(Also aphantasic)

I think about the axis on which the fold is performed. Because of “square” and “diagonal”, for the two corners to touch, the fold axis must lie on both of the “other two corners”. A reflection about an axis does not change points on the axis, both points are unchanged, so their distance is also unchanged.

▲ logicallee 3 hours ago | parent [-]

Yes that's correct. How do you answer this one: a hollow cube lying with one face on the horizontal like a die on a table has a hollow pyramid inside with the same base horizontal on the bottom like a usual pyramid. Inside the pyramid is as large a sphere as will fit. A horizontal ray is cast from outside of the cube through the entire structure, passing through each surface it would hit. The location of the ray is not specified, only that it is horizontal.

1) State the order of the surfaces it hits/passes through if you choose to cast it at a point where it passes through the maximum number of surfaces in the structure.

2) Is it possible to choose a different point where it would still pass through the cube but would pass through a different number of surfaces? (such as by missing the pyramid or sphere or both). State the distinct counts of how many surfaces it could pass through depending on where you cast it horizontally, like how many different counts exist. (Being tangent does not count as passing through a surface.)

I would answer this question by just picturing it and moving the ray around in my mind. How do you do it?

	▲	fwlr 2 hours ago \| parent [-]
		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.