MathDB

Let

k \in \mathbb{N},

S_k = \{(a, b) | a, b = 1, 2, \ldots, k \}.

Any two elements

(a, b)

(c, d)

\in S_k

are called "undistinguishing" in

S_k

a - c \equiv 0

\pm 1 \pmod{k}

and

b - d \equiv 0

\pm 1 \pmod{k}

; otherwise, we call them "distinguishing". For example,

(1, 1)

and

(2, 5)

are undistinguishing in

S_5

. Considering the subset

A

S_k

such that the elements of

A

are pairwise distinguishing. Let

r_k

be the maximum possible number of elements of

A

. (i) Find

r_5

. (ii) Find

r_7

. (iii) Find

r_k

for

k \in \mathbb{N}

Problem Statement