This is actually quite fun and simple! Even if the problem and my following explanation look complicated :P
Let’s look at the three dimensional case. One can parametrize a 3 dimensional cube as the Cartesian product of intervals [0, 1] x [0, 1] x [0, 1]. This means a cube is a set of points (a, b, c) where a, b and c are real numbers between 0 and 1. The 2 dimensional sides of the cube are then given by fixing one coordinate. That is, the 6 sides are
{0} x [0, 1] x [0, 1], {1} x [0, 1] x [0, 1], [0, 1] x {0} x [0, 1], [0, 1] x {1} x [0, 1], [0, 1] x [0, 1] x {0} and [0, 1] x [0, 1] x {1}.Now we just start in the middle of a side at (0, 0.5, 0.5). To get to the next side we walk towards an edge (0, 0, 0.5) and then to the middle of the next side (0.5, 0, 0.5). We iterate this process until we run out of sides with a fixed 0, then walk towards a side with a fixed 1 and continue there. That is:
(0 , 0.5, 0.5) -> (0 , 0 , 0.5) -> (0.5, 0 , 0.5) -> (0.5, 0 , 0 ) -> (0.5, 0.5, 0 ) -> (1 , 0.5, 0 ) -> (1 , 0.5, 0.5) -> (1 , 1 , 0.5) -> (0.5, 1 , 0.5) -> (0.5, 1 , 1 ) -> (0.5, 0.5, 1 )This path basically spirals around the cube, going through every side only once. Here’s a visualization (sorry, I’m no artist :P)
The same procedure works on a 4 dimensional cube or any other higher dimension. For the 4 dimensional cube it goes like this:
(0 , 0.5, 0.5, 0.5) -> (0 , 0 , 0.5, 0.5) -> (0.5, 0 , 0.5, 0.5) -> (0.5, 0 , 0 , 0.5) -> ... -> (0.5, 0.5, 0.5, 0 ) -> (1 , 0.5, 0.5, 0 ) -> (1 , 0.5, 0.5, 0.5) -> (1 , 1 , 0.5, 0.5) -> ... -> (0.5, 0.5, 0.5, 1 )This works for arbitrary dimension except for the 1 dimensional cube (which is just a line) because the “sides” there are the two end points of the line and not connected at all. Additionally note, that it is never specified how edges count in this problem, whether they somehow count towards a face or whether you’re allowed to go back and fourth on edges. You could technically only walk along edges and step into the sides every now and then.
You owe me $14.50 for reading that.
I skipped all the blabla and looked at the drawing and was pleased to see the path I started visualising in my head was exactly like that. I do think I would’ve needed a cube in my hands to confirm it, or a bit longer thinking about it instead to complete it.
I don’t usually do this, but I’m gonna go out on a limb and say this didn’t happen.
Every cube is four dimensional, assuming time as the fourth dimension. So it would travel forward in time at a relatively constant rate (since ants don’t typically walk at relativistic speeds [citation needed]) and it would traverse the other three dimensions in normal ant ways.
Damnnn bro. They gonna start you at $15 with that kinda mind.
If the ant can only move a single direction in time, it cannot reach all the time corners. Every corner in 3 dimensional space has a twin corner, at the beginning and end of time. Since the ant can only walk forward in time, it will only reach 2 4D corners, where it started, and where it ended.
I’ve seen this site so many times, and yet open it again each time I come across a link, just to marvel at its unhingedness 🥴
I wonder what would have happened if someone had attempted to explain sinusoids to that man. Like, they’d probably be called a dumb evil bastard and some racial and homo/transphobic slurs followed by the sort of logic that only schizophrenics can follow. But still, a chunk of this really is just a man mapping squares on circles
iirc, the author goes out on tangents about how people trying to explain actual maths to him were “educated stupid”
It must have been difficult for him to be so insane.
Interviewer did not define time. I will define it as 0 seconds per second. The ant can not move as movement is impossible at this time scale.
Unfortunately I don’t think this is true. Every 3D face is the intersection of a 2D plane with the upper and lower bounds of the 3rd dimension. So I think a hypercube “face” would be every 3D “plane” at both the very start time AND the very end time. Meaning the ant would need to immediately accelerate to light speed - so no time would pass - and then (otherwise) normally traverse the faces, wait until the end time, and then repeat the process in reverse (still at light speed).
I was thinking 4 spatial dimensions and was trying to trace a hypercube
You were doing it for free?
Too many people are obsessing about 4d topology in this thread. The real difficulty in the question is the non -deterministic pathfinding of the ant, in the absence of pheromones.
This is a lot like when Boston PD was found to screen out all the smart applicants. Sometime the company wants an obedient idiot.
Might actually be the case, lol.
Answer this question correctly (or even intelligently at all) and your application is rejected.
making sure you cannot solve it, so you are perfect for the job
Possible candidate responses:
- Solves it (too smart for job)
- “That’s bullshit, who needs this for a $14.50/hr job?” (too intolerant of bullshit for job)
- Tries to solve it but fails (lacks self-awareness for job)
- Knows they can’t solve it so doesn’t even try (too lazy for job)
- Doesn’t understand the question/comprehend what a hypercube is (too dumb for job)
Maybe they’re trying to weed out all actual applicants because they’re hiring the boss’ kid.
You forgot option 6, spew a bunch of techno bubble at the HR person who will definitely not understand the problem themselves and wouldn’t be able to tell if you’d answered it or not.
That’s just response 1 from the perspective of the HR person scoring it.
I’d argue that 3 and 5 are actually selection qualities for a job paying that low, with a question like that. The rest are all dis-qualifiers of course.
I believe this is sometimes the case. I was called for an interview with a group of 15 other people ones. We were like a class, being interviewed as a group, and were supposed to solve some problems together. Nobody in that group could solve even the simple, obvious problems - we’re talking basic math and reading comprehension here. Got an email the next day informing me that they had I had not been selected for recruitment.
Am I fucking stupid? Just walk in a shallow spiral?
This is a direct appliacation of the hairy ball theorem.
I ain’t even kidding
Hairy ball theorem applies to even-dimensional spheres (the ordinary sphere is the 2D surface of the 3D solid), but a cube in four-dimensional space is a three-dimensional surface, so it doesn’t apply.
This is a question about graph theory, not topology; it’s asking for a Hamiltonian path on the surface of 4D cube (where faces are vertices, which is different than the normal polytope graph).
You are right.
However most proofs of the hairy ball theorem also prove the converse, so that there is a continous non vanishing tangent vector field on uneven dimensional sphere surfaces.
This can be extended to all 3 dimensional surfaces in 4 dimensions homomorphic to the sphere. The ant walking can follow the vector field and solve this problem topologically.
My point being that the HR goon following the expected leet code solution might not understand this because they might expect the “approved” graph theory solution rather than an alternative approach.
Why does following a tangent vector field visit all faces of the hypercube? Surely it’s not going to visit something like a dense subset of the hypersphere’s surface? (Or is it? My intuition comes from thinking about the torus)
I’m more interested in the maths ;)
My topology and maths are very rusty, am a software developer these days.
I think that there are both tangent vector fields that don’t and some that do. In the two dimnsional case (circle) certainly all do.
In n I intuitively would say that you should be able to have a vector field that does but I am now less confident to think about a proof on my bus rides while I answer here. I tried twice already.
I will try to think about this more, will ping here if I get more
You’re hired 🤝
Yaayyy, where’s my hypercubicle?
Sure. Draw the cube for me and I will plot it’s path.
Here you go:
That renders in 2d for me
No shit? Next thing you say that there are no 3d games, because there are no 3d monitors. And those that say they are 3d as well as VR are just faking it, by using two 2d projections instead of one.
They were being a smart ass.
Just code up a lemmy plugin that lets you embed basic interaction for navigating 4D shapes, my dude. It’s just basic eigenvectors.
Just wait until they figure out how eyes work
There are 2D monitors though.
You can project a 3d object into 2d space and you can do the same with 4d into 3d, but collapsing more than that generally loses too much information.
Your portrait is now just a colored line the height of your subject, and this “4D cube” doesn’t mean anything because it looks like a 3d cube with a smaller cube cut out of the middle of it. Unless you’re really into geometry I guess it you dropped a /s.
You must be fun at parties
I still don’t understand it. Can you rotate it along the W axis so I can visualize it better?
Sure thing boss.
Okay, if you can explain to me in detail how four dimensional topology is going to be important to me while I’m stocking the shelves of your grocery store, I’ll give you an answer.
Listen, once you get the job, you’ll discover the truth about those shelves. And all I’m saying is, it becomes relevant that you can find your way through four dimensional space. Okay?
They got the shelves from an old university library, the librarian who sold the shelves was an orangutan.
stocking the shelves of your grocery store
See that’s what’s so ragebaity about the post. There’s no mention of what the job was, which means people can just make up whatever bit of background allows them to feel the most superior.
It has a pay rate that is less than a living wage in many places. In fact, any job that could justify such a question could would be salaried. So it couldn’t even be described with an hourly rate.
in many places
You do realize you’re still doing the exact thing I just described yeah?
I think we can safely assume he was talking about the US.
deleted by creator
At $14.50 per hour, he’s going to take the shortest route.
Entry level positions to Gregg’s (fast food sausage roll chain) require 1000 word personal statements as part of online applications
Yeah but you also get equity in the company so I think that’s fair enough.
You have to be proven worthy before you are handed the recipe for the vegan sausage roll. I want to know what addictive substance they put in there.
Ever heard of ChatGPT?
tricky with only four dimentions, but I’d use a Grathenbour’s loop with a transverse Z axis movement if gimbal locks are ignored, naturally.
How does this compare in efficiency to casting Xagyg’s Planar Binding and simply using a standard verity geas to question a daemon from one of the higher hypergeometric dimensions?
Be glad you got the shitty interview instead of getting ghosted
Isn’t a cube by definition a 3 dimensional object? If it were 4 dimensional, it would no longer be a cube.
Its a generalisation. A 4d cube is a shape that has the same length in all 4 dimensions. You can also talk of 5d cubes, 6d cubes, etc. These are commonly called n-cubes: a 4-cube is a 4d cube.
There are also 4D spheres, even though spheres are definitionally 3D. They are called n-spheres.
robot voice:
how many 'd' is <USER_MATERNAL_PROGENITOR> a...cube...within? Ha ha. Ha ha.
Wait, isn’t this trivial?
If we’re talking about “faces” as in the cubic faces of a tesseract then each of the 8 faces are connected to all other faces except the opposite face. So just spiral around from your starting face (keeping the faces you’ve visited on the inside of the spiral) and you’re fine.
If you mean 2D faces connecting the 3D ones, then things get more difficult but not that much because you can do the exact same thing. Choose a 1D edge as your origin, pick a face touching that edge to start with, traverse that edge twice to get the next two faces. Then traverse three faces which share edges with those faces you already traversed (there are 6 faces with this property, 3 for each vertex of our origin edge, the set you pick determines the “direction” of your overall progress through/around the tesseract). Repeat that step again but for the faces that share edges with two of the three you just did. Repeat again and again and again until the last three faces share a vertex with the origin edge you started with. You’re done.
Am I missing something? Did the prompt mean to say you can only traverse each edge once?
Edit: the 2D face path I described would miss 6 faces. Those six faces should be traversed in the middle, so do the first three faces, the second three, then all six which touch both those three you just traversed and the three you would have done next on the original path. Then do the rest just like I originally mentioned.
I understood some of those words.
But the sentences continue to elude me
Have you ever seen one of those images of a tesseract where it’s like a cube in a cube? (You can just look up “tesseract” to find an image)
Now, pick one of the corners of the outer cube and find the line that connects it to a corner of the inner cube. That’s our origin “edge” and we’re basically just going to move in through the cube along that direction.
There are three “faces” which share that “edge” (line). We do those ones first.
Then we move deeper in and do the three faces of the inner cube which share the corner our origin line connects to.
Then we have to zig zag around the six “faces” that exist between inner and outer cubes which are roughly perpendicular to our origin edge. (Imagine you broke the tesseract in half by cutting halfway between your starting corner and the corner opposite it. The “faces” we need to traverse would intersect that plane)
After that, we do the three faces on the far side of the smaller cube. (The ones opposite our starting corner)
Then we do the three around the line which connects that far corner of the inner cube to the outer cube.
Then we do the three faces on the outside of the large cube at that corner.
Finally we do the three faces on the outside of the cube around our starting corner.
