We define a magic triangle of size n as a triangle with n rows, the first one with n elements, the second one with n − 1 elements, …, and the last one with one element, such that:
Write a program that, given an n, prints all the magic triangles of size n, or indicates that there is none.
Input
Input consists of a natural number n > 0.
Output
Print all the magic triangles of size n. Note that the numbers are always written with two digits (if needed, adding a zero to the left), and that they are separated with two spaces. Print an empty line after each triangle.
Ignore the triangles that are symmetrical of any other lexicographically smaller. If there is no possible triangle for the given n, indicate it following the format of the example.
Observation
It can be proved that there is no solution for n ≥ 6, but do not use this fact in your program.
You can print the solutions to this exercise in any order.
Input
2
Output
01 03 02 02 03 01
Input
5
Output
06 14 15 03 13 08 01 12 10 07 11 02 04 09 05
Input
6
Output
no solution for n = 6