## Task 4 - 270614

Kerstin cuts a rectangular piece of paper so that the cut runs from the midpoint of one rectangular side to the midpoint of the opposite side. This results in two smaller rectangles; Kerstin places them exactly on top of each other so that only a smaller rectangle is visible, consisting of two layers.

Kerstin cuts this two-layer rectangle in the same way and again places the resulting rectangular pieces exactly on top of each other. This process continues: overlapping rectangles are cut, and the resulting pieces are placed on top of each other.

(Of course, this cannot be done indefinitely; the stack of paper to be cut becomes thicker each time.)

(1) State the number of paper pieces that have been created after the

a) second, b) third, c) fourth cutting!

(2) Assuming that cutting could be done indefinitely, why would it still be impossible for exactly 160 paper pieces to ever result from a single cut?
(1) The number is a) 4, b) 8, c) 16.

(2) As further numbers after the fifth, sixth, ... cut, etc., appear: $32, 64, 128, 256, \ldots$ and then even larger numbers (namely, each time double the preceding number). Therefore, the number 160 does not appear among the numbers that occur.

Solutions of the I. Round 1987 adopted from [5]

### 3.28.2 II. Round 1987, Class 6