The last 4 digits of Graham’s number

In the spring we had a lot of fun talking about Graham’s number. If you haven’t seen anything about Ggraham’s number before, you might enjoy checking out that prior blog post:

An Attempt to explain Graham’s number to kids

Also definitely check out the excellent series on Graham’s number that Numberphile did with Ron Graham!

Our talk today assumes just a tiny little bit of knowledge of Graham’s number – (1) mainly that it is an outrageously tall tower of powers of three, and (2) so large, in fact, that it is nearly impossible to even imagine how large the number actually is.

We returned to Graham’s number today because my younger son just started a new chapter about last digits in his number theory book. He’ll be learning about how to find the last digit of numbers like 3^{1000} and some other similar numbers. It is a neat subject and a fun way to continue to build number sense.

Right at the beginning, though, he asked me why we were only taking about the last digit – why not the tens digit, or hundreds digit? Well . . . today we’ll talk about the last 4 digits of Graham’s number just for fun.

Our first talk is a quick review of Graham’s number. If you want to understand the “up arrow” notation check out the links above, but that notation isn’t important for today. All you really need to know is that Graham’s numbers is a huge tower of powers of 3:

With the review out of the way we turn our attention to the last digit of Graham’s number. After looking at the first few powers of 3 we see that the last digit appears to repeat every 4th number. Quite surprisingly that pattern gives us enough information to infer that the last digit of Graham’s number is either 3 or 7. We spend probably half of the movie arriving at that fact and then we perform a more detailed calculation to see what the last digit actually is. One point that caused a little bit of confusion is that we need to look at the power itself in cycles of 4 (or what the remainder is when you divide by 4) even though we are looking for the last digit (so remainder when divided by 10):

Next we moved to the computer to get a little help from Mathematica! We essentially repeat the calculations that we just did on the whiteboard, but looking at the last two digits rather than looking only at the last digit. When you look at the last two digits you see a pattern that repeats every 20 powers – hence why a computer is helpful! Once we know that there is a pattern that repeats every 20 numbers we can use the computer to perform the same computation that we did by hand in the last movie to find the last two digits of Graham’s number:

The next step was looking for the last 3 digits. It is essentially the same process. We found that the last digit of the powers of 3 repeat every 4 powers and the last two digits of powers of 3 repeat every 20, so I asked the boys what they thought the pattern in the last 3 digits would be. They both guessed the repetition would be every 100 powers, which turns out to be right. Again, the computer is your friend here!

Also, we made a little mistake in this video and got confused between 100 and 1,000 when the pattern was repeating. Luckily that just made more work for the computer to do rather than for us (which is probably why we didn’t notice), but the result is unchanged (luckily).

We wrapped up by wondering why we are seeing powers of 5 in the way the digits repeat. The units digit repeats every 4 powers, the last two digits repeat every 20, the last three digits repeat every 100 powers, and the last 4 repeat every 500 powers – why are we multiplying by 5 every step? We didn’t arrive at an answer for this problem, but rather left it as something to wonder about.

The last thing we did was check out the Wikipedia page about Graham’s number to see if we got the last three digits right. That page gives the last 500 digits and our last 3 do actually match! We also now have a procedure to use to (perhaps) find all 500 digits.

So, a fun little project. Kicking myself for the 100 vs 1,000 mistake, but I guess that happens. The project kept the kids engaged all the way through – both the math and the computer results are really interesting. It is amazing (especially for kids) to see that even though you can’t really say anything at all about the number itself, you can compute some of the final digits.

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

Yesterday (11/5/2014) Numberphile published a really neat “proof” that all triangles are equilateral:

My older son and I are in the middle of reviewing the chapter in our geometry book about similar triangles, so this new video from Numberphile was well-timed for me.  We watched the video last night and today I used it as a way to review a little bit about similar triangles and constructions.

As usual, all of this was done on the fly and is pretty raw.  One particular bit of unluckiness was my choice a 3-4-5 triangle for this exercise, but since the main point was review anyway I’m not dwelling on that poor choice too much.  Should you want to do a similar exercise, though, definitely choose a different triangle  🙂

Despite how raw this project is, it is a pretty good struggle.  After we finished my son said “that was hard.”  He’s right, but that was part of the point.   The Numberphile video flows really well but that great flow hides a lot of geometry that is not super easy to work through – especially for someone just learning the subject.  What makes this project especially fun and hopefully useful is that this math is accessible to a geometry student with just a little bit of work.

So, with the warning about the rawness out of the way,  we began by talking about the problem and attempting the first construction – construct a  (non-equilateral) triangle given three sides:

Next we talked about the construction of the perpendicular bisector of the long side of the triangle.    My son was able to remember how that construction worked which made this part go much more smoothly than the later parts of the exercise.  After the construction, though, I asked how he knew that the perpendicular bisector was an equal distance from two nearby vertices of the triangle.  That geometric insight was tough for him.   Tough enough, in fact, that I paused the recording to give him some more time to think about it.

Returning from the pause he is able to see the congruent triangles that make the lengths from the triangles vertices equal.  I’m glad that he was able to make it through this little struggle and eventually see these congruent triangles.  Until walking through this geometry book with my son I’d never really thought about how connected similar triangles were with constructions.

Following the discussion about the perpendicular bisector, we drew in the angle bisector using a protractor.   The reason I used a protractor here is that  I did not think we would have enough time to include the angle bisector construction in the 45 minute time period we had this morning.  Having completed the project and seen how difficult it was, I’m not sure that I’d make a different choice now if I had more time.

In an unlucky coincidence the angle bisector we draw intersects the perpendicular bisector at one of the points we’d drawn in previously.   You can see in the Numberphile video that they had faintly drawn in all of the lines they needed ahead of time.  You can see in my video what happens when you don’t do that 🙂

The next part is the critical piece in finding the flaw in the Numberphile “proof,” and it turns out to be the most difficult piece for my son to understand.  The math problem here is figuring out how to construct a perpendicular line segment from a point to a given line.  The construction itself is pretty similar to the construction of the perpendicular bisector of a line segment, but if you’ve not tried out this construction before this similarity is far from obvious.  In this video we discuss the problem but do not solve it.

We talked about how to do the construction for probably 10 minutes after I turned off the camera.

After our 10 minute discussion he was ready to try out the construction on his own.  Even with the long discussion leading up to this part of the project, the ideas are not totally clear in his mind.  It is nice to see him work them out on the fly, though.

More than anything else,  I think the lesson here is that what seems like a really simple step in the Numberphile proof , just dropping these perpendicular lines to the (extended) sides of the original triangle, has some difficult geometric ideas behind it.  Working through these ideas takes some time – much much more time than I expected, in fact – but hopefully leads to a better understanding of geometry and constructions.

Now we return to our original diagram to construct the perpendicular lines we’ve spent the last 15 minutes discussing.   As with the practice construction in the last video this one isn’t without a little difficulty.  But we get there.  Also, we do get a little unlucky with our 3-4-5 triangle and one of the perpendiculars doesn’t quite come from the construction we’d been doing.

The final step now (and this is the final step just because we were coming up on our 45 min time limit) is to notice the difference between our picture and the picture at this step in the Numberphile video.  He doesn’t see the difference at first, so we pause and re watch the relevant part of the Numberphile video:

Now we return and he notices that the perpendiculars are not in the same place that they were in the video.   Something is amiss.  I’m not super happy with how I explain the flaw in the proof at the end, but that’s fine.  The main point was working through the constructions rather than understanding the flaw.  I’m pretty sure that he didn’t believe that all triangles were equilateral anyway!

So a fun and challenging project.  With a bit more prep I think this could have been quite a bit better.  Even the relatively raw approach shows how useful this Numberphile video could be to kids learning geometry, though.  At least I hope so!

What do mathematicians do

Lots of interesting math floating around the internet this week:

(1) Numberphile had an incredibly cool set of videos featuring Ron Graham talking about Graham’s Number,

(2) The NY Times had two articles on math education:  Why do Americans Stink at Math by Elizabeth Green and Don’t Teach Math, Coach It by Jordan Ellenberg,

(3) Two really interesting blog articles:  Jordan Ellenberg describes progress on understanding the rank of elliptic curves:  Are Ranks Unbounded? and Cathy O’Neil produced a neat little python notebook to walk people through RSA’s encription algorithm:  Nerding out: RSA on an iPython Notebook, and

(4) The “Twitter Math Camp 2014” teacher conference was happening in Oklahoma, which make for 100’s (of not 1000’s) of interesting discussions on twitter about teaching math.

All of of the fun math plus all of the ideas about teaching math made me want to step back and talk to the boys about what mathematicians do.    The math theme of the week seemed to be the difference between bounded and unbounded sets, so I tried to let that idea shape the discussion today.

We began by talking about Platonic solids.    Before turning on the camera we built a few of the Platonic solids out of our Zometool set for props.  Then we talked about what these shapes are and if there are infinitely many of them:

Next we talked about the prime numbers.  Ellenberg’s book How not to be Wrong has a wonderful discussion for a general audience about the prime numbers and I’ve been meaning to use some of his ideas to talk about the primes with the boys.  Luckily for me, right off the bat the boys were asking some questions about primes that Ellenberg answers. The main topic in this part of the talk is about the of primes, though my younger son wonders about the gaps between primes that will discuss in the next video:

Next, gaps between the primes. The boys seemed pretty interested in how the primes spread out. Ellengerg’s idea of using the even numbers and powers of 2 as an example turns out to be a really nice hook, and provides a great framework for talking about the new bounded gaps result:

After spending 10 minutes talking about some fun results about prime numbers, I wanted to spend the last few minutes talking about one way that prime numbers come into play in our daily lives. This part was inspired by Cathy O’Neil’s piece this week. I sort of daydreamed for a bit about an “rank of elliptic curves for kids” talk, but, um . . . , no.

What I focused on instead was the idea from O’Neil’s python notebook that it is easy to multiply two numbers and not so easy to factor. This idea forms the basis of encryption algorithms. Elliptic curves come into play, too, and Ed Frenkel discusses that a little bit in this fascinating video: Elliptic Curves and Cryptography.  But again, that’s for another day.

Definitely a fun week. Neat to see some new and exciting ideas from math in some blogs, and fun to see so much spirited discussion about math education. I think that many of the ideas in theoretical math will appeal to kids – Graham’s number and cryptography are just the two that emerged this week – and it is fun to be able to talk about these ideas and why mathematicians find these ideas interesting with my own kids.

Now, in the spirit of teaching and coaching from Ellenberg’s NYT article, I’m off to Boston to coach Brute Squad.

A small mistake in Numberphile’s videos about Graham’s number

[ not about math with my kids, but about some cool Numberphile videos that came out yesterday, and sorry this one was a little rushed]

Yesterday (July 20, 2014) I saw two absolutely incredible Numberphile videos about Graham’s number.  Part of what makes them so amazing is that the explanation of the number comes from Graham himself!   I love Numberphile’s work in bringing math to the masses.

There is, unfortunately, a little mistake in the text overlay in one of the videos that I wanted to point out.  The somewhat humorous result of this mistake is that Graham’s number is actually larger – vastly larger, in fact – than what the video indicates.

Since the formatting a blog post with the arrow notation and power towers was going to take more time than I had this morning, I decided to just go to the whiteboard. The mistake, which I explain in the first video below, is easy to make since all of the numbers are so large.  It involves confusing the number 3 ↑↑↑ 3 with 3^3^3^3.

After my video are the links to new Numberphile videos (that you might want to watch first if you aren’t familiar with Graham’s number) and two a few other fun Graham number articles, including a Family Math project about Graham’s number I did with my kids last year (which is the only reason I noticed the mistake in the new Numberphile videos).

Here are the new Numberphile videos (the text overlay error is in the 2nd video):

and here’s Evelyn Lamb’s piece on Graham’s number for Scientific American:

Finally, here’s our old Family Math project on Graham’s number, which is a really fun project to work through with kids:

An attempt to explain Graham’s number to kids

Last week Evelyn Lamb posted a great piece about Graham’s number:

I’d never head of Graham’s number before, but Lamb’s piece (and the Numberphile piece linked in the piece) really grabbed me.  Here’s their video for completeness:

I spent the week thinking of how to talk about Graham’s number with the boys.  It was a fun and challenging week, and one conclusion for sure is that Lamb’s title is indeed the correct summary – Graham’s number really is too big to even talk about!

After a couple of false starts earlier in the week I choose a pretty well-known problem involving a chess board as the starting point.  The problem involves putting one penny on the first square, 2 pennies on the 2nd square, 4 pennies on the 3rd square, and proceeding all the way to the 64th square.  How many pennies are on the 64th square?  Seemed like a good starting point for kids because you got to talk about powers as well as some really large numbers:


The next step was trying to introduce the arrow notation used in describing Graham’s number.  It took me a really long time to understand the notation, and those struggles made me want to cover the notation with the boys  in only  a superficial way.  Lamb’s piece above links to the Wikipedia page on Graham’s number – that’s as good a place as any to start if you really want to dig into the notation.  So, this part on Graham’s number is a little bit of notation and a little bit on powers of 3:

The second part of the discussion of Graham’s number is understanding how to compute the actual number.  The up arrow notation makes things look easy, but the computations themselves aren’t so easy.  In fact, almost immediately you encounter numbers that are virtually impossible to describe because they are so large.  Off camera I explained that as you created towers of powers of 3, the number of digits in each successive number was roughly equal to half of the previous number.  I’m not sure how much that helps, though, when you are talking about numbers that have 10 to the 3.6 trillion digits.  Ha ha:


This is all pretty heavy stuff, so I wanted to come back a little bit closer to reality, so we wrapped up with a question that Steven Strogatz posted on twitter last year:

This is a pretty neat question and since it involves an infinite tower of powers it is sort of connected to what we were talking about.   Strogatz’s question essentially asks if there are numbers greater than 1 for which the infinite tower of powers does not go to infinity.  (Also, I just started logs with my older son last week, so it was also a nice example with logs. )

Finally, since we weren’t going to solve the equation that came up in the last video on our own, we moved over to Mathematica and found the neat surprise that the answerr to Strogatz’s question involves the number e:

So, special thanks to Evelyn Lamb, Numberphile, and Steven Strogtaz for providing the inspiration for another fun Family Math day.  Talking about Graham’s number was really fun.



Pi Day

I have to head in to work early tomorrow so I was looking for a little fun math  to do tonight with the boys.  Numberphile came to the rescue when they posted some of their  \pi related videos on Twitter.  Their Buffon Needle video caught my eye:

As usual, a really nice job by Numberphile.

This problem is one of my favorites from high school calculus and I thought it would be fun to try to do the experiment with the boys. We watched the first couple of minutes to get the hang of what we were supposed to do and then went off to throw some straws as construction paper.

There are, I’m sure, 100’s of fun little math related activities that people will do tomorrow for \pi day.  This seems like an especially fun one to do with kids since it just involves counting.  The appearance of \pi in this  problem shows one neat connection between counting and geometry and also shows that \pi isn’t just about pies 🙂

The Numberphile infinite series video

I was having a fun conversation about the Numberphile 1 + 2 + 3 + . . . . = -1/12 video on twitter yesterday, but the limitations of twitter were making the conversation difficult.  So I decided to spend a little time today gathering my thoughts on the video, and perhaps more to the point, the video’s potential impact on the public perception of math.  It is proving to be more difficult than I expected  to pin down my own thoughts, but I’ve spent all the time I can, so here goes nothing!

**Update** – February 10th, 2014 – Ed Frenkel has an amazing 10 minute video interview about his reaction to the Numberphile video.  Wonderful.  I put this interview at the top of my post here because he says much of what I wanted to but far more eloquently than I could have:

** End update **


It almost feels as though there must be some sort of uncertainty principle for popular math presentations -> rigor * popularity =  constant.   The more rigor, the less popularity, and the more popularity, the less the rigor will have to be.

Who knows if that uncertainty principle is true or not, but even if it is true, I’m not sure it would bother me all that much.  It would mean that occasionally there will be popular presentations of interesting math that fall short on rigor.   Maybe the difficulty that I’m having gathering all of my thoughts is simply because  the Numberphile video falling short on rigor doesn’t bother me all that much.    I do not believe that people who are out to tell stories about interesting or unusual math are out intentionally  trying to deceive anyone.  If the cost of bringing math to the masses is an occasional angry lecture from the internet about things like analytic continuation,  then so be it.

Essentially everything that could possibly be written about the Numberphile video is covered in this wonderful blog post from Aperiodical:

For the purest of the pure math people, the two Terry Tao pieces tell the story.  Evelyn Lamb’s post is probably the best place (by far) to start if you don’t live and breathe math everyday.   If the video made  you cringe, then Cathy O’Neil’s post will probably make you stand up and cheer.

Rather than rehashing all of the math, I want to focus more on why I liked the video despite the flaws, and also talk through my reaction to two other popular math videos for comparison.

Let’s start with a fun one – Vi Hart’s “Wind and Mr. Ug” :

I wanted to start with this one because it is one of the most brilliant, if not the most brilliant, presentation of math for the masses that I have ever seen.  The first time I saw it I thought she should win a genius grant.  The only difficulty that I have with this video is that I don’t really know what you should do with people (kids mostly) after watching it.  Sure, you can make a few Moebius strips – and I’ve done that every time I’ve watched this video with kids – but that seems to be about it.  It is a beautiful video, but it floats away like a balloon after it ends.  Truthfully, I’m  not at all sure how I’d use it in a class if I was still teaching.   The one and only thing I’ve seen that used this video in a clever way is hardly math-related at all.  It is still fun though, so check out this  great parody video that tells the story from Mr. Ug’s perspective:

Next up is a video I don’t particularly like – Khan Academy’s video about the Golden Ratio and the moon:

For me, the example in this video is so contrived that I have a hard time seeing how students  would find it interesting.  Unlike the two videos above that I love (and watched all the way through before posting them to this article),  I struggle to make it even 30 seconds in to this one.  There’s so much beautiful math behind the golden ratio, why in the world you’d want to spend time on example like this is beyond me.  While I am probably a bigger fan of Khan Academy than most folks, and happily use their exercises with my kids from time to time, I worry that videos like this make math seem lifeless.   These contrived ideas make my blood boil, and so before I get too far down the road of a full-blown rant, let me stop and simply say that I wrote about it here:

Finally, the Numberphile video.

As I write my post here, the video has been viewed just over 1.3 million times.  Wow.

Yes there are mathematical flaws, and yes some articles have point out more flaws than I had noticed myself the first time I saw it, but I liked it and still like it despite the criticism.  I watched it with my kids who screamed at the screen that the equation was wrong (especially my 7 year old!).  I asked them about it again today at breakfast.  My 7 year old just repeated that it was all wrong.  My 10 year old said that it was just a bunch of physics nonsense (ha!).  At least their emotional reactions stuck with them.

If I was still teaching college or high school kids I would definitely watch the video with them.   Spending a day or two with students talking through some of the ideas in this video would, I think, be absolutely amazing.   There are so many different, and so many interesting paths to go down to help understand this crazy looking sum.

There are a couple of reasons that It didn’t really bother me that they were cutting corners on the math details.  First, many of those loose ends / errors are very interesting things to talk about in their own right and would be great things to talk through with students after watching the video.  Many of the blog articles explain in varying amounts of detail, how to correct these errors.    Second, I’m sure that a lot of physicists think about the world exactly the way that these guys in the video do, and I believe that they were trying to explain the rather odd result here as best they could without getting too caught up in ideas of analytic continuation or unusual interpretations of infinite sums.

A lot of the criticism reminds me of the criticism I heard directed at a new theory from Seiberg and Witten back in the 90s.  They had found a completely new way of thinking about some ideas in physics that had amazing applications in pure math.  Some mathematicians were worried about the lack of rigor.  Others ran with the theory and some big problems that had been unsolved for 100’s of years fell like dominoes.  I remember attending a few standing room only lectures by Cliff Taubes at Harvard.  I’ll never forget the excitement and buzz in the math world then, and I’ll also  never forget the healthy skepticism either – mathematicians crave rigor.

Physicists aren’t nearly as worried about rigor, though.  For example, here’s a write up on the Numberphile video from a physics blog:

Two of the items emphasized in this blog are (1) the infinite sum result is well known going back to Euler and Ramanujan, and (2) it gives the right answers in Quantum Electro Dynamics (QED) so as strange as the bizarre some of the series seems, there must be something there.  I’m sure there are some physicists who would want more rigor, but my guess is that those two points of emphasis would carry the day most of the time in the physics community.

Away from the mathematical rigor, one other very important point of focus was mentioned by Patrick Honner in our twitter conversation yesterday:

I have a couple of reactions.  First,  if you want to talk about fun and interesting math, I do not think you can ever completely eliminate a this problem.   If you are worried that people won’t get the math, you can report the proof of Fermat’s Last Theorem, or note that Harald Helfgott proved the weak Goldbach conjecture, but you can never give any details.  With regard to the specific topic of this video, covering analytic continuation or non-standard techniques used in infinite sums would alleviate the problem only because no one would watch the video.  Both of the topics are too technical for inclusion in a 10 minute presentation to a general audience.

Second, push back against ideas in science isn’t new or unusual and I hope that a large part of the message to the public about math is that it is ok to ask questions and ok to not understand.  Sometimes the pushback can be terribly unhealthy – see the story of Georg Cantor presented in “The Mystery of Aleph” for example.   But it can be healthy, too.   “The Infinity Puzzle” gives an absolutely fascinating (and a little controversial) description of how ideas in quantum physics have evolved over time.  Gary Kasparov’s series “My Great Predecessors” provides an astonishing presentation of the evolution ideas in chess.    Of the 1.3 million people who have viewed the Numberphile video, there are probably at least 1 million who thought “I don’t get this.”   There were many resources that came to light for people looking to understand it, which I think is great.  I really hope that these resources found they way to anyone whose reaction to the video was similar to the reaction Mr. Honner is worried about.

Third,  I do share Mr. Honner’s concern in general, and worry that a lot of popular math presentations end up being pitched in ways that are not always the best.  One that I saw recently really struck me because the author has done, and I’m sure will continue to do, more for math in the US than just about anyone.  If you look at some of the promotional material for Jordan Ellenberg’s new book “How not to be Wrong” you’ll find that “Ellenberg learned algebra at the age of 8 and got a perfect score on his Math SATs as a 12 year old.”

Of all the accomplishments he’s had, and all of the amazing things he’s done, I wouldn’t have even dreamed about putting his age 12 SAT score near the top.  I worry about a subtle message to kids that reinforces the idea that math is simply about innate ability – you are either a math person or you aren’t, and you might know by age 12 either way.  I worry a lot about the reaction of someone who thinks “I don’t get math” to this message.

Finally,  if you are  looking to find new and interesting and certainly non-controversial ways to pitch fun math, look no further than . . . . Numberphile!  Check out their interview with Ed Frenkel:

Frenkel’s quote at the end really struck me:  “What if I told you there is this beautiful world out there and you don’t even have to travel anywhere to find it.  It’s right at your fingertips . . . . this is the coolest stuff in the world.”

At this point there are many people out there working in various different ways to communicate fun and interesting math to the public.  While those people might have differing views about Numberphile’s infinite series video, I’m sure that group broadly agrees with Frenkel’s position.  His note at the end inspired me to do a series about 4 dimensional spheres with my kids (see my blog post right before this one), and hopefully other people who agree with Frenkel will continue to put new and fun math resources out for public consumption.  Can’t wait for the next public math controversy 🙂

4 Dimensional Spheres

I hope that everyone has seen the nice video that Numberphile and Ed Frenkel did about why people hate math:

As I write this blog post the video has about 130,000 hits, so I guess there’s still a few views to go before everyone has seen it.  Too bad.

The statement that really struck me in the video comes near the end (around 8:00).  Talking about the beauty of math, Frenkel says – “What if I told you there is this beautiful world out there and you don’t even have to travel anywhere to find it.  It’s right at your fingertips . . . . this is the coolest stuff in the world.”
The day before Numberphile posted the video I had done a fun project with my kids about circles and spheres.  We found how to calculate that the area of a circle was \pi * R^2 (two ways, actually!) and also how to find that the volume of a sphere was \frac{4}{3} \pi * R^3.    That talk with my kids had been inspired by a paper that Stephen Strogatz had posted earlier in the week that discussed the volume of a sphere in any dimension.  Pretty neat stuff for sure, but I stopped with just the circle and sphere with my kids.

After hearing Frenkel’s message at the end of the video, I changed my mind and plan to spend the next week talking with them about the 4 dimensional sphere.  They seemed pretty excited about the idea, and I’m super excited to try it so we started tonight.

The first thing that I wanted to talk about was some basics about circles.  They remembered some of the discussion from the weekend  ( it was sort of neat to see that they remembered the methods more than they remembered the results) and we quickly reviewed the rest.

After the review I backed up – what is a circle in the first place?  How can you extend that definition to a sphere?  What is a one dimensional version of a circle?    All fun things to discuss.

Finally I showed how we could create a 2 dimensional circle from a bunch of 1 dimensional circles and then build a sphere out of a bunch of 2 dimensional circles.   Of course, this was what we’d done over the weekend, but it probably didn’t hurt for them to see it again since we’ll use the same method to build up our 4 dimensional sphere.

All in all, it seemed like a nice start to our little project.

to be continued  (Jan 20, 2014)

Jan 21, 2014

Thought I’d be shoveling snow tonight so we moved on to our next talk about 4D spheres this morning.  My goal today was to help them get a slightly better understanding of how we move from zero dimensions to 4 dimensions.    To help with the geometry we built some models using our Zometool set.  The models were of squares and cubes rather than circles and spheres, but that just gave us two examples in the higher dimensions.  I took advantage of the two different shapes in each dimension to remind them of how we calculated the area and volume for the circle and sphere and to show that we could use the same method for the square and cube.    Seemed like the had a good time and they spent about 30 minutes after me made the movie building other stuff with the Zometool set.    Haven’t decided what we’ll talk about tomorrow, but I’m happy with how these two talks have gone.

to be continued (Jan 21, 2014)

Jan 22, 2014

I decided that the next step would be reviewing two methods that we’d talked about previously for finding the area of a circle and the volume of a sphere ( see this blog post: .  I settled on this approach because I wanted the kids to see how you could extend the concepts in these methods even though you didn’t necessarily know exactly what a 4d Sphere looked like.  Unfortunately this talk didn’t go nearly as well as I was hoping and I feel that I could have tied the points together much better if we went through it again.  Oh well, teaching lesson learned (hopefully!).

The main idea I’m trying to emphasize is breaking up a problem into smaller problems that you already know how to solve.  The specific example in the problems at hand is chopping a circle up into rectangles and chopping a sphere up into discs.  We extend that idea into chopping a 4 dimensional sphere up into spherical discs.  As we found with the circle, this method for finding the volume of a 4D sphere leaves us with a complicated sum involving square roots, and we’ll use Wolfram Alpha to help us understand that sum a little better tomorrow.

So, although my explanations in this talk were a little clumsy, I hope that the main ideas did come across and I’m excited to move on to understanding the new sum that will tell us the volume of the 4D sphere.  There’s a big surprise waiting for them in that sum!

to be continued (Jan. 22, 2014)

We took a break from shoveling out this morning to finish off the problem of finding the area of a 4 dimensional sphere.  After a quick review of what we’d done the last few days, we took a final look at the equation for the volume of one of the slices of the 4d sphere and jumped over to Wolfram Alpha to help us with the sum.  I didn’t want to dwell too much on how to evaluate the sum anyway, but using Wolfram Alpha had also had a special purpose in the lesson.  Since the sum we were looking at has a value of \pi / 2, I was wondering if either of the kids would be able to recognize the number.  Turns out that my oldest son actually did guess the right number after I asked him a few questions.  My youngest son thought it was pretty neat that we found a value for the sum, and also really cool that the volume of the 4d sphere with radius 1 was \pi^2 / 2.

Anyway, that’s the end of the main journey.  This exercise turned out to be one of the most enjoyable math talks that I’ve had with the boys.  I’m really happy that we were able to make it all the way to this neat result.

Numberphile’s “Pebbling the Chessboard” game and Mr. Honner’s square

Yesterday I saw an amazing math post on Twitter by Dan Anderson:

I love seeing math games with surprising outcomes that are simple to explain.  NumberPhile’s video on the problem is a masterpiece:

Solving the problem in the game involves summing a fairly complicated infinite series:

3/4 + 4/8 + 5/16 + 6/32 + 7/64 + . . . . .

The Numberphile video shows one way to sum that series, and eariler this year Patrick Honner published a nice visual proof showing how to sum (nearly) the same series:

Here is his beautiful picture that 1/4 + 2/8 + 3/16 + 4/32 + 5/64 +  . . . . = 1

I thought the game in the Numberphile video would be a super fun project to work through with kids.  I spent a little time last night trying to figure out how to talk about it with my boys and then spent the morning today going through it.

The first thing we talked about was Mr. Honner’s visual proof.  I wanted to do that at the beginning so that we wouldn’t get too distracted by the series when it came up during the game:

Finally, introducing the concept of invariants and connecting the game with Mr. Honner’s series:

The “Pebbling the Chessboard” game is such an amazingly fun and instructive exercise for kids.  Wish I would have known about this game back when I was teaching!