The Winter Hike solution codeforces

The Winter Hike solution codeforces

 

Circular land is an 2n×2n2n×2n grid. Rows of this grid are numbered by integers from 11 to 2n2n from top to bottom and columns of this grid are numbered by integers from 11 to 2n2n from left to right. The cell (x,y)(x,y) is the cell on the intersection of row xx and column yy for 1x2n1≤x≤2n and 1y2n1≤y≤2n.

There are n2n2 of your friends in the top left corner of the grid. That is, in each cell (x,y)(x,y) with 1x,yn1≤x,y≤n there is exactly one friend. Some of the other cells are covered with snow. The Winter Hike solution codeforces

Your friends want to get to the bottom right corner of the grid. For this in each cell (x,y)(x,y) with n+1x,y2nn+1≤x,y≤2n there should be exactly one friend. It doesn’t matter in what cell each of friends will be.

You have decided to help your friends to get to the bottom right corner of the grid.

For this, you can give instructions of the following types:

  • You select a row xx. All friends in this row should move to the next cell in this row. That is, friend from the cell (x,y)(x,y) with 1y<2n1≤y<2n will move to the cell (x,y+1)(x,y+1) and friend from the cell (x,2n)(x,2n) will move to the cell (x,1)(x,1).
  • You select a row xx. All friends in this row should move to the previous cell in this row. That is, friend from the cell (x,y)(x,y) with 1<y2n1<y≤2n will move to the cell (x,y1)(x,y−1) and friend from the cell (x,1)(x,1) will move to the cell (x,2n)(x,2n).
  • You select a column yy. All friends in this column should move to the next cell in this column. That is, friend from the cell (x,y)(x,y) with 1x<2n1≤x<2n will move to the cell (x+1,y)(x+1,y) and friend from the cell (2n,y)(2n,y) will move to the cell (1,y)(1,y).
  • You select a column yy. All friends in this column should move to the previous cell in this column. That is, friend from the cell (x,y)(x,y) with 1<x2n1<x≤2n will move to the cell (x1,y)(x−1,y) and friend from the cell (1,y)(1,y) will move to the cell (2n,y)(2n,y).

Note how friends on the grid border behave in these instructions.

Example of applying the third operation to the second column. Here, colorful circles denote your friends and blue cells are covered with snow.
You can give such instructions any number of times. You can give instructions of different types. If after any instruction one of your friends is in the cell covered with snow he becomes ill.

In order to save your friends you can remove snow from some cells before giving the first instruction:

  • You can select the cell (x,y)(x,y) that is covered with snow now and remove snow from this cell for cx,ycx,y coins.

The Winter Hike solution codeforces You can do this operation any number of times.

You want to spend the minimal number of coins and give some instructions to your friends. After this, all your friends should be in the bottom right corner of the grid and none of them should be ill.

Please, find how many coins you will spend.

The Winter Hike solution codeforces Input

The first line contains a single integer tt (1t1001≤t≤100) — the number of test cases.

The first line of each test case contains the single integer nn (1n2501≤n≤250).

Each of the next 2n2n lines contains 2n2n integers ci,1,ci,2,,ci,2nci,1,ci,2,…,ci,2n (0ci,j1090≤ci,j≤109) — costs of removing snow from cells. If ci,j=0ci,j=0 for some i,ji,j than there is no snow in cell (i,j)(i,j). Otherwise, cell (i,j)(i,j) is covered with snow.

It is guaranteed that ci,j=0ci,j=0 for 1i,jn1≤i,j≤n.

It is guaranteed that the sum of nn over all test cases doesn’t exceed 250250.

Output

For each test case output one integer — the minimal number of coins you should spend.

Example

input

Copy The Winter Hike solution codeforces
4
1
0 8
1 99
2
0 0 0 0
0 0 0 0
9 9 2 2
9 9 9 9
2
0 0 4 2
0 0 2 4
4 2 4 2
2 4 2 4
4
0 0 0 0 0 0 0 2
0 0 0 0 0 0 2 0
0 0 0 0 0 2 0 0
0 0 0 0 2 0 0 0
0 0 0 2 2 0 2 2
0 0 2 0 1 6 2 1
0 2 0 0 2 4 7 4
2 0 0 0 2 0 1 6

output The Winter Hike solution codeforces

Copy
100
22
14
42
Note

In the first test case you can remove snow from the cells (2,1)(2,1) and (2,2)(2,2) for 100100 coins. Then you can give instructions

  • All friends in the first collum should move to the previous cell. After this, your friend will be in the cell (2,1)(2,1).
  • All friends in the second row should move to the next cell. After this, your friend will be in the cell (2,2)(2,2).

In the second test case you can remove all snow from the columns 33 and 44 for 2222 coins. Then you can give instructions

  • All friends in the first row should move to the next cell.
  • All friends in the first row should move to the next cell.
  • All friends in the second row should move to the next cell.
  • All friends in the second row should move to the next cell.
  • All friends in the third column should move to the next cell.
  • All friends in the third column should move to the next cell.
  • All friends in the fourth column should move to the next cell.
  • All friends in the fourth column should move to the next cell.

It can be shown that none of the friends will become ill and that it is impossible to spend less coins.

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