A knight at position (r, c) can move in an L-shaped pattern to any of eight positions: moving over one row and up/down two columns to positions (r± 1, c±2), or moving two rows over and one column up/down to positions (r ±2, c ± 1). A rook at position (r, c) can move in a straight line either horizontally or vertically, moving any integral number x of squares. The rook ends up in positions (r±x, c) for a vertical move, or (r, c + x) for a horizontal move. Figure 5.13 A chess board, and the legal moves for a knight (left) or rook (right).
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In chess, a walk for a particular piece is a sequence of legal moves for that piece, starting from a square of your choice, that visits every square of the board. A tour is a walk that visits every square only once. (See Figure 5.13.)
5.13 Prove by induction that there exists a knight’s walk of an n-by-n chessboard for any n ≥ 4. (It turns out that knight’s tours exist for all even n ≥ 6, but you don’t need to prove this fact.)
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- On a chess board of r rows and c columns there is a lone white rook surrounded by a group of opponent's black knights. Each knight attacks 8 squares as in a typical chess game, which are shown in the figure - the knight on the red square attacks the 8 squares with a red dot. The rook can move horizontally and vertically by any number of squares. The rook can safely pass through an empty square that is attacked by a knight, but it must move to a square that is not attacked by any knight. The rook cannot jump over a knight while moving. If the rook moves to a square that contains a knight, it may capture it and remove it from the board. The black knights. never move. Can the rook eventually safely move to the designated target square? The figure illustrates how the white rook can move to the blue target square at the top-right corner in the first sample case. The rook captures one black knight at the bottom-right of the board on its way. Rok nd kight lcoes by Chunen Input The first line…A tile of a monkey puzzle has four monkey halves that can be labelled as north(N), east (E), south (S), and west (W) half. In addition to the shape of the borderrectangle, these halves determine which edges can be placed next one other. There isalso another way to define how the tiles can be placed: Each tile corner (i.e. compassdirections NE, SE, SW, and NW) has a monkey quarter. If we abstract this quarter, forexample, with a letter, only the tiles with the same letter in their touching corners canbe adjacent. illustrates one valid solution for this quarter monkey puzzle.Are the two monkey puzzle representations equivalent in the sense that if we have apile of ‘half monkey’ tiles H, it is possible to define a pile of ‘quarter monkey’ tilesQ that gives exactly the same set of solutions for the puzzle (and vice versa)?This problem is on Computer Graphics and is based on the topic 'Perspective Projection'. Please provide a step-by-step solution to the problem and explain the steps. The solution must include the General purpose perspective projection matrix. Question: Let a 3D point (423, -423, 423) be projected on a projection plane. Given that the center of the projection plane is (0.0, 0.0, -423.0) and the coordinate of the COP is (4, 2, 3). Determine the coordinate of that 3D point on the projection plane using a general purpose perspective projection matrix.
- Correct answer will be upvoted else Multiple Downvoted. Computer science. one maneuver, the robot should move one cell to the left or right, given that it doesn't move beyond the field of play. As such, if the robot was in the cell I, it should move to either the cell i−1 or the cell i+1, as long as it lies among 1 and n (endpoints comprehensive). The cells, in the request they are visited (counting the cell the robot is set), together make a decent way. Every cell I has a worth computer based intelligence related with it. Let c0,c1,… ,ck be the succession of cells in a decent way in the request they are visited (c0 is the cell robot is at first positioned, c1 is the cell where the robot is after its first move, etc; all the more officially, ci is the cell that the robot is at after I moves). Then, at that point, the worth of the way is determined as ac0+ac1+⋯+ack. Your errand is to work out the amount of qualities over all conceivable great ways. Since this number can be…Suppose images f(x,y) and g(x,y) have histograms h; and hg. Suppose you form image k(x,y) as the sum of these two images: k(x,y) = f(x,y) + g(x,y). (a) Under what conditions (on images f and g) can you determine the histogram hų of image k in terms of the histograms hf and hg? (That is, under what conditions can you determine hu if you are only given hj and he? Hint: this turns out to be fairly restrictive.) (b) Explain how you would derive hỵ in this case.5. F has one-tó-one correspondence. F {(-5, p), (10, w), (15, n), (20, m), (25, u), (30, b)}. Draw an arrow diagram for F. %3D
- Submit solutions to the following problems in a single pdf document preferably completed using LaTeX. It is likely that you will need to use programming (Python) or a matrix calculator to solve most of the problems. You do not need to submit the code, but you should include explanations of all your conclusions. 1. Suppose a robot mouse is placed in a grid. Each room in the grid has doors that open to all other rooms. One room contains a trap so if the robot mouse enters that room, then the robot loses and the game ends. If the robot mouse makes it to the outside of the grid, then it wins and the game ends. Suppose the grid is 5x5 and the trap is in the third row, third column. When the robot mouse is placed in a room, it randomly selects a move left, right, up, or down until it is either trapped or reaches the outside. Assume that the starting point is never the room with the trap. What is the probability that the robot mouse will win the game? Does it depend on where the mouse begins?…Explain the concept of duality in projective geometry and its applications in computer graphics, such as the duality of points and lines in 2D transformations.1. The representation must be on a linear scale (not logarithmic) Linear scale means each year is given the same weight in your scaling. The student is to present the years in a way that really helps the audience to visualize just how long Earth has been around compared to how short an amount of time humans have been on earth, for example. 2. The must include a legend that allows the audience to interpret how they are representing the time since each event has occurred (e.g., 1cm = 1000 years, 1mL = 100,000 years, 1 second = 10,000 years) 4. The student is to include the following important events: Start of life (~3.8 GYA[1]) Origin of eukaryotes (~2–3.5 GYA - your choice within that range) Origin of multicellularity (~2 GYA) Origin of Homo sapiens(~200 KYA) Present time (0 YA) 5. The student is to include at least 3 other events. The events must relate to the history of Earth, not human history. The student will need to find the dates of these events Formation of Earth…
- The annual snake festival is upon us and all the snakes of the kingdom have gathered to participate in the parade. The chef was tasked with reporting on the parade, so he decided to watch all the snakes first. When he sees the snake first, it will be his head, so he marks the "H". Snakes are long, and when they see the snake finally scurry away, they mark its tail with a "T". In the interim, when the snake is moving around it, or in the time between one snake and the next snake, it marks a '.'. Since the snakes come in a procession and one by one, a valid message would be something like "..H..T...HTH....T.", or "...", or "HT", while "T ...H..H.T", "H..T..H", "H..H..T..T" would be invalid messages (see explanation below). Formally, a snake is represented by an 'H' followed by some (possibly null) '.' and then 'T'. A valid message is one that begins with the (probably zero-length) string '.' and then some (possibly zero) snakes, with some '.'s in between, and finally some (possibly zero)…Correct answer will be upvoted else Multiple Downvoted. Don't submit random answer. Computer science. You are given a parallel table of size n×m. This table comprises of images 0 and 1. You can make such activity: select 3 distinct cells that have a place with one 2×2 square and change the images in these cells (change 0 to 1 and 1 to 0). Your assignment is to make all images in the table equivalent to 0. You are permitted to make all things considered 3nm activities. You don't have to limit the number of activities. It tends to be demonstrated that it is consistently conceivable. Input The principal line contains a solitary integer t (1≤t≤5000) — the number of experiments. The following lines contain portrayals of experiments. The principal line of the depiction of each experiment contains two integers n, m (2≤n,m≤100). Every one of the following n lines contains a parallel line of length m, depicting the images of the following column of the table. It is…Here are the arrow diagrams of two maps f (on the left) and g (on the right): And here are two possible results for the composition fo g: A) B) 6. O The diagram in A) pictures fog. O The diagram in B) pictures fog.