A tale of two problem solvers (Average cube shadows)

What's the average area of a cube's shadow?
An equally valuable form of support is to simply share the videos.

There's a small error at 19:30, I say "Divide the total by 1/2", but of course meant to say "Multiply..."

Curious why a sphere's surface area is exactly four times its shadow?

If you liked this topic you'll also enjoy Mathologer's videos on very interesting cube shadow facts:

I first heard this puzzle in a problem-solving seminar at Stanford, but the general result about all convex solids was originally proved by Cauchy.
Mémoire sur la rectification des courbes et la quadrature des surfaces courbes par M. Augustin Cauchy

The artwork in this video was done by Kurt Bruns

Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Hindi: rajeshwar-pandey

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Timestamps
0:00 - The players
5:22 - How to start
9:12 - Alice's initial thoughts
13:37 - Piecing together the cube
22:11 - Bob's conclusion
29:58 - Alice's conclusion
34:09 - Which is better?
38:59 - Homework

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These animations are largely made using a custom python library, manim. See the FAQ comments here:

You can find code for specific videos and projects here:

Music by Vincent Rubinetti.

Download the music on Bandcamp:

Stream the music on Spotify:

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Various social media stuffs:

[spicemasterii]

At last, Alice and Bob are doing something other than sending cryptic messages to each other.

[Shivumgrover]

28:30 "And we can simplify that 2π/4π to simply be 1/2"
Me: Finally something that I could've done myself.

[YonatanZunger]

Another note: Alice's _result_ is more generalizable than Bob's, while Bob's _method_ is more generalizable than Alice's. (You can see this by thinking about the harder problem of a nearby light, where Bob's method keeps working while Alice's doesn't!)

This is one reason why combining the two approaches is so valuable. You can start with something you know will work but may not unlock a great mystery, and then look for patterns that clue you in to a wider story.

[MistaSkilla]

When he started turning the sphere into a band I was preparing myself emotionally for him to turn the sphere inside out without pinching any points

[ontheballcity]

I did a PhD in pure maths. The main result in my thesis had a very pretty Alice-like proof. The way it eventually dawned on me was spending a couple of years doing Bob style calculations of specific examples.

[Mrsparky]

Another thing to note about the two philosophies is that Alice's way is beautiful but it requires you to be clever or lucky to connect disparate ideas and exploit the general connection. Bob explores the space with calculation and uses the connections that he identifies. I think there is not a separate Alice and Bob but instead a bob thinker picks away at a problem until he is able to build up to a generalization that equals Alice's. Bob's next question should be what about other shapes? Followed by what about all shapes? Eventually he would come to the same conclusion and probably prove the problem in the same way as Alice.

One of my frustrations with learning (highschool/undergrad level) math was that we only see Alice's brilliant proofs and sometimes it appears as a magnificent logical leap that I would have no hope making if I was in their position.

[DelusionalLogic]

To me this seems like the difference between what we in software call "Top down" versus "Bottom up" problem solving. Bob takes the "bottom up" approach of looking at the specific problem he's attacking, going through the motions of solving it, and through that, he might stumble into some generality that he can later come back to. Alice on the other hand starts from the top. She notices that if she manipulates and connects the abstract pieces of information to finally arrive that the simplest form of the problem, which she then solves.

One of my teachers had a nice saying about it: "Always solve the problem top down, except the first time", echoing the conclusion hit here. Top down problem solving is fast and awesome, but it's really difficult (if not impossible) to solve real problems like that. It often while working through the bottom up tedium that we realize what top down abstractions we can manipulate.

[SeanStClair-crjl]

Your writing is so, so insanely good. It is a RARITY to find an educator so capable and devoted to the task of creating genuine understanding. You demonstrate an ability not just to expound upon every detail, but to minimize, order, and portion complexity in a way that can actually be digested. You make your motivations very clear, and you execute with a self-awareness that shows just how much you understand your audience. Not to mention the relevance and quality of your ANIMATIONS.

There are many famous video demonstrations that have gone down in history - in physics classrooms, on YouTube - as being particularly eye-opening, particularly effective at conveying a topic in isolation. Somehow, you manage to achieve this quality in every video. I've only chosen to write this here because this is your most recent!

[DrTrefor]

I really appreciate this video's focus on contrasting different problem solving styles. I think it is important that we all be a bit reflective of our own biases and what we enjoy and what we find natural, particularly because some problems lean themselves more one way than the other. I know for myself I always thought of myself more as an "Alice", but over time I've actually come to really enjoy more computation-centric approaches.

[danbornside]

I think a nice upside to "The Bob approach" that I'd like to emphaize, is that you can make forward progress on a problem without having any particular insight into the problem. Sometimes it's a lot easier to have insight into an answer once you already have a solution.

[BTAMSU]

I don't always understand what's being said, but I do enjoy when a particularly astute blue pi gets angy.

[aemmelpear]

What I found from studying physics for over 4 years now, is that often times (as with this problem) the Bob approach is what happens first. At least for me I often do the hardcore calculation first for something, because I have difficulties of finding these "nice" solutions, without having spent time on this problem already. It happened a few times myself, that after doing the hardcore calculation, I found ways so simplify it further and further until it became a very pretty "Alice-like" solution. However I couldn't have found the Alice solution without being Bob first.

[puzzLEGO]

I first watched 3blue1brown about 3 or 4 years ago, and even though I didn’t understand it I thoroughly enjoyed it. Now years later when I’ve gone through the majority of high school, I realise these videos are some of the best on youtube

[DiracComb.]

I feel like Bob’s approach acts as a launch pad for Alice’s method. If you solve a few special cases, you can then look for patterns that then allow you to hunt down the elegant solution later. It feels rare for someone to see the elegant solution on first sight. It’s something found in hindsight after some special calculations are made to provide a sketch of what is probably true, though math doesn’t have to care if things are pretty.

[patrickoberholzer]

Anyone else incredibly impressed just by the process of drawing Bob and Alice?

[savantshuia]

Now if only Alice and Bob had a way to share their proofs, maybe by sending messages that no one else is able to read?

[mathemaniac]

34:30 One of the reasons is that most people, even for those mathematically inclined and consuming mathematical content during their spare time, do NOT want to exercise their brain to a degree that those tedious calculations would demand, and let's be honest, I don't, unless I am REALLY interested in the problem at hand. As a result, those videos that actually dive deep into the calculations would get buried; and the "slick" methods can get people's attention or even shares. It's almost like natural selection that promotes this bias rather than any creator's fault.

[ciscoortega]

God, I just say through this entire video in one setting. You had me hooked the entire way.
I think this is one of the best videos you've made---if not THE best---and certainly it's the most relevant.
I loved the wonderful aha moments throughout the Alice portions (you had me screaming out loud at certain points because I was so excited about an insight), but the meta-commentary you provided at the end is just as, if not more, important.

[MPSmaruj]

Being a very Bob-minded person myself: to me the most dangerous thing about Alice's approach is how easy it is to miss hidden assumptions.
I am sure Alice in this story was fully aware of all the assumptions she made along the way, but someone with less expertise trying to follow her method might not realise it.
Conversely Bob was forced to explicitly address the problem of defining uniform distribution of rotations and from his calculations it is evident that for specific shapes the answer absolutely would depend on the probability distribution.

[zekecochran]

All of Grant’s videos are good, but this one really stands out to me. Absolutely incredible work.

There were about five times throughout the video where I had a question/objection, or simply had to pause and justify things in my own head, and make sure I was really on board. Without fail, as soon as I unpaused the video, Grant addressed exactly what I had been wondering, with exactly the best and most intuitive justification I had been able to come up with. The video followed my path of thought almost to an unnerving level of precision.

I have never before seen a video that could hold a candle to the layout quality here. The order in which topics were addressed was perfect, as was the level of detail, not to mention the beautifully constructed graphics. And I think that is a rather difficult task for this topic, because at any one point, there is more than one interesting question to be answered. The questions to be explored do not lead into one another single file, but branch out like a tree.

The comparison of the methods at the end was very good as well. A hearty congratulations to everyone who contributed to this video.