called a t-(v,k,λ)-design. Again, these four numbers determine b and r and the four numbers themselves cannot be chosen arbitrarily. The equations are where λi is the number of blocks that contain any i-element set of points and λt = λ. Note that and . Theorem: Any t-(v,k,λ)-design is also an s-(v,k,λs)-design for any s with 1 ≤ s ≤ t. (Note that the "lambda value" changes as above and depends on s.) A consequence of this theorem is that every t-design with t ≥ 2 is also a 2-design. A t-(v,k,1)-design is called a Steiner system. The term block design by itself usually means a 2-design. Derived and extendable t-designs Let D = (X, B) be a t-(v,k,λ) design and p a point of X. The derived design Dp has point set X − {p} and as block set all the blocks of D which contain p with p removed. It is a (t − 1)-(v − 1, k − 1, λ) design. Note that derived designs with respect to different points may not be isomorphic. A design E is called an extension of D if E has a point p such that Ep is isomorphic to D; we call D extendable if it has an extension. Theorem: If a t-(v,k,λ) design has an extension, then k + 1 divides b(v + 1). The only extendable projective planes (symmetric 2-(n2 + n + 1, n + 1, 1) designs) are those of orders 2 and 4. Every Hadamard 2-design is extendable (to an Hadamard 3-design). Theorem:. If D, a symmetric 2-(v,k,λ) design, is extendable, then one of the following holds: D is an Hadamard 2-design, v = (λ + 2)(λ2 + 4λ + 2), k = λ2 + 3λ + 1, v = 495, k = 39, λ = 3. Note that the projective plane of order two is an Hadamard 2-design; the projective plane of order four has parameters which fall in case 2; the only other known symmetric 2-designs with parameters in case 2 are the order 9 biplanes, but none of them are extendable; and there is no known symmetric 2-design with the parameters of case 3. Inversive planes A design with the parameters of the extension of an affine plane, i.e., a 3-(n2 + 1, n + 1, 1) design, is called a finite inversive plane, or Möbius plane, of order n. It is possible