
Simple Set Game Proof Stuns Mathematicians - retupmoc01
https://www.quantamagazine.org/20160531-set-proof-stuns-mathematicians/
======
Xcelerate
> The paper soon set off a cascade of what Ellenberg called “math at Internet
> speed.” Within 10 days, Ellenberg and Dion Gijswijt, a mathematician at
> Delft University of Technology in the Netherlands, had each independently
> posted papers showing how to modify the argument to polish off the original
> cap set problem in just three pages.

This is a very interesting trend I've noticed. I wanted to cite a few papers
on arXiv in one of my own research papers recently, but my advisor commented
that none of the articles had been peer reviewed (since arXiv is a preprint
server). I told him that in the last year alone, six papers on arXiv follow a
"research trail" (i.e. a paper is put on arXiv in May that builds on results
from a February paper that builds on results from December, etc.), and that
the most recent peer-reviewed article in a published journal is so far behind
the state-of-the-art that completing my paper without any mention to the arXiv
works would put me significantly behind the rest of the field.

Of course, these papers all relate to math and computer science — whether a
new algorithm or proof works is (usually) immediately evident upon
implementation, and the papers on arXiv include the complete algorithm and
often link to the author's code. Peer-reviewing their work yourself often
takes no longer than a half hour or so (unlike, say, a research article in
materials science, where a complete replication study could take over a year).

~~~
Ankaios
That has been common in physics for a long time, too. For hot new topics, the
conversations are often happening through preprints rather than peer-reviewed
articles. (The papers are usually eventually published in peer-reviewed
journals, though.)

~~~
MichaelBurge
Building on research seems like it's a form of peer review, especially in math
where you'd need to understand the entire proof before making incremental
improvements to it.

~~~
abritinthebay
While this _can_ be true it's also common to have "assuming X is true" where X
is some very complicated hypothesis.

We saw this with Fermat's Last Theorem and it was with a great sigh of relief
that it _was_ finally proven in the 90s. If the inverse was true then entire
fields of Mathematics would have collapsed.

~~~
throwaway676565
> If the inverse was true then entire fields of Mathematics would have
> collapsed.

Honest question: what would've been the consequences of this?

~~~
btilly
The consequence is that a whole body of work winds up coming with an asterisk
until people figure out what they can and can't trust. Papers may be looked at
for inspiration, but won't be quoted for results. Eventually some of it gets
proved properly, and the rest is abandoned. After that the older papers become
mere historical curiosities.

A reasonably recent example of this is the
[https://en.wikipedia.org/wiki/Italian_school_of_algebraic_ge...](https://en.wikipedia.org/wiki/Italian_school_of_algebraic_geometry).

A possible place where this could happen is the classification of finite
groups. It has been "proven", but the proof is long, technical, and never was
adequately reviewed. Lots of papers these days start off using the
classification in interesting ways. However there is an open program to
produce an actual reviewed proof. If in the process of doing that, we found
that the original result was long, there would be a fairly large project to
figure out the consequences.

See
[https://en.wikipedia.org/wiki/Classification_of_finite_simpl...](https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups)
for more.

~~~
tamana
But when the results are useless anyway, it doesn't really matter if they are
right or wrong...they just may be speculative of some alternate universe, or
may still contain ideas that are applicable elsewhere.

~~~
Natanael_L
Prime numbers used to be useless when first researched (edit: during the
previous two centuries, when their properties was studied). We don't always
know in advance what will turn up useful.

~~~
kragen
You say, "Prime numbers used to be useless when first researched," but when
the Middle-Kingdom Egyptians were doing their initial research on prime
numbers, they needed them for the algorithms they used to calculate with
fractions. These were used in the Rhind Papyrus to calculate things like the
volumes of granaries. You could hardly have picked a worse example.

~~~
btilly
No, he picked a famous and perfect example. He just didn't specify it well
enough.

Over the last 2 centuries, number theorists developed the theory of large
prime numbers. The numbers that they were dealing with were so large that they
had no conceivable use in describing the physical universe.

Famously one prominent number theorist, G. H. Hardy, wrote _A Mathematician 's
Apology_, a book describing and justifying his life. In it he famously
described his field as being utterly useless with no practical applications.

Then cryptography came along, and the mathematics of finding large prime
numbers, and factoring hard to factor large numbers, turned out to have
practical applications of great importance!

~~~
kragen
Please don't blame me for refuting what he did say, instead of what he would
have said if he'd known what he was talking about.

~~~
btilly
From my point of view, both of you demonstrated a lop-sided knowledge of math
history.

Clearly you know more about the ancient history and origins. I'd be willing to
bet that you know that the ancient Greeks knew 2500 years ago what prime
numbers were, had proved that there were an infinite number of them, had
algorithms like the Euclidean algorithm for finding the greatest common
denominator, had proven unique factorization AND had demonstrated that sqrt(2)
was not a fraction. We don't actually know how much farther a lot of the
knowledge goes.

On the other side he had obviously encountered cryptography, and knew that a
whole lot of the necessary number theory dates back to Gauss, 200 years ago.
[https://en.wikipedia.org/wiki/Disquisitiones_Arithmeticae](https://en.wikipedia.org/wiki/Disquisitiones_Arithmeticae)
is the origin of concepts like modular arithmetic, quadratic residues, and so
on. But he was not familiar with the ancient history predating that, or else
he could not have thought that the study of primes only goes back 200 years!

He could have avoided the problem on his side by Googling for what he was
going to say before saying things with glaring and obvious errors. Very few of
us are so careful.

You could have avoided the problem on your side by giving him the benefit of
the doubt and assuming that he's probably not a complete idiot, then trying to
figure out what he might have meant. You might or might not have figured out
"cryptography", but you could have at least made your post in the form of a
much more pleasant question. However that is fairly rare to find, and doubly
so online.

As for me, I'm just lucky enough to know both halves of the history, so could
easily sort it out.

~~~
kragen
Ben, I'd've thought you'd known me long enough to know that I'm familiar with
RSA and the history of number theory. (Maybe I misremember; you seem to have
started doing Perl after I left clpm.) I read _A Mathematician 's Apology_
last year (most of it, anyway), and my friend Nadia keeps publishing papers
that factor large numbers of RSA keys in practice, the latest being
CacheBleed. Your understanding of cryptography is surely deeper than mine —
the most I've ever done myself is write an implementation of SRP — but it's
not as if I haven't heard of the field.

I had thought that it was common knowledge that (small) prime numbers had a
lot of practical uses (mental arithmetic in general, arithmetic with
fractions, including with vulgar fractions, gear train design, that kind of
thing) but apparently I was wrong. It turns out that lots of people don't know
about this. So my inference, that only a complete idiot would not know this,
he did not know this, and therefore he was a complete idiot, was ill-founded.

And so I came out looking like some kind of ignorant, arrogant know-it-all. I
really appreciate the feedback, Ben. Natanael_L, I'm sorry I was such a dick
to you.

~~~
btilly
I am familiar with your name, but hadn't tracked you well enough to remember
anything more than, "He knows Perl." I left Usenet about a year before I
learned Perl, so I was never in clpm.

So you just failed to register the cryptography reference and then backsolve
to what he really meant. If that's the worst thing that you did last month,
then you're a better person than I...

------
tokenadult
A lot of the history of mathematics is summed up in the last paragraph of this
very interesting article. "The fact that the cap set problem finally yielded
to such a simple technique is humbling, Ellenberg said. 'It makes you wonder
what else is actually easy.'" Mathematical progress comes about by some
mathematician noticing a pattern that makes solving a formerly difficult
problem more easy than solving it used to be. (For this purpose, innovations
like place-value decimal numeral notation fit the preceding statement.) Of
course the hard thing is being the first human being to figure out the easy
way to solve the problem, and then to embed that understanding in a technique
that other problem-solvers can learn. But the increasingly rapid collaboration
among mathematicians these days (mentioned in a comment posted earlier) speeds
the uptake of new techniques and makes more likely that new understanding will
rapidly spread among mathematical problem-solvers.

------
pnut
I did really well in school, and am a professional developer now, and my wife
utterly destroys me at this game.

She can find sets in real time as the cards are being dealt, while singing
along with music, it's maddening.

~~~
dhekir
Could it be somehow related to better peripheral vision?

I met a woman once who could gather four-leaf clovers nearly instantly, at the
same spot where me and my male friends had spent minutes searching.

I've seen written in some places (but could not find a worthy source) that
women might have better peripheral vision due to food gathering in prehistoric
times, but this might just be some old sexist construct (if anyone knows a
good source to confirm or refute this, please tell me).

~~~
mundo
Funny anecdata: in my circle of friends, the trait that best predicts Set
skill isn't IQ or gender or anything like that, it's musical ability.

This came up at a party. Six people playing Set, all of us from different
cities, and the three that took band in high school utterly dominated the
three that didn't. The reason? Field trips. The Band kids had all played Set
for hours on buses, the rest of us had only played it once or twice and lost
interest.

~~~
function_seven
I like how you introduce a correlating attribute in the beginning of your
comment, get me thinking about the relationship between musical ability and
set-identifying ability, then clobber whatever hypothesis I was putting
together just as fast with the confounding "field trips" variable.

Nice work :)

~~~
tamana
But why is set so popular on band trips, but not football trips?

~~~
function_seven
I'd guess that the players on a football team would be reviewing the playbook
all the way up until kickoff. Meanwhile the band isn't going to be practicing
their performance on the bus.

------
henrik_w
Nice to read about mathematics that involves Set! I found out about Set here
on Hacker News several years ago, when I read Peter Norvig's blog post on the
odds of not finding a set in the 12 cards [1]. I had never heard about the
game before, but tried it and really liked it.

The simplicity of the rules, the mathematical nature of it, and the fact that
adults and kinds can play together makes it such a great game. I continued
with the analysis of finding the odds of not set in each round of play, and
had fun doing so. It seems very difficult to find an analytical solution (and
quite beyond me), but simulation was a nice project that gave some good
insights into how the odds vary over the rounds played [2].

[1] [http://norvig.com/SET.html](http://norvig.com/SET.html)

[2] [https://henrikwarne.com/2011/09/30/set-probabilities-
revisit...](https://henrikwarne.com/2011/09/30/set-probabilities-revisited/)

~~~
zem
the similar but easier game "spot it!" works better for playing with kids, i
find. (i prefer it as an adult too - i thought i'd love set, but playing it
feels more like work than fun to me.)

~~~
dkurth
There's some interesting analysis of the math behind Spot It! on stackoverflow
[1].

That post mentions the Fano Plane, which, incidentally, I first read about in
the book How Not to Be Wrong by Jordan Ellenberg, a mathematician quoted in
the article about Set that started this thread. In the book, he uses the Fano
Plane to explain how to pick numbers for a specific kind of lottery.

[1] [http://stackoverflow.com/questions/6240113/what-are-the-
math...](http://stackoverflow.com/questions/6240113/what-are-the-mathematical-
computational-principles-behind-this-game)

~~~
zem
thanks, that was a beautiful thread! also a really good diagram explaining the
projective plane.

------
jerf
For those who check the comments first: Not clickbait. Title is literally
accurate.

~~~
joe_the_user
Oh come on,

The title and the article seem to be essentially accurate but are written in a
breathless "we'll make this math stuff exciting, darn it" tone that some might
like but which, as a math person, I find somewhat grating.

I think science/math writing of twenty or forty years ago was still reasonably
effective and with that, we could get-by with a headline such as "A impressive
result in the combinatorics field that uses the 'polynomial method'"

Edit: I appreciate that quantamagazine.org is doing a bunch of math articles.
But we gotta admit it's injecting about every bit of "this is simply
amaaaazing" rhetoric it can muster, which detracts somewhat from at least my
enjoyment of it and actually it a harder to figure what's happening (from MA-
level perspective).

~~~
tgb
As a grad student in math I read a lot of lifeless papers. Seeing the exact
opposite is fun. I thought it was a good article. Why does math need to avoid
breathlessness if we seem to embrace it with, say, a million breathless
articles about Stephen Curry's latest game? It only seems bad because there's
not a culture of excitement over math like there is for basketball, so it
stands out. But if we want there to be such a culture, maybe we need to start
by taking every bit of excitement we have and stretching it to the fullest.

~~~
schoen
Martin Gardner was very good at that, and I think Erica Klarreich (one of the
main Quanta math journalists and the author of this particular article) is
good at it too: she likes to focus on the history and context of a problem,
describing mathematicians' expectations and hopes for a solution, and then how
progress was made and what consequences it may have.

I guess there's some risk that many of these stories fall into similar
patterns (something was hard and mathematicians worked on it for a long time,
they didn't expect it to be solved soon, now it's been solved solved and
people are impressed and see various applications), but it's nice to see the
details and hear from some people in the field in their own words.

I feel like Klarreich tends to give more details than Gardner did when writing
about current research; maybe that partly has to do with the web form, because
she can include links to the actual papers and supplementary materials. It's
also a nice counterpoint to the enthusiasm that her articles convey because
the applications and consequences are _specific_ ; it's not like journalism
that says "maybe now we'll get a pony/interstellar spaceflight/spooky quantum
communications/faster computers!" without really showing how the discovery is
going to enable that.

The fact is that mathematicians are excited by unexpected progress in math
research, so hopefully other people can be too! :-)

------
bdamm
Wonderful piece; what I like is that I remember this game being taught to me
by the mathematics professors at my school, and I enjoyed the game but didn't
ask any deeper questions about the game, such as its mathematical properties.
Now here I see this way of looking at the game, perhaps even the reason for
its existence, and it's like rediscovering an old friend.

------
esturk
To me the most exciting part is that this may lead to a faster fast matrix
multiplication algorithm which will be huge. The current bound is O(n^~2.372).
They could very feasibly hit the theoretical limit of O(n^2+e).

------
aaron695
Maximal size of cap sets for games with up to six attributes -

2 4 9 20 45 112

[https://oeis.org/A090245](https://oeis.org/A090245)

------
delhanty
Another good link concerned with this problem is from OEIS:

[https://oeis.org/A090245](https://oeis.org/A090245)

"Maximum numbers of cards that would have no SET in an n-attribute version of
the SET card game"

The OEIS links appear to have been updated to reflect the recent breakthrough
of Ellenberg and Gijswijt, including their unified paper arXiv:1605.09223 from
May 30.

------
dmd
For your entertainment... [http://3e.org/set](http://3e.org/set)

------
furyofantares
I don't understand the Alice imagery (nor the witch, not sure if that is also
Alice imagery somehow) -- any ideas?

~~~
MaysonL
The witch appears to be the Red Queen. (Rotate her head about 135 degrees ccw
and you get what is close to the symbol for a chess queen).

------
syastrov
Can anyone explain how these polynomials are generated? Are the constants or
exponents based on the card characteristics? And if it is not understood why
they work, then how can we use them in a proof?

------
twic
Again with the Ramsey theory! Every day, i'm reading about some amazing new
result in Ramsey theory! It's like the microservices of discrete mathematics!
Enough already!

~~~
andrewflnr
I've been bumping into it a lot, too. I think it started with the ZFC-
independent turing machines a while back.

------
graycat
IIRC "Elegance in mathematics is directly proportional to what you can see in
it and inversely proportional to the effort required to see it" \-- S.
Eilenberg

------
OJFord

       > a different design with four attributes — color (which can be red, purple or green),
    

Was this game invented to annoy the colour-blind?

~~~
schoen
It was invented based on the file-marking symbols used by a geneticist for
genetics research:

[http://www.setgame.com/founder-inventor](http://www.setgame.com/founder-
inventor)

[https://web.archive.org/web/20130313190253/http://www.setgam...](https://web.archive.org/web/20130313190253/http://www.setgame.com/set/history.htm)

You can play a variant of Set in a single color (which the rules suggest is
easier -- but it's also more feasible for color-blind players).

------
ctdonath
iOS app of the referenced game:
[https://appsto.re/us/EXmoU.i](https://appsto.re/us/EXmoU.i)

Physical version of the game is available, I suggest adding it to your
collection.

~~~
MontagFTB
Variants exist on the web as well, e.g.,
[http://thebreretons.com/trifecta/](http://thebreretons.com/trifecta/)

