Comment by fwlr
19 hours ago
This is a bit more stream of thought, hopefully that is helpful.
The pyramid can’t take up all the space inside the cube, only another cube could do that. So there must be some points that are “inside cube, outside pyramid” and other points “inside cube, inside pyramid”. The pyramid is given as wholly inside the cube so there are no “outside cube, inside pyramid” points. Same for sphere inside the pyramid. So it’s
{ out-cube | in-cube out-pyramid | in-cube in-pyramid out-sphere | in-cube in-pyramid in-sphere }
I struggle parsing the ray description, “horizontal” must meaningfully constrain the ray somehow (otherwise it wouldn’t have been mentioned), but I don’t know what that changes. Then I think about facts I know about rays - they can be uniquely specified with two points, or one point and one direction - oh, and we are given the direction is always “horizontal”, so we only need one point. The ray goes on forever so every “in” is “enter+exit”, so add 2 for every “in”. So it’s
{ out-cube (=0) | in-cube(2) out-pyramid (=2) | in-cube(2) in-pyramid(2) out-sphere (=4) | in-cube(2) in-pyramid(2) in-sphere(2) (=6 }
1) Maximum, that’s 6, from the last option, so “cube, pyramid, sphere…”, have to exit the innermost first, so “…sphere, pyramid, cube”. I don’t know where this ray is positioned, “halfway” might be a good guess, or maybe pyramid is half of square and circle is half of pyramid so “a quarter way up”.
2) Distinct, that’s `0 2 4 6`. `0` doesn’t count because we want “still passes through the cube”, so 3 counts exist. Thankful that “tangents don’t count” so I don’t have to think about them, because nothing immediately comes to mind on how to handle them.
yes this is correct. Thanks for writing that out, it's really interesting to read how you solved this without picturing it.
It's a bit surprising to me that it was unclear what I meant by horizontal/level, I thought this word is well-known and easy to understand. Here's what I mean by horizontal: For me, it means that with respect to the ground, the ray isn't rising or falling as it travels, i.e. it's level (like a plane flying in level flight with respect to the ground, not rising or falling). This constraint doesn't really matter I guess, except that I said that the cube is also level on the ground on one of its surfaces, so since you have to cast the ray in a horizontal/level way, you can't cast the ray at an angle through one of the corners so it goes through the opposite corner of the cube, since - if you think of the ray like a plane - you can only move the angle of the ray side to side not up and down, it stays in level flight = horizontal, so you can't cast it through both opposite corners if the cube is lying flat on the horizontal on one of its surfaces. As I just wrote this paragraph, I pictured what my description of casting through both corners the would look like and why it's not level with the ground anymore. I guess it's harder to follow my words without picturing it.
You mentioned it's not clear how being horizontal constrains the ray, based on what I wrote and what I meant about it being level, is it now clearer how this constrains some of the rays that can be cast? (Like how you can't cast a ray through the opposite ends of the cube now)? Or is that hard for you to follow? For me I think maybe my words could be written in a bit confusing way but I think someone can understand them if they can picture what I referred to. Without that mental picture it could be difficult though.
Your words are not unclear! You are doing a fantastic job of using words that work very easily with mental imagery, and but not at all without. I said I didn’t understand horizontal because without mental imagery I have no “floor” for the ray to be parallel to - I could have found something to be the referent for the horizontal if I had to, but once I thought I had explained why “horizontal” was mentioned I stopped trying to figure it out further.
That's really interesting, thanks for your answers!