MathDB

Let

p

be a given prime number. For positive integers

n,k\geq2

let

S_1, S_2,\dots, S_n

be unit square sets constructed by choosing exactly one unit square from each of the columns from

p\times k

chess board. If

|S_i \cap S_j|=1

for all

1\leq i < j \leq n

and for any duo of unit squares which are located at different columns there exists

S_i

such that both of these unit squares are in

S_i

find all duos of

(n,k)

in terms of

p

.
Note: Here we denote the number of rows by

p

and the number of columns by

k

Problem Statement