adding one square at a time, one adds a cluster of squares at a time. This is often described as adding twigs. By proving that every n-omino is a sequence of twigs, and by proving limits on the combinations of possible twigs, one obtains an upper bound on the number of n-ominoes. For example, in the algorithm outlined above, at each step we must choose a larger number, and at most three new numbers are added (since at most three unnumbered squares are adjacent to any numbered square). This can be used to obtain an upper bound of 6.75. Using 2.8 million twigs, Klarner and Rivest obtained an upper bound of 4.65, which was subsequently improved by Barequet and Shalah to 4.5252. Free polyominoes Approximations for the number of fixed polyominoes and free polyominoes are related in a simple way. A free polyomino with no symmetries (rotation or reflection) corresponds to 8 distinct fixed polyominoes, and for large n, most n-ominoes have no symmetries. Therefore, the number of fixed n-ominoes is approximately 8 times the number of free n-ominoes. Moreover, this approximation is exponentially more accurate as n increases. Special classes of polyominoes Exact formulas are known for enumerating polyominoes of special classes, such as the class of convex polyominoes and the class of directed polyominoes. The definition of a convex polyomino is different from the usual definition of convexity, but is similar to the definition used for the orthogonal convex hull. A polyomino is said to be vertically or column convex if its intersection with any vertical line is convex (in other words, each column has no holes). Similarly, a polyomino is said to be horizontally or row convex if its intersection with any horizontal line is convex. A polyomino is said to be convex if it is row and column convex. A polyomino is said to be directed if it contains a square, known as the root, such that every other square can be reached by movements of up or right one square, without leaving the polyomino. Directed polyominoes, column (or row) convex polyominoes, and convex polyominoes have been effectively enumerated by area n, as well as by some other parameters such as perimeter, using generating functions. A polyomino is equable if its area equals its perimeter. An equable polyomino must be made from an even number of squares; every even number greater than 15 is possible. For