101654: [AtCoder]ABC165 E - Rotation Matching

Problem Statement

You are going to hold a competition of one-to-one game called AtCoder Janken. (Janken is the Japanese name for Rock-paper-scissors.) $N$ players will participate in this competition, and they are given distinct integers from $1$ through $N$. The arena has $M$ playing fields for two players. You need to assign each playing field two distinct integers between $1$ and $N$ (inclusive). You cannot assign the same integer to multiple playing fields. The competition consists of $N$ rounds, each of which proceeds as follows:

• For each player, if there is a playing field that is assigned the player's integer, the player goes to that field and fight the other player who comes there.
• Then, each player adds $1$ to its integer. If it becomes $N+1$, change it to $1$.

You want to ensure that no player fights the same opponent more than once during the $N$ rounds. Print an assignment of integers to the playing fields satisfying this condition. It can be proved that such an assignment always exists under the constraints given.

Constraints

• $1 \leq M$
• $M \times 2 +1 \leq N \leq 200000$

Input

Input is given from Standard Input in the following format:

$N$ $M$

Output

Print $M$ lines in the format below. The $i$-th line should contain the two integers $a_i$ and $b_i$ assigned to the $i$-th playing field.

$a_1$ $b_1$
$a_2$ $b_2$
$:$
$a_M$ $b_M$

Sample Input 1

4 1

Sample Output 1

2 3

Let us call the four players A, B, C, and D, and assume that they are initially given the integers $1$, $2$, $3$, and $4$, respectively.

• The $1$-st round is fought by B and C, who has the integers $2$ and $3$, respectively. After this round, A, B, C, and D have the integers $2$, $3$, $4$, and $1$, respectively.

• The $2$-nd round is fought by A and B, who has the integers $2$ and $3$, respectively. After this round, A, B, C, and D have the integers $3$, $4$, $1$, and $2$, respectively.

• The $3$-rd round is fought by D and A, who has the integers $2$ and $3$, respectively. After this round, A, B, C, and D have the integers $4$, $1$, $2$, and $3$, respectively.

• The $4$-th round is fought by C and D, who has the integers $2$ and $3$, respectively. After this round, A, B, C, and D have the integers $1$, $2$, $3$, and $4$, respectively.

No player fights the same opponent more than once during the four rounds, so this solution will be accepted.

Sample Input 2

7 3

Sample Output 2

1 6
2 5
3 4