Questions: Latin Squares and Orthogonal Structures

Question 1

A researcher needs an experimental design to test 6 treatments while controlling for two independent blocking factors, each with 6 levels. She plans to use two mutually orthogonal Latin squares of order 6. What is the fundamental problem with this plan?

• A. No Latin square of order 6 exists, so the design cannot be constructed at all
• B. No pair of mutually orthogonal Latin squares of order 6 exists — their nonexistence for n = 6 was proved by Tarry in 1901
• C. Two orthogonal Latin squares of order 6 exist, but three are required for two blocking factors
• D. Orthogonal Latin squares require prime order, and since 6 is composite, the required balance property cannot hold

Answer: (B)

Latin squares exist for every order n ≥ 1. But orthogonal Latin squares (OLS) — two squares whose superposition gives every ordered pair exactly once — do not exist for all orders. Euler conjectured that no OLS exist for orders of the form 4k+2 (including n = 6). For n = 6, Tarry confirmed this by exhaustive enumeration in 1901. The plan fails because the required orthogonal structure cannot be built at order 6, regardless of how the squares are arranged. Option D is a common misconception: Latin squares (not just OLS) exist for composite orders.

Question 2

Which property of a Latin square makes it effective as a statistical design for simultaneously controlling two nuisance variables?

• A. Each symbol appears at least once in every row, ensuring row-based balance while column assignment is flexible
• B. The symbols form a quasigroup under composition, guaranteeing statistical independence between treatment levels
• C. Each symbol appears exactly once in every row and exactly once in every column, so each treatment is automatically balanced across both blocking factors
• D. The square can be partitioned into transversals, each representing an independent experimental replicate

Answer: (C)

The double constraint — exactly once per row AND exactly once per column — is what enables simultaneous control of two nuisance variables. Rows represent one blocking factor (e.g., time period), columns represent another (e.g., location or batch). Because each treatment appears exactly once in each row and once in each column, treatment comparisons are balanced across both sources of variation — neither factor can confound the treatment effect. Option A describes only half the constraint, which would control only one blocking factor.

Question 3

A Sudoku puzzle is a constrained Latin square: a 9×9 Latin square using symbols 1–9 with the additional constraint that each symbol also appears exactly once in each of the nine 3×3 sub-grids.

• T. True
• F. False

Answer: True

Every valid Sudoku grid satisfies the Latin square condition (each digit 1–9 appears exactly once in each row and exactly once in each column). The 3×3 box constraint is an additional regional requirement layered on top. This makes Sudoku a proper constrained Latin square — all valid Sudoku grids are Latin squares, but not all Latin squares are valid Sudoku grids. The extra constraint dramatically restricts the solution space compared to the full set of 9×9 Latin squares.

Question 4

Two Latin squares are orthogonal if, when superimposed cell by cell, at least one ordered pair of symbols appears in nearly every cell of the resulting grid.

• T. True
• F. False

Answer: False

Orthogonality requires that *every* ordered pair appears *exactly once* across all n² cells — not merely that every pair appears somewhere. If n = 3, there are 9 ordered pairs from a 3-symbol alphabet, and the 3×3 superposition has exactly 9 cells; each pair must fill exactly one cell. If any pair appears twice or is absent, the squares are not orthogonal. This precise balance — every combination equally represented — is what makes orthogonal Latin squares powerful for experimental design and their nonexistence for certain orders significant.

Question 5

What makes two orthogonal Latin squares particularly useful for experimental design, and why doesn't this property hold for all pairs of Latin squares?

Model answer: When two orthogonal Latin squares are superimposed, every combination of one symbol from each square appears exactly once across all cells. In an experiment, this means every treatment (coded by the first square) is paired with every level of a second factor (coded by the second square) exactly once — no combination is over- or under-represented. This perfect balance enables unconfounded estimation of treatment effects. Not all pairs of Latin squares are orthogonal: superimposing arbitrary Latin squares may leave some ordered pairs missing and others repeated, creating imbalance that confounds comparisons.

The existence of OLS is deeply tied to number theory — complete sets of n−1 mutually orthogonal Latin squares exist when n is a prime power, but fail for n = 6 (and the status at n = 10 required computer search). The connection to finite projective planes explains why OLS existence is a profound combinatorial question, not just an engineering one.