KöMaL 2011 K313

Zsolti has nine identical drinking glasses. If one glass is placed onto another, the following different results are possible: If two tops or two bottoms touch, the two glasses will stay in that position. If a top meets a bottom, one glass will slide into the other and the combined height of the two glasses will be one and a half times the height of a glass. Zsolti is making a tower of glasses. He starts by arranging the nine glasses on the table in a 3×3 pattern, all with their tops facing upwards. Then he takes one row or column of glasses at the edge of the arrangement and places them, inverted, onto the glasses of the adjacent row or column. Thus he obtains a 2×3 arrangement. Then he inverts one of the rows or columns at the edge onto the adjacent one again, and so on, until a single tower of glasses is built. What may be the height of the tower?