ere are five separate sets of grid squares.Let S be the set of finite sets of grid squares which form connected shapes. We do not count squares touchingdiagonally just at a vertex as being connected. The sets a, b, c and e on the above picture are elements of S. Set dis not an element of S.Let ∼ be a binary relation on S. Say that s ∼ t if and only if t can be obtained from s by moving the whole setby some sequence of any finite number (including none) of the following moves: move horizontally left one square,move horizontally right one square, move up one square, move down one square, rotate ninety degrees clockwise(about any square).So b ∼ c in the diagram above.Show that ∼ is an equivalence relation

My First Geometry ProblemYou are on a 22-dimensional grid, where you start at (0,0)(0,0).You are given a binary string 𝑆S of length 44 where:𝑆1S 1​ refers to left direction;𝑆2S 2​ refers to right direction;𝑆3S 3​ refers to up direction;𝑆4S 4​ refers to down direction.𝑆𝑖=1S i​ =1 denotes that you are allowed to make a move in the respective direction and vice-versa.Find the number of cells (𝑥,𝑦)(x,y) you can possibly visit which satisfy −10≤𝑥,𝑦≤10−10≤x,y≤10.Note that:You always include the cell (0,0)(0,0) in your answer.If you can visit (𝐴1,𝐵1)(A 1​ ,B 1​ ) and (𝐴2,𝐵2)(A 2​ ,B 2​ ) individually, but not both at the same time, you will still include both of them in your answer.Moves are defined as:A move in left direction is a move from cell (𝐴,𝐵)(A,B) to (𝐴−1,𝐵)(A−1,B).A move in right direction is a move from cell (𝐴,𝐵)(A,B) to (𝐴+1,𝐵)(A+1,B).A move in up direction is a move from cell (𝐴,𝐵)(A,B) to (𝐴,𝐵+1)(A,B+1).A move in down direction is a move from cell (𝐴,𝐵)(A,B) to (𝐴,𝐵−1)(A,B−1).Input FormatThe first line of input will contain a single integer 𝑇T, denoting the number of test cases.Each test case consists of a binary string 𝑆S of length 44 - denoting the directions in which moves are allowed.Output FormatFor each test case, output on a new line, the number of cells you can visit as mentioned in statement.Constraints1≤𝑇≤151≤T≤15∣𝑆∣=4∣S∣=4𝑆𝑖∈{0,1}S i​ ∈{0,1}𝑆𝑖=1S i​ =1 for at least one 1≤𝑖≤41≤i≤4.Sample 1:InputOutput5001011000110111011111121121231441Explanation:Test case 11: The only allowed direction is up. Thus, you can only visit cells (0,0),(0,1),(0,2),…,(0,10)(0,0),(0,1),(0,2),…,(0,10); which are a total of 1111 cells.Test case 22: The allowed directions are left and right. Thus, you can visit cells (−10,0),(−9,0),…,(0,0),…,(9,0),(10,0)(−10,0),(−9,0),…,(0,0),…,(9,0),(10,0), which are a total of 2121 cells.Test case 33: The allowed directions are right and up. You can visit all cells (𝑥,𝑦)(x,y) such that 𝑥≥0x≥0 and 𝑦≥0y≥0, which are a total of 121121 cells.Accepted195Submissions293

You are given four integers sx, sy, fx, fy, and a non-negative integer t.In an infinite 2D grid, you start at the cell (sx, sy). Each second, you must move to any of its adjacent cells.Return true if you can reach cell (fx, fy) after exactly t seconds, or false otherwise.A cell's adjacent cells are the 8 cells around it that share at least one corner with it. You can visit the same cell several times.

There is an m x n grid with a ball. The ball is initially at the position [startRow, startColumn]. You are allowed to move the ball to one of the four adjacent cells in the grid (possibly out of the grid crossing the grid boundary). You can apply at most maxMove moves to the ball.Given the five integers m, n, maxMove, startRow, startColumn, return the number of paths to move the ball out of the grid boundary. Since the answer can be very large, return it modulo 109 + 7. Example 1:Input: m = 2, n = 2, maxMove = 2, startRow = 0, startColumn = 0Output: 6

You are on a 22-dimensional grid, where you start at (0,0)(0,0).You are given a binary string 𝑆S of length 44 where:𝑆1S 1​ refers to left direction;𝑆2S 2​ refers to right direction;𝑆3S 3​ refers to up direction;𝑆4S 4​ refers to down direction.𝑆𝑖=1S i​ =1 denotes that you are allowed to make a move in the respective direction and vice-versa.Find the number of cells (𝑥,𝑦)(x,y) you can possibly visit which satisfy −10≤𝑥,𝑦≤10−10≤x,y≤10.Note that:You always include the cell (0,0)(0,0) in your answer.If you can visit (𝐴1,𝐵1)(A 1​ ,B 1​ ) and (𝐴2,𝐵2)(A 2​ ,B 2​ ) individually, but not both at the same time, you will still include both of them in your answer.Moves are defined as:A move in left direction is a move from cell (𝐴,𝐵)(A,B) to (𝐴−1,𝐵)(A−1,B).A move in right direction is a move from cell (𝐴,𝐵)(A,B) to (𝐴+1,𝐵)(A+1,B).A move in up direction is a move from cell (𝐴,𝐵)(A,B) to (𝐴,𝐵+1)(A,B+1).A move in down direction is a move from cell (𝐴,𝐵)(A,B) to (𝐴,𝐵−1)(A,B−1).Input FormatThe first line of input will contain a single integer 𝑇T, denoting the number of test cases.Each test case consists of a binary string 𝑆S of length 44 - denoting the directions in which moves are allowed.Output FormatFor each test case, output on a new line, the number of cells you can visit as mentioned in statement.Constraints1≤𝑇≤151≤T≤15∣𝑆∣=4∣S∣=4𝑆𝑖∈{0,1}S i​ ∈{0,1}𝑆𝑖=1S i​ =1 for at least one 1≤𝑖≤41≤i≤4.Sample 1:InputOutput5001011000110111011111121121231441Explanation:Test case 11: The only allowed direction is up. Thus, you can only visit cells (0,0),(0,1),(0,2),…,(0,10)(0,0),(0,1),(0,2),…,(0,10); which are a total of 1111 cells.Test case 22: The allowed directions are left and right. Thus, you can visit cells (−10,0),(−9,0),…,(0,0),…,(9,0),(10,0)(−10,0),(−9,0),…,(0,0),…,(9,0),(10,0), which are a total of 2121 cells.Test case 33: The allowed directions are right and up. You can visit all cells (𝑥,𝑦)(x,y) such that 𝑥≥0x≥0 and 𝑦≥0y≥0, which are a total of 121121 cells.

This is the net of a square-based pyramid. How many such nets are there in total, not counting those which are equal by rotations/mirror symmetry?