A8636 | Cube Snake

You've got an $n×n×n$ cube, split into unit cubes. Your task is to number all unit cubes in this cube with positive integers from $1$ to $n^{3}$ so that:

- each number was used as a cube's number exactly once;
- for each $1<=i<n^{3}$ , unit cubes with numbers $i$ and $i+1$ were neighbouring (that is, shared a side);
- for each $1<=i<n$ there were at least two different subcubes with sizes $i×i×i$ , made from unit cubes, which are numbered with consecutive numbers. That is, there are such two numbers $x$ and $y$ , that the unit cubes of the first subcube are numbered by numbers $x$ , $x+1$ , $...$ , $x+i^{3}-1$ , and the unit cubes of the second subcube are numbered by numbers $y$ , $y+1$ , $...$ , $y+i^{3}-1$ .

Find and print the required numeration of unit cubes of the cube.

The first line contains a single integer $n$ ( $1<=n<=50$ ) — the size of the cube, whose unit cubes need to be numbered.

Print all layers of the cube as $n$ $n×n$ matrices. Separate them with new lines. Print the layers in the order in which they follow in the cube. See the samples for clarifications.

It is guaranteed that there always is a solution that meets the conditions given in the problem statement.