Something about 4 colors makes sense to me intuitively because you've got 0 or 1 color for each of the two cartesian axes (2 * 2 = 4). Not sure there's anything behind that though.
Show HN: I made a puzzle game that gently introduces my favorite math mysteries
101–110 of 177 posts
Re: Show HN: I made a puzzle game that gently introduces my favorite math mysteries
#102Hi, my two cents; you claim "Although mathematicians believe their proof is correct, it is too complex to verify without computer assistance", but I'm not sure "believe" is the correct verb since the proof has been formally verified (see for instance https://github.com/coq-community/fourcolor for a formal verification in Coq). I understand that you want to emphasize the fact that no human can understand the proof wit…
I would consider a proof to be a "repeatable argument". One you could show to someone and expect them to be convinced by it. I think it is a defensible viewpoint that a proof in coq is not 100% convincing. If you think otherwise, then how can you reconcile the existence of falso (a coq verified proof of "false")?
https://github.com/clarus/falso
Proof and belief I think are pretty strongly intertwined, but I'm not going to pretend to have a particularly rigorous philosophy on the matter. Similarly, when the proof of Fermat's last theorem was published, I don't know if I should consider that to be a proof because it is well beyond my comprehension. I have no reason to question it, but should I consider it a proof? I know that people smarter than me (e.g. Wiles) thought the original version of it was a proof, but it had a subtle error in it which required a fix. While I haven't looked at the proof and revision, I would be surprised if I could look at the two versions as labelled and tell which one is the correct version.
Re: Show HN: I made a puzzle game that gently introduces my favorite math mysteries
#103So addictive! And it's great how you tie it in to the greater mathematical concept. I'm loving it!!
Re: Show HN: I made a puzzle game that gently introduces my favorite math mysteries
#104> Step 1. I put a , , or color dot on each region, but hide it under a post-it
> Step 2. You click any two bordering regions, and I pull off their post-its
> Step 3. You check that the revealed colors are different
I would add explicitly in step 1 that you tell me what 3 colors you used, and in step 3 that I check that the revealed colors are different from each other and are both from that set of 3. Similar in the explanation of why this should convince me.
Re: Show HN: I made a puzzle game that gently introduces my favorite math mysteries
#105Had a great time with this! Great break to get my mental gears moving
Re: Show HN: I made a puzzle game that gently introduces my favorite math mysteries
#106Loved the interactions and flow overall but I'm a bit lost on the zero knowledge proof example. I'm familiar with the concept but I don't follow how the example is one. E.g. "By repeating the process enough times, the probability that you never catch me becomes smaller than, say, getting struck by lightning" doesn't seem to show it's a proof? If I pick a hundred numbers it'll look like I just proved some black box fu…
OP: Although I'm not really in the target audience for this demo (I already knew all the punchlines), it does occur to me that it might be helpful to readers like the parent-commenter — and even perhaps thought-provoking to us know-it-alls — if you provided a mode at the end of the demo where the graph was in fact not three-colorable, and the computer actually would lie about its being three-colorable. So it would generate a "three-coloring" with a flaw somewhere, and display its representation as products of primes, and you'd get to choose two adjacent products and receive their factors... and so you could see for yourself exactly how long it took for you to luck into catching the computer in one of its lies.
And the demo could also tell you the expected number of iterations to catch the lie with (50/90/99%) probability. It'd be a pretty large number even for such a small graph, I'd bet.
(Of course the computer could also lie about the factorizations, since it's unlikely a human would bother to catch it in that kind of lie; but let's assume it doesn't ever do that.)
Readers might also be interested in the https://mathworld.wolfram.com/McGregorMap.html (reported, on 1 April 1975, to require five colors!)
Re: Show HN: I made a puzzle game that gently introduces my favorite math mysteries
#107Something about 4 colors makes sense to me intuitively because you've got 0 or 1 color for each of the two cartesian axes (2 * 2 = 4). Not sure there's anything behind that though.
Well it does not generalize into 3 dimensions: https://math.stackexchange.com/a/4767802
Re: Show HN: I made a puzzle game that gently introduces my favorite math mysteries
#108I showed this to my two kids, and we all three enjoyed it. The zero knowledge proof portion didnt really click for me, but we liked the four color map theorem stuff. This led me to download some maps for my kids to attempt coloring on paper, and also got me wondering about how this holds or doesn't on non-euclidian spaces. Turns out the maximum is four colors on a sphere, but 7 colors on a torus! More details here: h…
The best analogy of zk-proofs I've heard is to suppose you have found Waldo in "Where's Waldo," and want to prove that you have done this without revealing the location. You could take a piece of paper (much larger than the picture/book), and cut out a waldo-shaped hole it and position the paper such that he is shown in the hole. Then, when you show it to the challenger, they know that you have found him without you…
It took me a minute to fully get this, so I'm adding this so it's a bit more obvious for anyone else: the piece of paper is much larger than the picture/book so that it can hide the book's relative position underneath it.
Re: Show HN: I made a puzzle game that gently introduces my favorite math mysteries
#109Just a quick note: Rahul Ilango is a phenomenal theoretical CS researcher who has made great progress towards understanding the "Minimum Circuit Size Problem" [MCSP], long believed to be, but not yet proven, NP hard. Needless to add, "the username checks out".
Re: Show HN: I made a puzzle game that gently introduces my favorite math mysteries
#110Loved the interactions and flow overall but I'm a bit lost on the zero knowledge proof example. I'm familiar with the concept but I don't follow how the example is one. E.g. "By repeating the process enough times, the probability that you never catch me becomes smaller than, say, getting struck by lightning" doesn't seem to show it's a proof? If I pick a hundred numbers it'll look like I just proved some black box fu…
> E.g. "By repeating the process enough times, the probability that you never catch me becomes smaller than, say, getting struck by lightning" doesn't seem to show it's a proof?
That's fine though, because the point isn't really to publish math papers without disclosing proofs. For example, presenting a valid digital signature is sometimes colloquially called a proof that you had the private key, even though there is 1 in gazillion chance that you didn't. For such practical tasks, very high chance tends to be good enough.