codeforces#P1719B. Mathematical Circus
Mathematical Circus
Description
A new entertainment has appeared in Buryatia — a mathematical circus! The magician shows two numbers to the audience — $n$ and $k$, where $n$ is even. Next, he takes all the integers from $1$ to $n$, and splits them all into pairs $(a, b)$ (each integer must be in exactly one pair) so that for each pair the integer $(a + k) \cdot b$ is divisible by $4$ (note that the order of the numbers in the pair matters), or reports that, unfortunately for viewers, such a split is impossible.
Burenka really likes such performances, so she asked her friend Tonya to be a magician, and also gave him the numbers $n$ and $k$.
Tonya is a wolf, and as you know, wolves do not perform in the circus, even in a mathematical one. Therefore, he asks you to help him. Let him know if a suitable splitting into pairs is possible, and if possible, then tell it.
The first line contains one integer $t$ ($1 \leq t \leq 10^4$) — the number of test cases. The following is a description of the input data sets.
The single line of each test case contains two integers $n$ and $k$ ($2 \leq n \leq 2 \cdot 10^5$, $0 \leq k \leq 10^9$, $n$ is even) — the number of integers and the number being added $k$.
It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.
For each test case, first output the string "YES" if there is a split into pairs, and "NO" if there is none.
If there is a split, then in the following $\frac{n}{2}$ lines output pairs of the split, in each line print $2$ numbers — first the integer $a$, then the integer $b$.
Input
The first line contains one integer $t$ ($1 \leq t \leq 10^4$) — the number of test cases. The following is a description of the input data sets.
The single line of each test case contains two integers $n$ and $k$ ($2 \leq n \leq 2 \cdot 10^5$, $0 \leq k \leq 10^9$, $n$ is even) — the number of integers and the number being added $k$.
It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.
Output
For each test case, first output the string "YES" if there is a split into pairs, and "NO" if there is none.
If there is a split, then in the following $\frac{n}{2}$ lines output pairs of the split, in each line print $2$ numbers — first the integer $a$, then the integer $b$.
Samples
<div class="test-example-line test-example-line-even test-example-line-0">4</div><div class="test-example-line test-example-line-odd test-example-line-1">4 1</div><div class="test-example-line test-example-line-even test-example-line-2">2 0</div><div class="test-example-line test-example-line-odd test-example-line-3">12 10</div><div class="test-example-line test-example-line-even test-example-line-4">14 11</div><div class="test-example-line test-example-line-even test-example-line-4"></div>
YES
1 2
3 4
NO
YES
3 4
7 8
11 12
2 1
6 5
10 9
YES
1 2
3 4
5 6
7 8
9 10
11 12
13 14
Note
In the first test case, splitting into pairs $(1, 2)$ and $(3, 4)$ is suitable, same as splitting into $(1, 4)$ and $(3, 2)$.
In the second test case, $(1 + 0) \cdot 2 = 1 \cdot (2 + 0) = 2$ is not divisible by $4$, so there is no partition.