#diverta20192f. [diverta2019_2_f]Diverta City

[diverta2019_2_f]Diverta City

Problem Statement

Diverta City is a new city consisting of NN towns numbered 1,2,...,N1, 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!/2N! / 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 101110^{11}.

Constraints

  • NN is a integer between 22 and 1010 (inclusive).

Input

Input is given from Standard Input in the following format:

NN

Output

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

w1,1w1,2w1,3...w1,Nw_{1, 1} \\ w_{1, 2} \\ w_{1, 3} \\ ... \\ w_{1, N} w2,1w2,2w2,3...w2,Nw_{2, 1} \\ w_{2, 2} \\ w_{2, 3} \\ ... \\ w_{2, N} :: :: :: wN,1wN,2wN,3...wN,Nw_{N, 1} \\ w_{N, 2} \\ w_{N, 3} \\ ... \\ w_{N, N}

where wi,jw_{i, j} is the length of the road connecting Town ii and Town jj, which must satisfy the following conditions:

  • wi,i=0w_{i, i} = 0
  • wi,j=wj,i(ineqj)w_{i, j} = w_{j, i} \\ (i \\neq j)
  • 1leqwi,jleq1011(ineqj)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.


Sample Input 1

Sample Output 1

0 6 15
6 0 21
15 21 0

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

  • 1231 → 2 → 3: The total length is 6+21=276 + 21 = 27.
  • 1321 → 3 → 2: The total length is 15+21=3615 + 21 = 36.
  • 2132 → 1 → 3: The total length is 6+15=216 + 15 = 21.

They are distinct, so the objective is met.


Sample Input 2

Sample Output 2

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

There are 1212 Hamiltonian paths, with distinct total lengths.