Black and white tiles in the shower at a Palo Alto Hyatt got me working on random polyominoes. I think it's a great topic for high school algebra class. Paraphrasing from a short, unpublished paper I wrote in 1988: b = number of tiles in body of figure (= order of polyomino) n = number of tiles in neighborhood to define figure r = number of distinct orientations p = probability of a tile being figure color q = 1-p = probability of a tile being background color P = probability of finding a given figure in any orientation and "centered" on a specified tile = r p^b q^n A convenient set to analyze is the polyominoes through order 5. Some have identical b, n and r values, and therefore have the same chance of occurring for all p. For example, the T and Z tetrominoes have b=4, n=8, and r=4. Because of cases like this, the 21 polyominoes through order 5 are described by only 16 probability equations. For each of the 16 classes, it is not hard to find the p and P where the chance of occurrence peaks, and where its curve touches or crosses the curves of the other classes. It's easy to generalize the method to several colors, or to non-connected figures. Some problems, most of which I don't know the answer to, are: 1. If we look only for black polyominoes on a white background, the P curve for most polyominoes is asymmetric. For polyominoes with b=n, it is symmetric. An easy example is the 4X4 16-omino. Is there a lower-order polyomino whose P curve is symmetric? Are there any other 16-ominoes with a symmetric P curve? What is the smallest irregular (define: r=8) polyomino with a symmetrical curve? 2. Is there any polyiamond or polyhex whose b, n and r values are the same as those of some polyomino? 3. The point at p=1/2 where the P curves for four classes of polyomino intersect is difficult to graph, because of the relative values of the curves. What scaling of the axes would show the area around this point clearly? (Members of those 4 classes are: straight tetromino; L pentomino; T, U & Z pentominoes; X pentomino.) 4. For polyominoes of any given order, there is a minimum and a maximum n. Consider a histogram of the n values for a given b. Is this histogram always unimodal, for every b? If so, what is the most common n, as a function of b, as b increases without bound? If not, what is the smallest order of polyomino with a multimodal histogram? 5. What is the smallest order polycube whose P curve is symmetric? What are the two smallest polycubes whose P curves intersect? 6. It is easy to show that p of the peak = b / (b+n). Using this, it is easy to show that the peak likelihood is P_peak = r (b^b n^n) / (b+n)^(b+n). The symmetry of this formula in b and n leads to the following conclusion. Consider two polyominoes, the b of one equal to the n of the other, and vice versa. The symmetry of the P_peak equation tells us that the likelihood curves of the two shapes have the same maximum value. Can you find any such pairs of polyominoes? Better yet, can you find a way of generating an infinite family of such pairs? One idea is pairs of rectangles, but this does not work to make an infinite family. With some effort, you can show there are only 5 integer rectangle pairs where the area of one is the perimeter of the other, and vice versa: 1 X 34 and 7 X 10 1 X 38 and 6 X 13 1 X 54 and 5 X 22 2 X 10 and 4 X 6 2 X 13 and 3 X 10 (The related 3-dimensional problem of surface-volume pairs of bricks is much harder. Rich Schroeppel and Dean Hickerson worked some theory on this. My computer search found there are 246 pairs of different bricks, plus 10 bricks whose S = V.)