5. Consider the grid shown below; one can imagine that it is a map of roads...
5. Consider the grid shown below; one can imagine that it is a map of roads in a city. In this
problem we consider the problem of moving from one corner of the grid to another.
A
B
C
1
(a) Suppose that, starting at the point labeled A, you can go one step up or one step to the
right at each move, as long as you stay on the grid. If you end up at the point labeled
B, how many different paths are there?
(b) Now suppose that we must go through the point labeled C. How many paths are there?
If we must instead avoid the point labeled C how many paths are there?
Homework Answers
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5. Consider the grid shown below; one can imagine that
it is a map of roads...
5. The below grid represents the map of your neighborhood. Your home is at the corner...
5. The below grid represents the map of your neighborhood. Your home is at the corner labeled A, while your work is at the corner labeled B. The corner marked with . is your favorite coffee shop (a) How many different paths of minimal length are there from your home to your work? (b) How many different paths of minimal length are there from your home to your work that let you stop at your favorite coffee shop?
The below grid represents the map of your neighborhood. Your home is at the corner labeled...
The below grid represents the map of your neighborhood. Your home is at the corner labeled A, while your work is at the corner labeled B. The corner marked with is your favorite coffee shop. (a) How many different paths of minimal length are there from your home to your work? b) How many different paths of minimal length are there from your home to your work that let you stop at your favorite coffee shop?
4. Consider a chessboard, shown below. At starting at the square al (the lower right hand...
4. Consider a chessboard, shown below. At starting at the square al (the lower right hand corner), you can go one step up or one step right at each move. The procedure stops until the point h8 (the upper right corner) is reached. a) How many different paths from al to h8 are possible? (b) How many differnt paths from al to h8 are possible if each path must pass through the square e4? 5. If 10 new teachers are...
8. (Extra Credits: You can get up to 60% more!) A grid in Cartesian coordinates with...
8. (Extra Credits: You can get up to 60% more!) A grid in Cartesian coordinates with size 6 x 4 is shown in Figure 2a. We start from the original point (0,0) and repeat moving 1 unit length to the next grid point by either moving up (denoted as U-action or right (denoted as R-action), until we reach the destination point (6,4). Each sequence of such R- and U-actions forms a path. Figure 2b shows an example path of this...
1) Consider an ant that is walking on a Cartesian grid, starting at (0,0) and ending...
1) Consider an ant that is walking on a Cartesian grid, starting at (0,0) and ending at (15, 18). The ant always chooses to walk exactly one unit either up or to the right (towards his destination) whenever he arrives at a Lattice point. (A Lattice point is a point with integer coordinates.) Thus, from (0,0) he either walks to (1,0 or (0). If the ant is not allowed to go to the points (6, 8) and (, 15), how...
Consider an ant that is walking on a Cartesian grid, starting at (0,0) and ending at...
Consider an ant that is walking on a Cartesian grid, starting at (0,0) and ending at (20, 12). The ant always chooses to walk exactly one unit either up or to the right (towards his destination) whenever he arrives at a Lattice point. (A Lattice point is a point with integer coordinates.) Thus, from (0,0) he either walks to (1, 0) or (0, 1). If the ant is not allowed to go to the points (10, 5) and (12, 8),...
4. A region contains a number of towns connected by roads. Each road is labeled by...
4. A region contains a number of towns connected by roads. Each
road is labeled by the average number of minutes required for a
fire engine to travel to it. Each intersection is labeled with a
circle. Suppose that you work for a city that has decided to place
a fire station at location G. (While this problem is small, you
want to devise a method to solve much larger problems).
(a) What algorithm would you recommend (use
Dykstra's) be...
Under the standard (x, y)-coordinate system, a grid of points with integer coordinates is given i...
Under the standard (x, y)-coordinate system, a grid of points
with integer coordinates is given in the picture below. Here, the
bottom-left point is the origin (0, 0) and the topright point has
coordinate (6, 3). The coordinates of the remaining points can
easily be computed accordingly. 2 During the 74th Hunger Game,
Katniss and Peeta are separated inside a grid maze. Katniss is
currently at the point (0, 0) and Peeta is (badly injured) at the
point (x, y)...
Question 2 In the minimum grid-path problem we are given a two-dimensional grid, where each point...
Question 2 In the minimum grid-path problem we are given a two-dimensional grid, where each point on the grid has a value, which is a non-negative integer. We have to find a path from the top left corner of the grid to the bottom right corner, where each single step on the path must lead from one grid point to its neighbour to the right or below. The cost of the path is the sum of all values of the...
Treasure Hunt: The map shown accompanies the following directions to buried treasure: "Start at the old...
Treasure Hunt: The map shown accompanies the following directions to buried treasure: "Start at the old oak tree, walk due north for 500 paces, then due east for 100 paces. Dig. Unfortunately, when you arrive, you find an angry dragon just north of the tree. To avoid the dragon, you set off along the yellow brick road at an angle 60 east of north. After walking 300 paces, you see an opening in the hedges bordering it. What direction should...
5. The below grid represents the map of your neighborhood. Your home is at the corner labeled A, while your work is at the corner labeled B. The corner marked with . is your favorite coffee shop (a) How many different paths of minimal length are there from your home to your work? (b) How many different paths of minimal length are there from your home to your work that let you stop at your favorite coffee shop?
The below grid represents the map of your neighborhood. Your home is at the corner labeled A, while your work is at the corner labeled B. The corner marked with is your favorite coffee shop. (a) How many different paths of minimal length are there from your home to your work? b) How many different paths of minimal length are there from your home to your work that let you stop at your favorite coffee shop?
4. Consider a chessboard, shown below. At starting at the square al (the lower right hand corner), you can go one step up or one step right at each move. The procedure stops until the point h8 (the upper right corner) is reached. a) How many different paths from al to h8 are possible? (b) How many differnt paths from al to h8 are possible if each path must pass through the square e4? 5. If 10 new teachers are...
8. (Extra Credits: You can get up to 60% more!) A grid in Cartesian coordinates with size 6 x 4 is shown in Figure 2a. We start from the original point (0,0) and repeat moving 1 unit length to the next grid point by either moving up (denoted as U-action or right (denoted as R-action), until we reach the destination point (6,4). Each sequence of such R- and U-actions forms a path. Figure 2b shows an example path of this...
1) Consider an ant that is walking on a Cartesian grid, starting at (0,0) and ending at (15, 18). The ant always chooses to walk exactly one unit either up or to the right (towards his destination) whenever he arrives at a Lattice point. (A Lattice point is a point with integer coordinates.) Thus, from (0,0) he either walks to (1,0 or (0). If the ant is not allowed to go to the points (6, 8) and (, 15), how...
4. A region contains a number of towns connected by roads. Each
road is labeled by the average number of minutes required for a
fire engine to travel to it. Each intersection is labeled with a
circle. Suppose that you work for a city that has decided to place
a fire station at location G. (While this problem is small, you
want to devise a method to solve much larger problems).
(a) What algorithm would you recommend (use
Dykstra's) be...
Under the standard (x, y)-coordinate system, a grid of points
with integer coordinates is given in the picture below. Here, the
bottom-left point is the origin (0, 0) and the topright point has
coordinate (6, 3). The coordinates of the remaining points can
easily be computed accordingly. 2 During the 74th Hunger Game,
Katniss and Peeta are separated inside a grid maze. Katniss is
currently at the point (0, 0) and Peeta is (badly injured) at the
point (x, y)...
Question 2 In the minimum grid-path problem we are given a two-dimensional grid, where each point on the grid has a value, which is a non-negative integer. We have to find a path from the top left corner of the grid to the bottom right corner, where each single step on the path must lead from one grid point to its neighbour to the right or below. The cost of the path is the sum of all values of the...
Treasure Hunt: The map shown accompanies the following directions to buried treasure: "Start at the old oak tree, walk due north for 500 paces, then due east for 100 paces. Dig. Unfortunately, when you arrive, you find an angry dragon just north of the tree. To avoid the dragon, you set off along the yellow brick road at an angle 60 east of north. After walking 300 paces, you see an opening in the hedges bordering it. What direction should...
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