Problem Statement

There is a grid with H rows from top to bottom and W columns from left to right. Each square has a piece placed on it or is empty.

The state of the grid is represented by H strings S_1, S_2, \ldots, S_H, each of length W.
If the j-th character of S_i is #, the square at the i-th row from the top and j-th column from the left has a piece on it;
if the j-th character of S_i is ., the square at the i-th row from the top and j-th column from the left is empty.

How many squares in the grid have pieces on them?

Constraints

• 1\leq H,W \leq 10
• H and W are integers.
• S_i is a string of length W consisting of # and ..

Sample Input 1

3 5
#....
.....
.##..

Sample Output 1

3

The following three squares have pieces on them:

• the square at the 1-st row from the top and 1-st column from the left;
• the square at the 3-rd row from the top and 2-nd column from the left;
• the square at the 3-rd row from the top and 3-rd column from the left.

Thus, 3 should be printed.

Sample Input 2

1 10
..........

Sample Output 2

0

Since no square has a piece on it, 0 should be printed.

Sample Input 3

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

Sample Output 3

16

Problem Statement

Given integers A and B, print the value A^B.

Constraints

• 1 \leq A, B \leq 9
• All values in the input are integers.

Sample Input 1

4 3

Sample Output 1

64

4^3 = 64, so 64 should be printed.

Sample Input 2

5 5

Sample Output 2

3125

Sample Input 3

8 1

Sample Output 3

8

Problem Statement

The AtCoder amusement park has an attraction that can accommodate K people. Now, there are N groups lined up in the queue for this attraction.

The i-th group from the front (1\leq i\leq N) consists of A_i people. For all i (1\leq i\leq N), it holds that A_i \leq K.

Takahashi, as a staff member of this attraction, will guide the groups in the queue according to the following procedure.

Initially, no one has been guided to the attraction, and there are K empty seats.

1. If there are no groups in the queue, start the attraction and end the guidance.
2. Compare the number of empty seats in the attraction with the number of people in the group at the front of the queue, and do one of the following:
   • If the number of empty seats is less than the number of people in the group at the front, start the attraction. Then, the number of empty seats becomes K again.
   • Otherwise, guide the entire group at the front of the queue to the attraction. The front group is removed from the queue, and the number of empty seats decreases by the number of people in the group.
3. Go back to step 1.

Here, no additional groups will line up after the guidance has started. Under these conditions, it can be shown that this procedure will end in a finite number of steps.

Determine how many times the attraction will be started throughout the guidance.

Constraints

• 1\leq N\leq 100
• 1\leq K\leq 100
• 1\leq A_i\leq K\ (1\leq i\leq N)
• All input values are integers.

Sample Input 1

7 6
2 5 1 4 1 2 3

Sample Output 1

4

Initially, the seven groups are lined up as follows:

Part of Takahashi's guidance is shown in the following figure:

• Initially, the group at the front has 2 people, and there are 6 empty seats. Thus, he guides the front group to the attraction, leaving 4 empty seats.
• Next, the group at the front has 5 people, which is more than the 4 empty seats, so the attraction is started.
• After the attraction is started, there are 6 empty seats again, so the front group is guided to the attraction, leaving 1 empty seat.
• Next, the group at the front has 1 person, so they are guided to the attraction, leaving 0 empty seats.

In total, he starts the attraction four times before the guidance is completed. Therefore, print 4.

Sample Input 2

7 10
1 10 1 10 1 10 1

Sample Output 2

7

Sample Input 3

15 100
73 8 55 26 97 48 37 47 35 55 5 17 62 2 60

Sample Output 3

8

Problem Statement

Takahashi is playing a game called Chess960. He has decided to write a code that determines if a random initial state satisfies the conditions of Chess960.

You are given a string S of length eight. S has exactly one K and Q, and exactly two R's, B's , and N's. Determine if S satisfies all of the following conditions.

• Suppose that the x-th and y-th (x < y) characters from the left of S are B; then, x and y have different parities.

• K is between two R's. More formally, suppose that the x-th and y-th (x < y) characters from the left of S are R and the z-th is K; then x< z < y.

Constraints

• S is a string of length 8 that contains exactly one K and Q, and exactly two R's, B's , and N's.

Sample Input 1

RNBQKBNR

Sample Output 1

Yes

The 3-rd and 6-th characters are B, and 3 and 6 have different parities. Also, K is between the two R's. Thus, the conditions are fulfilled.

Sample Input 2

KRRBBNNQ

Sample Output 2

No

K is not between the two R's.

Sample Input 3

BRKRBQNN

Sample Output 3

No

Problem Statement

You are given patterns S and T consisting of # and ., each with H rows and W columns.
The pattern S is given as H strings, and the j-th character of S_i represents the element at the i-th row and j-th column. The same goes for T.

Determine whether S can be made equal to T by rearranging the columns of S.

Here, rearranging the columns of a pattern X is done as follows.

• Choose a permutation P=(P_1,P_2,\dots,P_W) of (1,2,\dots,W).
• Then, for every integer i such that 1 \le i \le H, simultaneously do the following.
  • For every integer j such that 1 \le j \le W, simultaneously replace the element at the i-th row and j-th column of X with the element at the i-th row and P_j-th column.

Constraints

• H and W are integers.
• 1 \le H,W
• 1 \le H \times W \le 4 \times 10^5
• S_i and T_i are strings of length W consisting of # and ..

Sample Input 1

3 4
##.#
##..
#...
.###
..##
...#

Sample Output 1

Yes

If you, for instance, arrange the 3-rd, 4-th, 2-nd, and 1-st columns of S in this order from left to right, S will be equal to T.

Sample Input 2

3 3
#.#
.#.
#.#
##.
##.
.#.

Sample Output 2

No

In this input, S cannot be made equal to T.

Sample Input 3

2 1
#
.
#
.

Sample Output 3

Yes

It is possible that S=T.

Sample Input 4

8 7
#..#..#
.##.##.
#..#..#
.##.##.
#..#..#
.##.##.
#..#..#
.##.##.
....###
####...
....###
####...
....###
####...
....###
####...

Sample Output 4

Yes

Problem Statement

We have a multiset of integers S, which is initially empty.

Given Q queries, process them in order. Each query is of one of the following types.

• 1 x: Insert an x into S.

• 2 x c: Remove an x from S m times, where m = \mathrm{min}(c,( the number of x's contained in S)).

• 3 : Print ( maximum value of S)-( minimum value of S). It is guaranteed that S is not empty when this query is given.

Constraints

• 1 \leq Q \leq 2\times 10^5
• 0 \leq x \leq 10^9
• 1 \leq c \leq Q
• When a query of type 3 is given, S is not empty.
• All values in input are integers.

Sample Input 1

8
1 3
1 2
3
1 2
1 7
3
2 2 3
3

Sample Output 1

1
5
4

The multiset S transitions as follows.

• 1-st query: insert 3 into S. S is now \lbrace 3 \rbrace.
• 2-nd query: insert 2 into S. S is now \lbrace 2, 3 \rbrace.
• 3-rd query: the maximum value of S = \lbrace 2, 3\rbrace is 3 and its minimum value is 2, so print 3-2=1.
• 4-th query: insert 2 into S. S is now \lbrace 2,2,3 \rbrace.
• 5-th query: insert 7 into S. S is now \lbrace 2, 2,3, 7\rbrace.
• 6-th query: the maximum value of S = \lbrace 2,2,3, 7\rbrace is 7 and its minimum value is 2, so print 7-2=5.
• 7-th query: since there are two 2's in S and \mathrm{min(2,3)} = 2, remove 2 from S twice. S is now \lbrace 3, 7\rbrace.
• 8-th query: the maximum value of S = \lbrace 3, 7\rbrace is 7 and its minimum value is 3, so print 7-3=4.

Sample Input 2

4
1 10000
1 1000
2 100 3
1 10

Sample Output 2

If the given queries do not contain that of type 3, nothing should be printed.

Problem Statement

We have an infinite hexagonal grid shown below. Initially, all squares are white.

A hexagonal cell is represented as (i,j) with two integers i and j.
Cell (i,j) is adjacent to the following six cells:

• (i-1,j-1)
• (i-1,j)
• (i,j-1)
• (i,j+1)
• (i+1,j)
• (i+1,j+1)

Takahashi has painted N cells (X_1,Y_1),(X_2,Y_2),\dots,(X_N,Y_N) black.
Find the number of connected components formed by the black cells.
Two black cells belong to the same connected component when one can travel between those two black cells by repeatedly moving to an adjacent black cell.

Constraints

• All values in the input are integers.
• 1 \le N \le 1000
• |X_i|,|Y_i| \le 1000
• The pairs (X_i,Y_i) are distinct.

Sample Input 1

6
-1 -1
0 1
0 2
1 0
1 2
2 0

Sample Output 1

3

After Takahashi paints cells black, the grid looks as shown below.

The black squares form the following three connected components:

• (-1,-1)
• (1,0),(2,0)
• (0,1),(0,2),(1,2)

Sample Input 2

4
5 0
4 1
-3 -4
-2 -5

Sample Output 2

4

Sample Input 3

5
2 1
2 -1
1 0
3 1
1 -1

Sample Output 3

1

Problem Statement

You are given a tree with N vertices. The vertices are numbered 1 to N, and the i-th edge connects vertices A_i and B_i.

You are also given a sequence of positive integers C = (C_1, C_2, \ldots ,C_N) of length N. Let d(a, b) be the number of edges between vertices a and b, and for x = 1, 2, \ldots, N, let \displaystyle f(x) = \sum_{i=1}^{N} (C_i \times d(x, i)). Find \displaystyle \min_{1 \leq v \leq N} f(v).

Constraints

• 1 \leq N \leq 10^5
• 1 \leq A_i, B_i \leq N
• The given graph is a tree.
• 1 \leq C_i \leq 10^9

Sample Input 1

4
1 2
1 3
2 4
1 1 1 2

Sample Output 1

5

For example, consider calculating f(1). We have d(1, 1) = 0, d(1, 2) = 1, d(1, 3) = 1, d(1, 4) = 2.
Thus, f(1) = 0 \times 1 + 1 \times 1 + 1 \times 1 + 2 \times 2 = 6.

Similarly, f(2) = 5, f(3) = 9, f(4) = 6. Since f(2) is the minimum, print 5.

Sample Input 2

2
2 1
1 1000000000

Sample Output 2

1

f(2) = 1, which is the minimum.

Sample Input 3

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

Sample Output 3

56

Problem Statement

Takahashi and a cargo are on a coordinate plane.

Takahashi is currently at (X_A,Y_A), and the cargo is at (X_B,Y_B). He wants to move the cargo to (X_C,Y_C).

When he is at (x,y), he can make one of the following moves in a single action.

• Move to (x+1,y). If the cargo is at (x+1,y) before the move, move it to (x+2,y).
• Move to (x-1,y). If the cargo is at (x-1,y) before the move, move it to (x-2,y).
• Move to (x,y+1). If the cargo is at (x,y+1) before the move, move it to (x,y+2).
• Move to (x,y-1). If the cargo is at (x,y-1) before the move, move it to (x,y-2).

Find the minimum number of actions required to move the cargo to (X_C,Y_C).

Constraints

• -10^{17}\leq X_A,Y_A,X_B,Y_B,X_C,Y_C\leq 10^{17}
• (X_A,Y_A)\neq (X_B,Y_B)
• (X_B,Y_B)\neq (X_C,Y_C)
• All input values are integers.

Sample Input 1

1 2 3 3 0 5

Sample Output 1

9

Takahashi can move the cargo to (0,5) in nine actions as follows.

• Move to (2,2).
• Move to (3,2).
• Move to (3,3). The cargo moves to (3,4).
• Move to (3,4). The cargo moves to (3,5).
• Move to (4,4).
• Move to (4,5).
• Move to (3,5). The cargo moves to (2,5).
• Move to (2,5). The cargo moves to (1,5).
• Move to (1,5). The cargo moves to (0,5).

It is impossible to move the cargo to (0,5) in eight or fewer actions, so you should print 9.

Sample Input 2

0 0 1 0 -1 0

Sample Output 2

6

Sample Input 3

-100000000000000000 -100000000000000000 100000000000000000 100000000000000000 -100000000000000000 -100000000000000000

Sample Output 3

800000000000000003