MathDB

Let

n

be a nonzero natural number and let

S = \{1, 2, . . . , n\}

. A

3 \times n

board is called beautiful if it can be completed with numbers from the set

S

like this as long as the following conditions are met:

\bullet

on each line, each number from the set S appears exactly once,

\bullet

on each column the sum of the products of two numbers on that column is divisible by

n

(that is, if the numbers

a, b, c

are written on a column, it must be

ab + bc + ca

be divisible by

n

). For which values of the natural number

n

are there beautiful tables ¸and for which values do not exist? Justify your answer.

Problem Statement