The Maths Thread

[SillyCowCorner1]

A comment on the video by

The thing about these "less than a minute" problems is that it all comes down to how quickly you see how to solve the problem. I'd bet that the kids who solved this problem in under a minute had been working with geometry of shapes and were very familiar with this sort of problem.

A puzzle-solving competition for kids sounds like something that Ferrari would have done to solve the quartic equation back in the day. And this:

Fiore, believing Tartaglia’s claim to be a bluff, challenged him
to a public problem-solving contest. Each contestant was to propose 30 problems, the
victor being the one who could solve the greatest number within 50 days. Tartaglia was
aware that his rival had inherited the solution of some form of cubic equation from a
deceased master, and he worked frantically to find the general procedure. Shortly before
the appointed date, he devised a scheme for solving cubics that lacked the second-degree
term. Thus, Tartaglia entered the competition prepared to handle two types of cubics,
whereas his opponent was equipped for but one. Within two hours, Tartaglia had reduced
all 30 problems posed to him to particular cases of the equation x^3 + px = q, for which
he knew the answer.

From The History of Mathematics by Burton

 

[vcs]

Find the probability that a stick, when broken randomly into three pieces, a triangle can be formed out of the 3 pieces.

 

[harsh.ag]

Find the probability that a stick, when broken randomly into three pieces, a triangle can be formed out of the 3 pieces.

100% probability

 

[vcs]

No

 

[harsh.ag]

No

A triangle can be formed with 100% probability. A triangle with the sides of the exact length of those 3 sticks has a lower probability

 

[vcs]

I meant to ask the probability that a triangle can be formed using those exact 3 pieces.

 

[Bijed]

I won't be able to answer anyway, but I'm assuming that the three pieces have to fit 'neatly' into a triangle, and that my technical example below would not be a valid solution?

Edit: Ah, clarified anyway in the 15 minutes it took for me to render my image

 

[vcs]

Yep. Basically no point in me trying to dance around the obvious.

The three segment lengths have to obey the triangle inequality. The problem is basically restating that in a different way and asking what is the probability that this will happen if the stick is randomly broken.

 

[vcs]

I won't be able to answer anyway, but I'm assuming that the three pieces have to fit 'neatly' into a triangle, and that my technical example below would not be a valid solution?

Edit: Ah, clarified anyway in the 15 minutes it took for me to render my image

View attachment 25167

Nice out of the box thinking though

 

[Spark]

haven't actually solved it yet so here's my go:

let x be any real in (0,1)

if x > 1/2 then the triangle inequality can never be satisfied for any partition x:y:z of [0,1] (here i'm denoting x:y:z to denote a partition of [0,1] into three sub-intervals of length x, y, z respectively, which obviously means that x+y+z=1) as y+z < 1/2 by definition so y+z < x. so assume x <= 1/2.

(by extension, y,z<1/2 as well)

so let x be an arbitrary real in (0, 1/2]. we only need to fix some y in order to fix the partition x:y:z of [0,1]. so we need to find all the possible y such that all three xyz triangle inequalities are satisfied.

by the same logic, y < 1/2 and z < 1/2 as well. however, if x + y < 1/2, then z > 1/2 as x + y + z = 1 so the triangle inequality fails. so x + y > 1/2.

consider this information translated to the xy plane. the region of valid partitions is that bounded by x > 0, y > 0, and x + y < 1 (because otherwise you don't have a partition into x + y + z). this bounds a triangle of area 1/2.

the region within the xy plane that defines a partition x:y:z that will satisfy the triangle inequalities is thus that within this aforementioned triangle that is also bounded by the lines x = 1/2, y = 1/2 and x + y > 1/2. this is a triangle of area (1/2)*(1/2)*(1/2) = 1/8, i.e. 1/4 of the region that defines valid partitions.

so the answer is 1/4, which isn't really surprising.

 

[vcs]

Yeah, correct. I did it the same way. I saw a nice alternative solution online that helps to visualize what you posted above.

https://www.youtube.com/watch?v=qTRW7ELhhME

 

[vcs]

It would be nice to get a spoiler function on here BTW.

 

[Spark]

Yeah, correct. I did it the same way. I saw a nice alternative solution online that helps to visualize what you posted above.

https://www.youtube.com/watch?v=qTRW7ELhhME

that frankly seems way more complicated than simply noticing that you have to have x < 1/2 and x + y > 1/2, which is basically the crux of the issue

 

[vcs]

I'm a sucker for a good bijection though.

 

[vcs]

https://www.math.utah.edu/~cherk/mathjokes.html

 

[Magrat Garlick]

https://www.math.utah.edu/~cherk/mathjokes.html

what, no mention of the anagram of Banach-Tarski?

 

[RossTaylorsBox]

Is this Mochizuki paper a thing or nah

 

[vcs]

Is this Mochizuki paper a thing or nah

From Wikipedia:

Mochizuki proved Grothendieck's conjecture on anabelian geometry in 1996. He was an invited speaker at the International Congress of Mathematicians in 1998.[7] In 2000-2008 he discovered several new theories including the theory of frobenioids, mono-anabelian geometry and the etale theta theory for line bundles over tempered covers of the Tate curve.

On August 30, 2012 Mochizuki released four preprints, whose total size was about 500 pages, that develop inter-universal Teichmüller theory and apply it to attempt to prove several very famous problems in Diophantine geometry.[8] These include the strong Szpiro conjecture, the hyperbolic Vojta conjecture and the abc conjecture over every number field. The preprints have not been published. In September 2018, Mochizuki posted a report on his work by Peter Scholze and Jakob Stix asserting that the third preprint contains an irreparable flaw; he also posted several documents containing his rebuttal of their criticism.[9] The majority of number theorists have found Mochizuki's preprints very difficult to follow and have not accepted the conjectures as settled, although there are a few prominent exceptions, including Go Yamashita, Ivan Fesenko, and Yuichiro Hoshi, who vouch for the work and have written expositions of the theory.[

Yea I think it's safe to say none of us here will be qualified to comment on this

EDIT : I was thinking of saying "with the exception of Spark" when I posted that

 

[Spark]

I've been low-key following that on and off for a while. It's more accurate to say not a single soul on earth with the possible exception of maybe Ivan Fesenko understands the underlying structure of the proof's argument, and without that you have nothing; it is deliberately written in such an obtuse, non-standard and self-referential way as to be virtually opaque to any sort of attempt at independent analysis and verification. I do know that there was a paper claiming to have found a significant problem with the proof in 2018. Mochizuki replied that they hadn't put in the time to really understand IUT theory -- maybe, but it's up to him to explain it in a way that's comprehensible to other people in the first place, that he's failed to do.

My current understanding is that the general view in the professional algebraic geometry community is that, until he bothers to actually do the work to make his proof and theory comprehensible to mere mortals, whether by way of a conference or a series of review papers or a lecture course or something, his proof is considered unverified.

EDIT: https://inference-review.com/article/a-crisis-of-identification This is a good semi-recent piece on the whole mess. That Mochizuki seems to have responded to criticism - completely par for the course in this line of work - with ad-homs is not a good sign for the work itself.

“Two mutually alien copies of conventional scheme theory,” he writes, “are glued together.”12 He describes his technique as, “dismantling the two underlying combinatorial dimensions of a ring [emphasis original].”

Like I've read enough maths papers to know that, if you don't know the jargon, the words can seem almost random and the meaning completely opaque, but nonetheless this verges on nonsensical. I actually know what a fair few of these words mean individually; put together I can't even begin to describe what this could possibly be talking about.

 

[Spark]

Incidentally, algebraic geometry - particular EGA and SGA and what it led to, since I already know enough of the classical stuff to move on to that - is high on my list of "things to teach myself when I have time".