Torsten Sillke, 1995-11-24, Frankfurt Update 2000-12-20: X I3 problem solved Rectangles with X5 and I3: -------------------------- rectangles where both polyominoes are used. 12 * 13: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x . . . x . . . x . . . x x x X x x x X x x x X . . x X X X x X X X x X X X . . . X . . . X . . . X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . unique solution. Only the Xs are shown. 10 * 15: (16.11.95) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x . . . x . . . x . . . . . x x x X x x x X x x x X . . . . x X X X x X X X x X X X . . . . . X . . . X . . . X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . almost unique solution. The right upper corner: . . . . . . . . . . . . . . . . . . This figure has 5 solutions with I3. No 9*n rectangle is possible. Each 9*N strip can contain at most one X. The resulting boundaries are: + + 2 2 2 + + 1 1 1 + 2 2 2 + 1 1 1 + + enlargement forces 2 2 2 + + 1 1 1 + + + 2 2 2 + + + + 1 1 1 + + + 2 2 2 + + 1 1 1 + 2 2 2 + 1 1 1 + + enlargement forces 2 2 2 + + 1 1 1 + + + 2 2 2 + + + + 1 1 1 + + + 2 2 2 + + 2 2 2 + 2 2 2 + 1 1 1 + + enlargement forces 2 2 2 + + 1 1 1 + + + 2 2 2 + + + + 1 1 1 + + + 2 2 2 + + 2 2 2 + + 1 1 1 + 1 1 1 + + enlargement forces 2 2 2 + + 1 1 1 + + + 2 2 2 + + + + 1 1 1 + + + 2 2 2 + + 2 2 2 + + 1 1 1 + 2 2 2 Problem: -------- Are rectangles possible where the area is not a multiple of 3? Answer: no (Mike Reid 2000) General X and I_n with n>=4 problem: ------------------------------------ Are there rectanges possible for X and I_n for n>=4. (I_n is an n-omino.) I think they are impossible. This would follow from the following more general conjecture: --------------- Conjecture: X-I (24.11.95) --------------- Each tiling of k*Z (an double open strip of width k) with the X-pentomino and the I_n polyomino (for n>=4) contain only a finite number of X-pentominoes. More general is the following conjecture. --------------- Conjecture: X-I (07.07.96, corrected Dec.96) --------------- Each tiling of N*Z (the halfplane) with the X-pentomino and the I_n polyomino (for n>=4) has the property: The number of infinite-X-columns is finite or the number of infinite-X-rows is finite. (What is an infinite-X-column? Paint all X in a tiling black and the rest white. If a column of this tiling contains an infinite number of black squares this column is named an infinite-X-column.) ----- Note: ----- Pairs of pentominoes which tile rectangles have been analyzed for some time. Only the combination (X,I) is unsolved yet. Tables of these result can be get from Anton Hanegraaf, Heemskerstraat 9, 6662 AL Elst, NL, +31-481-372402 Frits G"obel and Anneke Treep made a list of (n-omino, pentomino) combinations which tile rectangles. (published in JoRM) -- mailto:Torsten.Sillke@uni-bielefeld.de http://www.mathematik.uni-bielefeld.de/~sillke/