Grid Xor

2000ms

262144K

Description:

Note: The XOR-sum of set $$$\{s_1,s_2,\ldots,s_m\}$$$ is defined as $$$s_1 \oplus s_2 \oplus \ldots \oplus s_m$$$, where $$$\oplus$$$ denotes the bitwise XOR operation.

After almost winning IOI, Victor bought himself an $$$n\times n$$$ grid containing integers in each cell. $$$n$$$ is an even integer. The integer in the cell in the $$$i$$$-th row and $$$j$$$-th column is $$$a_{i,j}$$$.

Sadly, Mihai stole the grid from Victor and told him he would return it with only one condition: Victor has to tell Mihai the XOR-sum of all the integers in the whole grid.

Victor doesn't remember all the elements of the grid, but he remembers some information about it: For each cell, Victor remembers the XOR-sum of all its neighboring cells.

Two cells are considered neighbors if they share an edge — in other words, for some integers $$$1 \le i, j, k, l \le n$$$, the cell in the $$$i$$$-th row and $$$j$$$-th column is a neighbor of the cell in the $$$k$$$-th row and $$$l$$$-th column if $$$|i - k| = 1$$$ and $$$j = l$$$, or if $$$i = k$$$ and $$$|j - l| = 1$$$.

To get his grid back, Victor is asking you for your help. Can you use the information Victor remembers to find the XOR-sum of the whole grid?

It can be proven that the answer is unique.

Input:

The first line of the input contains a single integer $$$t$$$ ($$$1 \le t \le 100$$$) — the number of test cases. The description of test cases follows.

The first line of each test case contains a single even integer $$$n$$$ ($$$2 \leq n \leq 1000$$$) — the size of the grid.

Then follows $$$n$$$ lines, each containing $$$n$$$ integers. The $$$j$$$-th integer in the $$$i$$$-th of these lines represents the XOR-sum of the integers in all the neighbors of the cell in the $$$i$$$-th row and $$$j$$$-th column.

It is guaranteed that the sum of $$$n$$$ over all test cases doesn't exceed $$$1000$$$ and in the original grid $$$0 \leq a_{i, j} \leq 2^{30} - 1$$$.

Hack Format

To hack a solution, use the following format:

The first line should contain a single integer t ($$$1 \le t \le 100$$$) — the number of test cases.

The first line of each test case should contain a single even integer $$$n$$$ ($$$2 \leq n \leq 1000$$$) — the size of the grid.

Then $$$n$$$ lines should follow, each containing $$$n$$$ integers. The $$$j$$$-th integer in the $$$i$$$-th of these lines is $$$a_{i,j}$$$ in Victor's original grid. The values in the grid should be integers in the range $$$[0, 2^{30}-1]$$$

The sum of $$$n$$$ over all test cases must not exceed $$$1000$$$.

Output:

For each test case, output a single integer — the XOR-sum of the whole grid.

Sample Input:

Sample Output:

4
9
5

Note:

For the first test case, one possibility for Victor's original grid is:

$$$1$$$	$$$3$$$
$$$2$$$	$$$4$$$

For the second test case, one possibility for Victor's original grid is:

$$$3$$$	$$$8$$$	$$$8$$$	$$$5$$$
$$$9$$$	$$$5$$$	$$$5$$$	$$$1$$$
$$$5$$$	$$$5$$$	$$$9$$$	$$$9$$$
$$$8$$$	$$$4$$$	$$$2$$$	$$$9$$$

For the third test case, one possibility for Victor's original grid is:

$$$4$$$	$$$3$$$	$$$2$$$	$$$1$$$
$$$1$$$	$$$2$$$	$$$3$$$	$$$4$$$
$$$5$$$	$$$6$$$	$$$7$$$	$$$8$$$
$$$8$$$	$$$9$$$	$$$9$$$	$$$1$$$

Informação

Provedor Codeforces

Código CF1628C