MathDB

3 \times n

grid is filled as follows. The first row consists of the numbers from

1

n

arranged in ascending order. The second row is a cyclic shift of the top row: i,i\plus{}1,...,n,1,2,...,i\minus{}1 for some

i

. The third row has the numbers

1

n

in some order so that in each of the

n

columns, the sum of the three numbers is the same. For which values of

n

is it possible to fill the grid in this way? For all such

n

, determine the number of different ways of filling the grid.

Problem Statement