Score : $900$ points

Problem Statement

Diverta City is a new city consisting of $N$ towns numbered $1, 2, ..., N$.

The mayor Ringo is planning to connect every pair of two different towns with a bidirectional road. The length of each road is undecided.

A Hamiltonian path is a path that starts at one of the towns and visits each of the other towns exactly once. The reversal of a Hamiltonian path is considered the same as the original Hamiltonian path.

There are $N! / 2$ Hamiltonian paths. Ringo wants all these paths to have distinct total lengths (the sum of the lengths of the roads on a path), to make the city diverse.

Find one such set of the lengths of the roads, under the following conditions:

• The length of each road must be a positive integer.
• The maximum total length of a Hamiltonian path must be at most $10^{11}$.

Constraints

• $N$ is a integer between $2$ and $10$ (inclusive).

Input

Input is given from Standard Input in the following format:

$N$

Output

Print a set of the lengths of the roads that meets the objective, in the following format:

$w_{1, 1} \ w_{1, 2} \ w_{1, 3} \ ... \ w_{1, N}$

$w_{2, 1} \ w_{2, 2} \ w_{2, 3} \ ... \ w_{2, N}$

$:$ $:$ $:$

$w_{N, 1} \ w_{N, 2} \ w_{N, 3} \ ... \ w_{N, N}$

where $w_{i, j}$ is the length of the road connecting Town $i$ and Town $j$, which must satisfy the following conditions:

• $w_{i, i} = 0$
• $w_{i, j} = w_{j, i} \ (i \neq j)$
• $1 \leq w_{i, j} \leq 10^{11} \ (i \neq j)$

If there are multiple sets of lengths of the roads that meet the objective, any of them will be accepted.

3

0 6 15
6 0 21
15 21 0

There are three Hamiltonian paths. The total lengths of these paths are as follows:

• $1 \to 2 \to 3$: The total length is $6 + 21 = 27$.
• $1 \to 3 \to 2$: The total length is $15 + 21 = 36$.
• $2 \to 1 \to 3$: The total length is $6 + 15 = 21$.

They are distinct, so the objective is met.

4

0 111 157 193
111 0 224 239
157 224 0 258
193 239 258 0

There are $12$ Hamiltonian paths, with distinct total lengths.