Mathematics desk | ||
---|---|---|
< July 20 | << Jun | July | Aug >> | Current desk > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
(This is a question inspired by and abstracted from various adventure computer games involving navigating a map of rooms.)
Consider a grid of squares. The grid has total width W and total height H. You start on an entrance square in the lower-left or southwestern corner. Your goal is to reach an exit square in the fewest steps possible, where a "step" means to move orthogonally (vertically or horizontally only; diagonal moves are not allowed) into an adjacent square.
If the exit square is in the top-right or northeast corner, this would take W + H - 2 steps: you would move W - 1 steps to the right/east, and then H - 1 steps to the top/north to reach the exit square, for a total of W + H - 2 steps.
But, there is a twist: whenever you take a step, there is a constant probability P that you will be teleported to a random square instead of to the square you were trying to step to.
The two parts to my question:
(1) In terms of W, H, and P, what is the expected number of steps to reach that exit square in the top-right corner?
(2) Same question, but now W and H are both guaranteed to be odd, and the exit square is now the center square of this grid rather than the top-right/northeast square of this grid.
— SeekingAnswers ( reply) 06:28, 21 July 2023 (UTC)
In calculus, for every smooth function there exists a clear meaning of It's simply the derivative of at , i.e. the slope of the tangent of the graph of at
For every given smooth function , does also the expression have an intuitive meaning, in any branch of mathemstics? 2A06:C701:7471:3000:39AA:1A85:25C2:975B ( talk) 12:50, 21 July 2023 (UTC)
Mathematics desk | ||
---|---|---|
< July 20 | << Jun | July | Aug >> | Current desk > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
(This is a question inspired by and abstracted from various adventure computer games involving navigating a map of rooms.)
Consider a grid of squares. The grid has total width W and total height H. You start on an entrance square in the lower-left or southwestern corner. Your goal is to reach an exit square in the fewest steps possible, where a "step" means to move orthogonally (vertically or horizontally only; diagonal moves are not allowed) into an adjacent square.
If the exit square is in the top-right or northeast corner, this would take W + H - 2 steps: you would move W - 1 steps to the right/east, and then H - 1 steps to the top/north to reach the exit square, for a total of W + H - 2 steps.
But, there is a twist: whenever you take a step, there is a constant probability P that you will be teleported to a random square instead of to the square you were trying to step to.
The two parts to my question:
(1) In terms of W, H, and P, what is the expected number of steps to reach that exit square in the top-right corner?
(2) Same question, but now W and H are both guaranteed to be odd, and the exit square is now the center square of this grid rather than the top-right/northeast square of this grid.
— SeekingAnswers ( reply) 06:28, 21 July 2023 (UTC)
In calculus, for every smooth function there exists a clear meaning of It's simply the derivative of at , i.e. the slope of the tangent of the graph of at
For every given smooth function , does also the expression have an intuitive meaning, in any branch of mathemstics? 2A06:C701:7471:3000:39AA:1A85:25C2:975B ( talk) 12:50, 21 July 2023 (UTC)