular bar without holes. How many whole pieces can be taken out in this way?

Let the number of whole pieces removed from the bar—out of the ten pieces labeled 1 to 10 below—be \(n,\) and let the number written on the last piece removed be \(x.\)

What is \(nx?\)

Details and Assumptions:

  • At the start, the sloping cut passes through the bottom right corner of piece 9, so that all the pieces below it stays the same each time the cut pieces are reassembled.
  • Each reassembly is done with 3 cut pieces (out of 4), as in the animation, along and above the red, dotted line. Remember, 1 labeled piece is always eaten up after each reassembly.
  • If after taking out a whole piece, the remaining pieces cannot be reassembled into a solid bar without holes, that piece does not count.
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