A collection of random problems I've come up with over the past few days... [please use spoilers when answering!]
(1) A large sphere of radius 'r' is rolling around the interior of a massive cube-shaped room, chasing around a smaller sphere. The room's volume is thousands of times bigger than the larger sphere and completely empty (other than the spheres). The smaller sphere's radius is some constant multiplied by 'r'. The smaller sphere is rolling away from the larger sphere, trying to escape... it quickly realizes that the only possible escape is to wait it out against the edge or in the corner.
(1a) What is this constant's upper limit if the smaller sphere can fit against the edge of the floor and a wall and not touch the larger sphere?
(1b) What is this constant's upper limit if the smaller sphere can fit snugly in the corner of the room (ie, the floor and two walls) and not touch the larger sphere?
(2) You have a 10 by 10 by 10 cube, made of 1000 unit cubes. You also have a "punching device", which can be aimed at a specific surface unit square and pointed perpindicular to that surface square. This "punches out" and gets rid of all of the cubes in that column. If this is hard to understand, I can elaborate further. The question is, what is the maximum number of "punches" you can use to completely eliminate all of the cube?
(3) You are playing hide-and-go-seek-tag with a friend (problems of this type also translate into scenarios where two separated parties are looking to meet up and are simultaneously moving around). You and your friend are in an area that can be described as a circle with a circumference of 100. You are at one point on the circle, your friend is somewhere else. Assume that if someone is going in one direction at a constant speed, they won't change directions. Also, your speed is 'v', your friend's speed is 'w', and v>w>0. All of this applies for both sub-problems.
(3a) It's a tossup whether your friend is going to stay put, move counterclockwise or move clockwise. Because you don't know which of those three options your friend will choose, and the game has to end sometime, you HAVE to move, either clockwise or counterclockwise (you intend to flip a coin to choose which way to go). What is the average time it will take for you to catch your friend?
(3b) In this scenario, you know that your friend is NOT going to stay put... so you have to choose, between moving (either counterclockwise or clockwise) of course, or staying put and waiting. What is the average time it will take for you and your friend to meet up in both cases (ie, with you moving too or with you staying put)? So, what should you do - stay put or move? And what is your answer to that if v=5 and w=3? For what set of speeds (v,w) does it not matter whether you go or stay (ie, both options have the same avg time) and what does this have to do with the golden ratio?
Question
unreality
A collection of random problems I've come up with over the past few days... [please use spoilers when answering!]
(1) A large sphere of radius 'r' is rolling around the interior of a massive cube-shaped room, chasing around a smaller sphere. The room's volume is thousands of times bigger than the larger sphere and completely empty (other than the spheres). The smaller sphere's radius is some constant multiplied by 'r'. The smaller sphere is rolling away from the larger sphere, trying to escape... it quickly realizes that the only possible escape is to wait it out against the edge or in the corner.
(1a) What is this constant's upper limit if the smaller sphere can fit against the edge of the floor and a wall and not touch the larger sphere?
(1b) What is this constant's upper limit if the smaller sphere can fit snugly in the corner of the room (ie, the floor and two walls) and not touch the larger sphere?
(2) You have a 10 by 10 by 10 cube, made of 1000 unit cubes. You also have a "punching device", which can be aimed at a specific surface unit square and pointed perpindicular to that surface square. This "punches out" and gets rid of all of the cubes in that column. If this is hard to understand, I can elaborate further. The question is, what is the maximum number of "punches" you can use to completely eliminate all of the cube?
(3) You are playing hide-and-go-seek-tag with a friend (problems of this type also translate into scenarios where two separated parties are looking to meet up and are simultaneously moving around). You and your friend are in an area that can be described as a circle with a circumference of 100. You are at one point on the circle, your friend is somewhere else. Assume that if someone is going in one direction at a constant speed, they won't change directions. Also, your speed is 'v', your friend's speed is 'w', and v>w>0. All of this applies for both sub-problems.
(3a) It's a tossup whether your friend is going to stay put, move counterclockwise or move clockwise. Because you don't know which of those three options your friend will choose, and the game has to end sometime, you HAVE to move, either clockwise or counterclockwise (you intend to flip a coin to choose which way to go). What is the average time it will take for you to catch your friend?
(3b) In this scenario, you know that your friend is NOT going to stay put... so you have to choose, between moving (either counterclockwise or clockwise) of course, or staying put and waiting. What is the average time it will take for you and your friend to meet up in both cases (ie, with you moving too or with you staying put)? So, what should you do - stay put or move? And what is your answer to that if v=5 and w=3? For what set of speeds (v,w) does it not matter whether you go or stay (ie, both options have the same avg time) and what does this have to do with the golden ratio?
~~~
thanks and good luck
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