be symmetric, if the labels or blocks are sorted the right way: Symmetric 2-designs (SBIBDs) The case of equality in Fisher's inequality, that is, a 2-design with an equal number of points and blocks, is called a symmetric design. Symmetric designs have the smallest number of blocks among all the 2-designs with the same number of points. In a symmetric design r = k holds as well as b = v, and, while it is generally not true in arbitrary 2-designs, in a symmetric design every two distinct blocks meet in λ points. A theorem of Ryser provides the converse. If X is a v-element set, and B is a v-element set of k-element subsets (the "blocks"), such that any two distinct blocks have exactly λ points in common, then (X, B) is a symmetric block design. The parameters of a symmetric design satisfy This imposes strong restrictions on v, so the number of points is far from arbitrary. The Bruck–Ryser–Chowla theorem gives necessary, but not sufficient, conditions for the existence of a symmetric design in terms of these parameters. The following are important examples of symmetric 2-designs: Projective planes Finite projective planes are symmetric 2-designs with λ = 1 and order n > 1. For these designs the symmetric design equation becomes: Since k = r we can write the order of a projective plane as n = k − 1 and, from the displayed equation above, we obtain v = (n + 1)n + 1 = n2 + n + 1 points in a projective plane of order n. As a projective plane is a symmetric design, we have b = v, meaning that b = n2 + n + 1 also. The number b is the number of lines of the projective plane. There can be no repeated lines since λ = 1, so a projective plane is a simple 2-design in which the number of lines and the number of points are always the same. For a projective plane, k is the number of points on each line and it is equal to n + 1. Similarly, r = n + 1 is the number of lines with which a given point is incident. For n = 2 we get a projective plane of order 2, also called the Fano plane, with v = 4 + 2 + 1 = 7 points and 7 lines. In