Diverta City

AtCoder

IDdiverta2019_2_f

Time3000ms

Memory256MB

Difficulty—

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: $N$ [samples]

Samples

Input #1

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:

*   $1 → 2 → 3$: The total length is $6 + 21 = 27$.
*   $1 → 3 → 2$: The total length is $15 + 21 = 36$.
*   $2 → 1 → 3$: The total length is $6 + 15 = 21$.

They are distinct, so the objective is met.

Input #2

Output #2

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.

API Response (JSON)

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      "statement_type": "Markdown",
      "content": "Diverta City is a new city consisting of $N$ towns numbered $1, 2, ..., N$.\nThe mayor Ringo is planning to connect every pair of two different towns with a bidirectional road. The length of each road ...",
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