You are given two grids of numbers a𝑎 and b𝑏, with n𝑛 rows and m𝑚 columns. All the values in the grid are 0, 1 or 2 You can perform the following operation on a𝑎 any number of times: • Pick any subrectangle in the grid with length and width ≥2. You are allowed to choose the entire grid as a subrectangle. • The subrectangle has four corners. Take any pair of diagonally opposite corners of the chosen subrectangle and add 1 to their values modulo 3. • For the pair of corners not picked, add 2 to their values modulo 3. Note that the operation only changes the values of the corners of the picked subrectangle. Is it possible to convert the grid a𝑎 into grid b𝑏 by applying the above operation any number of times (possibly zero)?
Question
You are given two grids of numbers a𝑎 and b𝑏, with n𝑛 rows and m𝑚 columns. All the values in the grid are 0, 1 or 2 You can perform the following operation on a𝑎 any number of times: • Pick any subrectangle in the grid with length and width ≥2. You are allowed to choose the entire grid as a subrectangle. • The subrectangle has four corners. Take any pair of diagonally opposite corners of the chosen subrectangle and add 1 to their values modulo 3. • For the pair of corners not picked, add 2 to their values modulo 3. Note that the operation only changes the values of the corners of the picked subrectangle. Is it possible to convert the grid a𝑎 into grid b𝑏 by applying the above operation any number of times (possibly zero)?
Solution
Yes, it is possible to convert grid a into grid b by applying the above operation any number of times. Here's a step-by-step guide on how to do it:
Step 1: Start by comparing the top left corner of both grids. If the values are not the same, perform the operation on the entire grid to make the values the same. This will also change the values of the other three corners.
Step 2: Move to the top right corner and compare the values. If they are not the same, choose a subrectangle that includes the top right corner and the bottom left corner. Perform the operation to make the values the same. This will also change the values of the other two corners.
Step 3: Repeat the process for the bottom left corner and the bottom right corner. Choose subrectangles that include the corner you are trying to change and one of the corners that already has the correct value.
Step 4: Once all the corners have the correct values, move to the next row or column and repeat the process. Always choose subrectangles that include a corner with the correct value.
Step 5: Continue this process until all the values in grid a are the same as in grid b.
Note: This process assumes that the operation can be performed any number of times. If there is a limit to the number of operations, it may not be possible to convert grid a into grid b.
Similar Questions
ProblemSubmissionsLeaderboardDiscussionsYou are given two grids of numbers a𝑎 and b𝑏, with n𝑛 rows and m𝑚 columns. All the values in the grid are 0, 1 or 2 You can perform the following operation on a𝑎 any number of times: • Pick any subrectangle in the grid with length and width ≥2. You are allowed to choose the entire grid as a subrectangle. • The subrectangle has four corners. Take any pair of diagonally opposite corners of the chosen subrectangle and add 1 to their values modulo 3. • For the pair of corners not picked, add 2 to their values modulo 3. Note that the operation only changes the values of the corners of the picked subrectangle. Is it possible to convert the grid a𝑎 into grid b𝑏 by applying the above operation any number of times (possibly zero)?Input FormatThe first line contains two integers n𝑛 and m𝑚, the number of rows and columns in the grid (2≤n,m≤500). Each of the next n lines contain m characters — the j-th character of the i-th line represents ai,j. Each of the next n lines contain m characters — the j-th character of the i-th line represents bi,j(0≤ai,j,bi,j≤2). It is guaranteed that the sum of n𝑛 over all test cases and the sum of m𝑚 over all test cases do not exceed 500.Constraints(2≤n,m≤500).Output FormatFor each test case print "YES" (without quotes) if it is possible to convert grid a into grid b and "NO" (without quotes) otherwise.
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