Problem Statement

N people numbered 1,2,\ldots,N were in M photos. In each of the photos, they stood in a single line. In the i-th photo, the j-th person from the left is person a_{i,j}.

Two people who did not stand next to each other in any of the photos may be in a bad mood.

How many pairs of people may be in a bad mood? Here, we do not distinguish a pair of person x and person y, and a pair of person y and person x.

Constraints

• 2 \leq N \leq 50
• 1 \leq M \leq 50
• 1 \leq a_{i,j} \leq N
• a_{i,1},\ldots,a_{i,N} contain each of 1,\ldots,N exactly once.
• All values in the input are integers.

Sample Input 1

4 2
1 2 3 4
4 3 1 2

Sample Output 1

2

The pair of person 1 and person 4, and the pair of person 2 and person 4, may be in a bad mood.

Sample Input 2

3 3
1 2 3
3 1 2
1 2 3

Sample Output 2

0

Sample Input 3

10 10
4 10 7 2 8 3 9 1 6 5
3 6 2 9 1 8 10 7 4 5
9 3 4 5 7 10 1 8 2 6
7 3 1 8 4 9 5 6 2 10
5 2 1 4 10 7 9 8 3 6
5 8 1 6 9 3 2 4 7 10
8 10 3 4 5 7 2 9 6 1
3 10 2 7 8 5 1 4 9 6
10 6 1 5 4 2 3 8 9 7
4 5 9 1 8 2 7 6 3 10

Sample Output 3

6

Problem Statement

Given a positive integer N, find the maximum integer k such that 2^k \le N.

Constraints

• N is an integer satisfying 1 \le N \le 10^{18}.

Sample Input 1

6

Sample Output 1

2

• k=2 satisfies 2^2=4 \le 6.
• For every integer k such that k \ge 3, 2^k > 6 holds.

Therefore, the answer is k=2.

Sample Input 2

1

Sample Output 2

0

Note that 2^0=1.

Sample Input 3

1000000000000000000

Sample Output 3

59

The input value may not fit into a 32-bit integer.

Problem Statement

You are given two sequences, each of length N, consisting of integers: A=(A_1, \ldots, A_N) and B=(B_1, \ldots, B_N).

Determine whether there is a sequence of length N, X=(X_1, \ldots, X_N), satisfying all of the conditions below.

• X_i = A_i or X_i = B_i, for every i(1\leq i\leq N).

• |X_i - X_{i+1}| \leq K, for every i(1\leq i\leq N-1).

Constraints

• 1 \leq N \leq 2\times 10^5
• 0 \leq K \leq 10^9
• 1 \leq A_i,B_i \leq 10^9
• All values in input are integers.

Sample Input 1

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

Sample Output 1

Yes

X=(9,6,3,7,5) satisfies all conditions.

Sample Input 2

4 90
1 1 1 100
1 2 3 100

Sample Output 2

No

No X satisfies all conditions.

Sample Input 3

4 1000000000
1 1 1000000000 1000000000
1 1000000000 1 1000000000

Sample Output 3

Yes

Problem Statement

There are zero or more sensors placed on a grid of H rows and W columns. Let (i, j) denote the square in the i-th row from the top and the j-th column from the left.
Whether each square contains a sensor is given by the strings S_1, S_2, \ldots, S_H, each of length W. (i, j) contains a sensor if and only if the j-th character of S_i is #.
These sensors interact with other sensors in the squares horizontally, vertically, or diagonally adjacent to them and operate as one sensor. Here, a cell (x, y) and a cell (x', y') are said to be horizontally, vertically, or diagonally adjacent if and only if \max(|x-x'|,|y-y'|) = 1.
Note that if sensor A interacts with sensor B and sensor A interacts with sensor C, then sensor B and sensor C also interact.

Considering the interacting sensors as one sensor, find the number of sensors on this grid.

Constraints

• 1 \leq H, W \leq 1000
• H and W are integers.
• S_i is a string of length W where each character is # or ..

Sample Input 1

5 6
.##...
...#..
....##
#.#...
..#...

Sample Output 1

3

When considering the interacting sensors as one sensor, the following three sensors exist:

• The interacting sensors at (1,2),(1,3),(2,4),(3,5),(3,6)
• The sensor at (4,1)
• The interacting sensors at (4,3),(5,3)

Sample Input 2

3 3
#.#
.#.
#.#

Sample Output 2

1

Sample Input 3

4 2
..
..
..
..

Sample Output 3

0

Sample Input 4

5 47
.#..#..#####..#...#..#####..#...#...###...#####
.#.#...#.......#.#...#......##..#..#...#..#....
.##....#####....#....#####..#.#.#..#......#####
.#.#...#........#....#......#..##..#...#..#....
.#..#..#####....#....#####..#...#...###...#####

Sample Output 4

7

Problem Statement

You are given an integer N and strings R and C of length N consisting of A, B, and C. Solve the following problem.

There is a N \times N grid. All cells are initially empty.
You can write at most one character from A, B, and C in each cell. (You can also leave the cell empty.)

Determine if it is possible to satisfy all of the following conditions, and if it is possible, print one way to do so.

• Each row and each column contain exactly one A, one B, and one C.
• The leftmost character written in the i-th row matches the i-th character of R.
• The topmost character written in the i-th column matches the i-th character of C.

Constraints

• N is an integer between 3 and 5, inclusive.
• R and C are strings of length N consisting of A, B, and C.

Sample Input 1

5
ABCBC
ACAAB

Sample Output 1

Yes
AC..B
.BA.C
C.BA.
BA.C.
..CBA

The grid in the output example satisfies all the following conditions, so it will be treated as correct.

• Each row contains exactly one A, one B, and one C.
• Each column contains exactly one A, one B, and one C.
• The leftmost characters written in the rows are A, B, C, B, C from top to bottom.
• The topmost characters written in the columns are A, C, A, A, B from left to right.

Sample Input 2

3
AAA
BBB

Sample Output 2

No

For this input, there is no way to fill the grid to satisfy the conditions.