On a Problem of Arrangements

The following problem was proposed by Dr. Vijayaraghavan to Gul Abdulla and Lal Bahadur who have established the possibility of its solution when 2n + 1 is a prime number, and have also given the solution of the problem when 2n + 1= 21, a composite number. Later Dr. Hansaraj Gupta has shown how from a solution for n = m-1, a solution for n = m can in general be constructed. In proving this, he assumed an unproved problem in permutations, whose solution 'seems always to exist'. By using Gupta's method, Dr. S. Chowla has proved that the problem is solvable for n=m + 1 where 2n-1 is a prime number. But Prof. F. W. Levi has obtained a simple proof of the general problem which is not yet published. The following is an independent proof for all 2n+1, composite or prime.