fix a positive constant k. consider a collection of polyominoes (located on the square grid) so that each n-omino is adjacent (along edges) to other polyominoes with total area kn. a little thought will convince you that: 1) when k=1, there are trivial solutions. 2) there are no examples unless k is an integer. 3) there are examples for arbitrarily large k. surprisingly, the solutions when k=2 have been completely classified by joe devincentis. the areas of the polyominoes involved in such a configuration have to be multiples of one of the following sets: {1}, {1,2}, {1,2,3}, {1,2,3,4}, or {1,2,3,4,5,6}. the proof is omitted, but the readers of this list should have no problem checking the various cases. the smallest known solutions (in terms of total area) for various values of k and various area sets are shown at the link below. improvements are welcome. http://www2.stetson.edu/~efriedma/mathmagic/0716.html <http://www2.stetson.edu/~efriedma/mathmagic/0716.html> is there any classification for k=3 as to what area sets are possible? when k=4, is there a solution using only 3- and 4-ominoes? it seems unlikely, but a proof has eluded me. what is the smallest solution (in terms of total area, or area set) for a given value of k? erich friedman