Gil Kalai’s Median game

Saw this neat tweet from Jordan Ellenberg yesterday:

The game itself is really simple enough for kids to understand, so I thought it would be fun to explain it to the boys and play. Here’s the explanation and a little bit of discussion about some math-related ideas that the boys had about the game:

Next we played the game with the boys writing down their numbers on the board and me selecting my number with an 8-sided die. We didn’t know the tie breaker rules so we just made up something at the end. It was fun watching them find a little bit of strategy on the fly:

Finally we wrapped up by talking about some strategy ideas they learned playing the game:

Definitely a fun game. It would be fun to play this in a room full of kids (of any age!) and see what creative strategies they would come up with.

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Choosing 3 million points on a 300-dimensional sphere

[sorry this post is a little sloppy – I had a hard stop at 7:00 pm and wanted to get it out the door.]

During the last day of my machine learning class we discussed “word2vec.” I’d heard of it before because of a couple of tweets and blog posts from Jordan Ellenberg. For example:

Jordan Ellenberg’s Messing Around with word2vec

I was reviewing this blog post during one of the breaks and stumbled on this passage:

Screen Shot 2016-07-22 at 8.48.38 PM.png

During class I couldn’t figure out how to think through this problem, but on the bike ride home I had an idea that seemed to work.

I don’t actually know if I’ve arrived at a “close enough” answer by coincidence because higher dimensional geometry is strange.  Here are a few example projects that I’ve done with the boys which range from fun to mind blowing:

Did you know that there is a 30-60-90 triangle in a hyper-cube?

Carl Sagan on the 4th dimension

Using snap cubes to talk about the 4th dimension

Sharing 4d shapes with kids

Counting Geometric Properties in 4 and 6 dimensions

A Strange Problem I overheard Bjorn Poonen discussing

Bjorn Poonen’s n-dimensional sphere problem with kids

A fun surprise in Bjorn Poonen’s n-dimensional sphere problem

One strange thing that comes to mind immediately in the statement from Ellenberg’s blog is that it must be somehow hard to find vectors in higher dimensions that meet at angles close to 0 degrees or 180 degrees. But why?

My solution to the 3 million points on a 300 dimensional sphere problem on the way home went something like this:

(1) Use a 2x2x . . . x2 cube with center at the origin and with all verticies having coordinates in every dimension that are either +1 or -1.

(2) Assume that the first vector is the 300-dimensional vector (1,1, . . . ,1)

(3) Now form 3 million 300-dimensional vectors with +1 or -1 randomly chosen for each position in the vector.

(4) By the dot product formula for vectors, the smallest dot product will product the largest angle, so now we just have to figure out how many -1’s the vector with the most -1’s should have.

(5) The number of -1’s in our vectors should be described by the binomial distribution with a mean of 150 and a standard deviation of 5\sqrt{3}

(6) To get to a 1 in 3 million chance we have to go out about 5 standard deviations (so about 43 away from the mean) but just to double check I ran this handy dandy Mathematica code:

Screen Shot 2016-07-23 at 6.54.35 PM.png

sure enough, with 3,000,000 trials we expect about 1 vector to have 192 to 193 -1’s .  Let’s say 192.

(7) So, that vector is going to have 84 more -1’s than +1’s so the dot product with our vector (1,1,1, . . . . , 1) is going to be -84.    That means the cosine of the angle between the two vectors will be -0.28.

So, close to what Ellenberg stated in his blog.

I’m not sure that this method is 100% right, but think it captures the general idea.  It certainly helped me understand why it is hard for random vectors to be parallel (or anti-parallel) in high dimensions.

 

A challenge for professional mathematicians

[March 24th, 2016 update – I’m going to link some articles at the end of the blog as I see them. There are two from today. I’m really happy that people are writing about this!]

I saw this article on gravity waves via a Steven Strogatz tweet this morning:

Seeing the article reminded me of the interview that Numberphile did with Ed Frenkel a while back – in particular, the part from roughly 5:00 to 7:00 when Frenkel discuses the need for mathematicians to do better at sharing their ideas with the public:

Frenkel’s point is that even though the ideas in fields such as biology and physics are just as complicated as the ideas in math, these other areas of science are much better at communicating with the public than mathematics is.

I was reminded of Frenkel’s point again this morning when I learned that earlier this month Maryna S. Viazovska solved the 8-dimensional sphere packing problem. Viazovska’s paper on arxiv.org is here:

The sphere packing problem in dimension 8

Maybe I’m a little biased – especially right now because I’ve been spending this week playing around with 4-dimensional shapes with my kids . . .

img_0947

but I think that the sphere packing problem (i) is something that can be explained to the public (it certainly seems less complicated than gravity waves) and (ii) is something that the public would find to be interesting. There’s not been much of any coverage of Viazovska’s result, though. Here’s what I found doing a simple Google news search:

Screen Shot 2016-03-22 at 2.24.44 PM

So, it sure seems this new result is something that would be great to share with the general public. There are, of course, many different directions an article could go – just off the top of my head:

(A) Jordan Ellenberg does a great job explaining the sphere packing problem and the connection to things like the Leech lattice and Hamming codes in How not to be Wrong,

(B) John Cook and Keith Devlin both have recent blog post with connections to higher dimensional spheres / cubes:

The empty middle: why no one is average by John Cook

Theorem: You are Exceptional by Keith Devlin

(C) Two years ago, Steven Strogatz shared this wonderful paper on N-dimensional spheres:

(D) The 2-dimensional problem of circle packing is something anyone can understand and is pretty fun to play with – here’s an old project I did with the boys using disc golf discs, for example:

 

Screen Shot 2016-03-22 at 2.51.32 PM

Sphere packing (well . . . circle packing)

Also, a version of the circle packing problem was in Jim Propp’s most recent blog post about mathematical thinking:

Believe it, then don’t: Toward a Pedagogy of Discomfort

So – come on professional mathematicians!! – here’s a great opportunity to promote a neat result and bring some really cool math to the public’s attention. Don’t let the physics crowd have all the fun!

A few articles that I’ve seen:

On Gil Kalai’s blog:

A Breakthrough by Maryna Viazovska lead to the long awaited solutions for the densest packing problem in dimensions 8 and 24

Kalai’s blog post also led to a question on Quora:

Why is the solution in dimension 8 such a breakthrough?

Amazing math from mathematicians to share with kids

About two years ago I saw this Numberphile interview with Ed Frenkel:

One of the ideas that Frenkel mentions in the interview is that professional mathematicians haven’t done a good job sharing math with the general public. Although I’m not really the kind of professional mathematician Frenkel was talking about, I took his words to heart and have been on the lookout for math to share – especially with kids.

It turns out that there are some fantastic ideas that are out there for kids to see. Some surprising fun I had sharing Larry Guth’s “no rectangles” problem with kids earlier this week (see below) made me want to share some of the ideas I’ve found in the last couple of years, so here are a few examples:

(1) One of the most incredible lectures that you’ll ever see is Terry Tao’s “Cosmic Distance Ladder” lecture at the Museum of Mathematics in New York City:

I used Tao’s video for three projects with my kids – but there are probably 20 math projects for kids you could get out of it.

Part 1 of using Terry Tao’s MoMath lecture to talk about math with kids – the Moon and the Earth

Part 2 of using Terry Tao’s MoMath lecture to talk about math with kids – Clocks and Mars

Terry Tao’s MoMath lecture part 3 – the speed of light and paralax

(2) The Museum of Math’s public lectures are a great source beyond Tao’s lecture.

Here’s a project based on Bryna Kra’s lecture:

Angry Birds and Snap Cubes – Using Bryna Kra’s MoMath public lecture to talk math with kids

Eric Demaine’s lecture was part of our Fold and Cut theorem project:

Fold and Cut part 3

and I can’t say enough good thinks about Laura Taalman’s work – she’s inspired dozens of our projects.  Just search for her name on the blog:

(3) and Speaking of Fold and Cut . . .

Katie Steckles and Numberphile put together an incredible video about the Fold and Cut theorem. I used the video this week for project with 2nd and 3rd graders at my younger son’s school earlier this week.  Steckles’s presentation is so incredible – this is the kind of math that really inspires kids:

We used it for three projects (including the Eric Demaine one above):

Our One Cut Project

The Fold and Cut Theorem is Awesome!

In prepping for the grades 2 and 3 projects I also totally coincidentally ran across a “fold and punch” exercise that is a great activity to try with kids before trying out fold and cut:

(4) Another great success with the 2nd and 3rd graders was Larry Guth’s “no rectangles” problem. I had a great time playing around with this problem with my kids, but nothing prepared me for how enthusiastic the kids in the two programs were about this problem.

Larry Guth’s “No Rectangles” problem

After the 3rd grade night, Patrick Honner sent me this picture that I used to wrap things up with the 2nd graders.

(5) The Surreal Numbers

I’d seen John Conway’s surreal numbers previously via an amazing Jim Propp blog post:

The Life of Games.

and I wanted to revisit them after finally reading Donald Knuth’s book:

Revisiting the Surreal Numbers

Infinity + 1 and other Surreal Numbers

Playing with the surreal numbers via checker stacks is an incredibly engaging way for kids to learn about mathematical thinking.

(6) Speaking of John Conway –

In the 2014 edition of the Best Writing in Mathematics Conway had an article about variations on the Collatz conjecture. It was a fascinating article that even gave us the idea to translate some of the math into music.

The Collatz Conjecture and John Conway’s “Amusical” variation

I’ve also talked with the boys about the standard version of the Collatz conjecture:

It is a great way to introduce kids to an unsolved problem in math while also sneaking in a little bit of arithmetic practice!

(7) Occasional contest math problems

I happened to run across another MoMath lecture yesterday – this one by Po-Shen Loh. He was talking about “Massive Numbers.” I thought maybe he’d be talking about the book “Really Big Numbers” by Richard Evan Schwartz:

A few projects for kids from Richard Evan Schwartz’s “Really Big Numbers”

or maybe Graham’s Number:

An attempt to explain Graham’s number to kids

The last 4 digits of Graham’s number

but instead he talked about a neat problem from the 2010 International Mathematics Olympiad:

His presentation is fascinating and I even talked through the first version of the problem with my younger son:

Another math contest-like problem I really enjoyed talking about with the kids was this one:

Show that any positive integer n has a (positive) multiple which has only the digits 1 and 0 when represented in base 10.

A challenging arithmetic / number theory problem

(8) Building off of popular books by mathematicians as well as public lectures

I was surprised at how much great math writing and speaking there has been for the general public in the last couple of years.

Jordan Ellenberg’s “How not to be Wrong” inspired several projects – probably my favorite was using his idea of “algebraic intimidation” to talk about the famous 1 + 2 + 3 + . . . = -1/12 video by Numberphile. :

Jordan Ellenberg’s Algebraic Intimidation

Jacob Lurie’s Breakthrough Prize public lecture inspired two projects about a year apart from each other:

 

Using Jacob Lurie’s Breakthrough Prize Lecture to Inspire Kids

Using Jacob Lurie’s Breakthrough Prize talk with kids

And, Ed Frenkel, who got me thinking about sharing advanced math with kids in the first place has inspired a few projects, too:

Fine Ed Frenkel – you convinced me

Ed Frenkel, the square root of 2, and i

and one of my all time favorites:

A list Ed Frenkel will love

(9) Finally, it would be impossible to write a post like this one without mentioning the work that Evelyn Lamb is doing writing math articles for the general public. I’ve lost count of how many projects she’s inspired, but it is probably well over 20. I’m especially grateful for her talk about topology which have generated really fun conversations with the boys. For example:

Using Evelyn Lamb’s Infinite Earring with kids

Evelyn Lamb’s fun torus tweet

and

Henry Segerman’s Flat Torus

which arose after Lamb pointed out this video:

 

So, I’m really happy that mathematicians are sharing so many amazing ideas. I think this is the sort of math promotion that Frenkel had in mind. Hopefully it continues for many years to come 🙂

10 fun math things from 2014

I’ve been paying attention to math a little more in 2014 than I have in previous years and thought it would be fun to put together a list of fun math-related things I’ll remember from this year:


(10) Dan Anderson’s “My Favorite” post

Dan asks his students to talk about things they would like to learn more about in math class, and the students talked about subjects ranging from topology to diving scoring. I was really happy to see the incredibly wide range of topics that the kids thought would be interesting. Beautiful post by Dan and a fantastic list of topics chosen by his students – this one made a big impression on me:

Dan Anderson’s “My Favorite” post

My initial reaction to Dan’s post is here:

A list Ed Frenkel will love

(9) Laura Taalman’s Makerhome blog:

We bought a 3D printer early in the year and it allowed us to do a bunch of math projects that wouldn’t have occurred to me in a million years. Most of those projects came either directly or indirectly from reading Laura Taalman’s 3D printing blog. As 3D printing becomes cheaper and hopefully more available in schools, Taalman’s blog is going to become the go to resource for math and 3D printing. It is an absolute treasure:

Laura Taalman’s Makerhome blog

An early post of mine about the possibilities of 3D printing in education is here:

Learning from 3D Printing

and one of our later projects is here:

Klein Bottles and Möbius Strips

(8) Numberphile

It has been nearly a year since Numberphile’s fun infinite series video hit the web. I know people had mixed feelings about it, but I loved seeing a math video spark so many discussions:

 

I’ve used so many of their videos to talk math with my kids, I’m not even sure which of them to pick for examples. Here are two:

Using Numberphile’s “All Triangles are Equilateral” video to talk about constructions

Some fun with Numberphile’s Pythagorean Theorem video

(7) Fields Medals and the Breakthrough Prizes

Erica Klarreich’s coverage of the Fields Medals over at Quanta Magazine was absolutely amazing. Two of her articles are below, but all of them (including the videos) are must reads. Her work her made it possible for anyone to meet the four 2014 Fields medal winners:

Erica Klarreich on Manjul Bhargava

Erica Klarreich on Maryam Mirzakhan

A really cool opportunity to understand the work of one of the Fields Medal winners came when the Mathematical Association of America made an old Manjul Bhargava’s paper available to the public. I had a lot of fun playing around with this paper (that he wrote as an undergraduate, btw). It made me feel sort of connected to math research again:

A fun surprised with Euler’s identity coming from Manjul Bhargava’s generalized factorials

The Breakthrough Prizes in math didn’t seem to get as much attention as the Fields Medals did, which is too bad. The Breakthrough Prize winners each gave a public lecture about math. Jacob Lurie’s lecture was absolutely wonderful and a great opportunity to show kids a little bit of fun math and a little bit about the kinds of problems that mathematicians think about:

Using Jacob Lurie’s Breakthrough Prize talk with kids

I’m glad to see more and more opportunities for the general public to see and appreciate the work of the mathematical community. Speaking of which . . . .

(6) Jordan Ellenberg’s “How Not to be Wrong”

Jordan Ellenberg’s book How not to be Wrong is one of the best books about math for the general public I’ve ever read. I have it on audiobook and have been through it probably 3 times in various trips back and forth to Boston. My kids even enjoy listening to it – “consider the set of all integers plus a pig” always gets a laugh.

One of the more mathy takeaways for me was his discussion of infinite series and what he calls “algebraic intimidation.” Both led to fun (and overlapping) discussions with my kids:

Talking with about Infinite Series

Jordan Ellenberg’s “Algebraic Intimidation”

(5) The Mega Menger Project

The Mega Menger project was a world wide project that involved building a “level 4” Menger sponge out of special business cards. We participated in the project at the Museum of Math in NYC. The kids had such a good time that they asked to go down again the following weekend to help finish the build.

Menger Boys

It was nice to see so many kids involved with the build in New York. It also made for another fun opportunity to explore the math behind the project a little more deeply:

The Museum of Math and Mega Menger

(4) People having a little fun with math and math results

For some family fun, check out the new game Prime Climb:

our review is here:

A review of Prime Climb by Math for Love

Also, don’t forget to have a little fun when tweeting about new and important math results. Like Jordan Ellenberg tweeting about the solution of an old Paul Erdos conjecture:

Erica Klarreich’s Quanta Magazine article on the same result was just published yesterday by coincidence:

Erica Klarreich on prime gaps

For me the math laugh of the year was Aperiodical announcing the results of an 8 year search confirming the 44th Mersenne Prime:


(3) Evelyn Lamb’s writing

Evelyn Lamb’s blog is a must read for me. I love the wide range of topics and am pretty jealous of her incredible ability to communicate abstract math ideas with ease. Her coverage of the Heidelberg Laureate Forum was sensational (ahem Breakthrough Prize folks, take note!). This post, in particular, gave me quite a bit to think about:

A Computer Scientist Tells Mathematicians How To Write Proofs

My thoughts on proof in math are here:

Proof in math

Away from her blog, if you want a constant source of fun and interesting math ideas just follow her on Twitter. For instance this tweet:

led to a great little project with the boys:

Irrationality of the Square root of 2


(2) Terry Tao’s public lecture at the Museum of Math

On of the most amazing lectures that I’ve ever seen is Terry Tao’s public lecture at the Museum of Math. I don’t know how it had escaped my attention previously, but I finally ran across it about a month ago. What an incredible – probably unparalleled – opportunity to learn from one of the greatest mathematicians alive today:

Explaining a few bits of his talk in more detail led to three super fun projects with the boys:

Part 1 of using Terry Tao’s MoMath lecture to talk about math with kids – the Moon and the Earth

Part 2 of using Terry Tao’s MoMath lecture to talk about math with kids – Clocks and Mars

Part 3 of using Terry Tao’s MoMath lecture to talk about math with kids – the speed of light and paralax


(1) Fawn Nguyen’s work

When one of the top mathematicians around is tweeting about projects going on in a 6th grade classroom 2000 miles away, the world is working the right way!

Fawn is producing and sharing some of the most interesting math projects for kids that I have ever seen, and I’m super happy that her work is getting recognized. She’s probably inspired more than 20 projects with the boys, and I can’t wait for the next 20 in 2015. Here are two from this year:

Fawn Nguyen’s Geometry Problem

A 3d Geometry proof without words courtesy of Fawn Nguyen

If you have even a passing interest in fun, exciting, and generally kick-ass math projects for kids – you have to follow Fawn.

Jordan Ellenberg’s “Algebraic Intimidation”

One of the ideas that seems to have stuck in my mind from reading “How not to be Wrong” a couple of times is the concept of “algebraic intimidation.”   Ellenberg uses this phrase to describe one of the standard ways to “prove”  that 0.9999….. = 1.   I go through the proof that he’s talking about in the first video below if you’ve not seen it before.

The idea of algebraic intimidation is, I suppose, pretty simple:  the math all looks right, therefore the result must be right because you **better** believe the math!

This concept obviously generalizes to all sorts of situations.   As the maybe useful / maybe harmful (depending on what year it is) quantatitive ideas seem to be creeping back into the financial markets, I feel like I’m seeing the old algebraic intimidation hammer at work a lot more frequently these days.  But, hey, we all miss 2008, right?

While a post about martingales might more more relevant to the attempts at using math to intimidate in the financial markets, I think Ellenberg’s example is infinitely more interesting.  Particularly for students, and I’d love to use the examples below in a room full of kids who are interested in math.

The idea of talking about algebraic intimidation once again came up this past weekend in our Family Math project.  I asked the boys what they wanted to talk about  and they gave me a surprising answer – “Infinite Series.”   The entire set of talks from this weekend is here:

The two conversations relevant to algebraic intimidation  are below and came when one of the examples that they wanted to talk about was “the -1/12 series.”  Say what you want about that old Numberphile video, but the ideas in it sure stuck with my kids!

I led off this part of our project with the standard proof of why 0.999…. = 1 and then, following some examples in Ellenberg’s book, extended the ideas in that proof to a few other areas where you get some rather odd results.  We then moved on to the “-1/2 series” and followed the ideas in the original Numberphile video.

You’ll see that both kids are quite skeptical of the results.  My younger son in particular is almost physically upset.  That’s good.  I want them to learn to question results rather than just blindly trusting the math, and I especially want them to feel free to question results that seem odd.  You certainly won’t find many results that seem more goofy than the ones below 🙂

Talking about “Infinite Series”

[Note:  I’m in a little rush to get up to Boston for Brute Squad practice today, so I’m just getting the videos up now and will expand this blog entry either later today after practice or later in the week.  I haven’t even proof read this, but it was so fun I just wanted to get it out there!]

Today I told the boys that we could cover whatever they wanted for today’s Family Math project and they chose infinite series as the topic.  In particular they wanted to talk about

(1) Fibonacci Numbers,

(2) Pascal’s triangle,

(3) the sum 1/2 + 1/4 + 1/8 + 1/16 + . . . ., and

(4) “the -1/12 series”

I’m not sure that I could have been more excited about this list of topics!

We started with the Fibonacci numbers.  The idea here was to review the idea of how you create the list of Fibonacci numbers, see what the boys remembered about this sequence, and show them how the Fibonacci numbers arise in a simple continued fraction.  The boys remembered that you could use the numbers to make a spiral, so we spent a little bit of time talking about the spiral, too.

I wanted to show the continued fraction example because the Fibonacci numbers occur in both the numerator and the denominator of the continued fraction convergents, but the numbers are shifted over 1 in the numerators.  That shift of an infinite sequence will come into play in our last videos when we discuss “the -1/12 series”

The next topic was Pascal’s Triangle, which turns out to be an absolutely perfect next step by luck.  We started by reviewing how you create the triangle and then moved on to looking at some other sequences that are hiding in the triangle. We found several fun patterns hiding in the triangle including some patterns that describe some fun geometry. At the end I showed them that even the Fibonacci numbers are hiding in the triangle in sort of a sneaky way. I wanted to talk more about this but a bee flew into the room, oh well . . . :

The third topic was the sum 1/2 + 1/4 + 1/8 + . . . and why this series sums up to be 1.    This was also really fun and I got a nice surprise as each kid had a slightly different geometric way of showing why this series summed to 1.  I showed them a 3rd slightly different idea and then showed them a second neat series that also sums to 1 ->  1/4 + 2/8 + 3/16 + 4/32 + . . . Patrick Honner gave a really cool visual proof of this fact here  and show them how his visual proof works.

The last topic is the “-1/12 series” made famous by this Numberphile video:

After an introductory talk about this series and the seemingly (or perhaps, “actually”) crazy sum, I backed up a little by talking about a question that seems to be a tiny bit easier -> does 0.999…. = 1?  Following the line of reasoning in Jordan Ellenberg’s “How Not to be Wrong” I showed that the standard way of proving this also can produce some strange results.  I really like Ellenberg’s description of this standard proof as “algebraic intimidation” and you can see how that algebraic intimidation plays out in the next two videos as both kids really don’t believe that the original sum is -1/12, but also seem to be convinced by the math that it does.

Finally, I followed the ideas in the Numberphile video above and showed how you get the result that the sum of 1 + 2 + 3 + 4 + . . . . = -1/12.  I love that this result seems to actually physically bother my younger son.

This was a super fun project.  Shows the fun you can have when you let the kids pick the topics 🙂