In the Fano plane S(2, 3, 7), you pick any two of the 7 points. How many blocks contain both of those points?
A0 — the Fano plane has lines, not blocks, so pairs are not guaranteed to share a line
B1 — every 2-element subset appears in exactly λ = 1 block, by definition of a (v, k, λ)-design
C3 — each point belongs to 3 lines, so any two points must share 3 blocks
DIt depends on which two points are chosen — the design is not balanced for all pairs
The defining property of a (v, k, λ)-design is that every 2-element subset of the point set appears in exactly λ blocks. For the Fano plane S(2, 3, 7), λ = 1. This holds for every pair without exception — this is precisely what 'balanced' means. Option C confuses the replication number r (how many blocks contain a single point, which is 3 for the Fano plane) with λ (how many blocks contain any specific pair of points, which is 1). Option D denies the balancedness that is built into the definition.
Question 2 Multiple Choice
A combinatorialist verifies that a proposed parameter set (v, k, λ) = (21, 5, 1) satisfies all necessary conditions: Fisher's inequality holds and both r and b are positive integers. What can she conclude?
AThe design exists — satisfying all necessary parameter conditions guarantees existence
BThe design may or may not exist — the necessary conditions are not sufficient for existence
CThe design does not exist — (21, 5, 1) parameters are provably impossible for Steiner systems
DShe must compute the determinant of the incidence matrix to determine whether the design exists
This is a fundamental distinction in combinatorial design theory: necessary conditions are not sufficient for existence. The parameter conditions (Fisher's inequality, integrality of r and b) must hold for any design to exist, but passing these tests does not guarantee a design can be constructed. A parameter set can satisfy every necessary condition and still have no design — existence proofs require explicit constructions, while non-existence despite valid parameters requires algebraic or combinatorial arguments. This asymmetry between necessary and sufficient conditions is a defining challenge of the field.
Question 3 True / False
In any balanced incomplete block design, every point appears in exactly the same number of blocks.
TTrue
FFalse
Answer: True
This follows from the 'balanced' condition by a straightforward counting argument. Since every pair of points appears in exactly λ blocks, and any fixed point forms pairs with (v−1) other points, the total number of (point, block) incidences for that point is λ(v−1). Since each block containing the point contributes k−1 such pairs, the replication number r = λ(v−1)/(k−1) is the same for every point. The design is balanced with respect to individual points as well as pairs, which is what makes BIBDs useful in experimental design: every treatment receives equal representation.
Question 4 True / False
If a parameter set (v, k, λ) satisfies most necessary divisibility conditions, a balanced incomplete block design with those parameters is very likely to exist.
TTrue
FFalse
Answer: False
The necessary conditions — λ(v−1) divisible by k−1, λv(v−1) divisible by k(k−1), Fisher's inequality b ≥ v — are required for existence but not sufficient. The existence question for specific parameters is often a deep open problem. There exist parameter sets satisfying all necessary conditions for which no design exists, demonstrated only by non-existence proofs (often using eigenvalue methods or algebraic constraints on the incidence matrix). Existence typically requires an explicit construction; the gap between necessary and sufficient conditions is central to the theory.
Question 5 Short Answer
What does 'balanced' mean in 'balanced incomplete block design,' and why is this property essential for the design's applications in experimental statistics?
Think about your answer, then reveal below.
Model answer: 'Balanced' means that every 2-element subset of the point set (every pair of treatments, in the statistical context) appears together in exactly λ blocks. No pair is favored over another — every pair receives the same number of direct comparisons across the experiment. 'Incomplete' means each block contains only k out of v elements (k < v), making the design practical when including all treatments in every block is impossible or too costly. The balance property is essential for statistics because it ensures unbiased estimation: every pair of treatment effects can be estimated with equal precision, eliminating systematic confounding. If some pairs appeared together more often than others, comparisons involving those pairs would be more informative than others, introducing bias into the experiment's conclusions.
Fisher developed BIBD theory precisely to provide a mathematical guarantee of statistical fairness. The combinatorial symmetry directly translates into the statistical property that no treatment is systematically advantaged or disadvantaged by the experimental arrangement.