# A neat problem from my son’s math team

I saw this problem on a sheet my son brought home from math team practice:

A couple of people point out on twitter than the problem isn’t worded so well. That’s my fault. The actual problem was a little longer but I was trying to capture the main point of the problem and did so in a slightly sloppy way.

Anyway, my son struggled with the problem at the math team tryouts, and the problem seemed so cool to me that I wanted to do a little more than just explain it to him.

Here’s our discussion. In the first part we just try to understand what’s going on – his first question was how you could get different shapes?

The second part of the discussion was about how to determine the largest difference in volume. This turned into a nice discussion about numbers and arithmetic. How could we tell which of the two numbers we had was bigger?

Fun little problem – and definitely a heck of a challenging problem for 6th graders!

# Having the boys watch a Brian Greene video

This talk by Brian Greene was recommended to me by my friend Amy after she saw some of the 4-dimensional work we were doing last week. I was out the door really early today so I didn’t have a chance to work with the boys. Instead I had the boys watch the video and we talked about their ideas tonight.

Here’s the link to Greene’s talk:

Brian Greene’s talk on String Theory

Here’s my older son talking about what he saw:

And here’s my younger son’s thoughts:

Listening to them talk with interest about string theory really reinforces the idea that I first heard from Ed Frenkel – other science subjects do a much better job explaining their ideas to the public than math does. Makes me want to keep working to explain many more math ideas to kids!

# What I’ve learned playing around with primes

I’ve been playing around with the new result about primes by the Stanford mathematicians Robert J Lemke Oliver and Kannan Soundararajan. The result has been interesting to share with my kids:

Looking at the new discovery about primes with kids

Prime triples and the Sieve of Eratosthenes

and also to study on my own:

My fun interaction with prime numbers this week

I’ve kept going – slowly – after writing the last post. As I write this one I’ve counted ( via a simple Mathematica program) how often the last digits of consecutive prime triples occur up to the 14 billionth prime.

All of the data is in this google doc:

My prime triple counting data

On the “Prediction vs Actual” tab you can see the predicted results from the paper vs the actual counts for the last digits of consecutive primes (in base 10). I was able to understand the paper’s predictions for triples of the form (a,a,a) and (a,b,a).

For example (and assuming I’ve done the calculations right), in cell L9 of the Prediction vs. Actual tab, you’ll see that the new paper predicts 8,664,330 consecutive primes ending in (7,7,7) from the 7 billionth prime to the 8 billionth prime. There are actually 8,699,947 (cell L28). Not bad!

I’m interested to keep going to study the errors a bit more.

Here’s what I’ve learned so far:

(1) The new paper is clear enough for me to understand a few pieces even though I have virtually no background in number theory.

(2) Understanding those pieces took some work, though, and I’ve probably had to spend 5 hours to get the few pieces that I have gotten. I’m glad I didn’t give up ðŸ™‚

(3) In base 10, triples of consecutive primes end in (a,b,c) and (-c,-b,-a) with roughly equal frequency. I saw this result in the data and talked about it with the boys in the Sieve of Eratosthenes project above (after writing to the prime paper’s authors to ask for help understanding it).

(4) I have lots of ideas about how to understand all of this data about primes and essentially all of those ideas turn out to be totally wrong – ha ha.

(5) Working through the paper to calculate some of the paper’s predicted values for prime triples was one of the most satisfying activities that I’ve done in math this year . . . . even though I’m not sure that my calculations are right.

(6) One of the things that’s puzzling me now is that the actual vs predicted errors for the triples with equal digits – (1,1,1), (3,3,3), (7,7,7), and (9,9,9) – are too large. They seem to have size roughly proportional to 1/Log(n). The large errors make me think I’ve missed something in the paper – that’s the next thing I want to study.

(7) Working through any of the basic ideas about prime numbers from this paper is a great computer exercise for kids – especially if you have an easy way to identify primes (I was lucky to have Mathematica).

(8) I was really nervous about writing to the authors of the paper to ask about why the triples (a,b,c) and (-c,-b,-a) appear with equal frequency, but I’m glad I did. It was so cool that they wrote back.

So, this has been a really fun little side project for me. Many of the public results in math over the last few years have been great, but way way way over my head – the prime gap result from Zhang, the Fields Medal results, and the Breakthrough Prize results for example. I don’t know how frequently a new result in math or science can be understood by the public, but I sure am glad that this one landed on my desk!

# Some piano math – just for fun

About a week ago I was just goofing around and noticed that the sound you got from an old seltzer bottle was a D on the piano:

Today my younger son was a little tired from a hike up Mount Monadnock, so I was looking for something a little lighter than usual. We decided to explore the sounds from the bottles a bit more carefully:

The short conversation in the first part of the project led to my son guessing a procedure for making a few more notes, so we tried it out. We didn’t get it quite right, but after the video below he returned to the sink and was able to produce the note he was looking for!

Fun little project, and I was glad to see that he was interested enough to want to get the note from the 2nd movie right after we finished ðŸ™‚

# Prime triples and the Sieve of Eratosthenes

For the last week I’ve been looking at the last digits of triples of consecutive prime numbers. This exercise was inspired by the new paper of by Robert J. Lemke Oliver and Kannan Soundararajan of Stanford who just made some surprising discoveries about last digits of pairs of consecutive primes. I wrote about my week-long project here:

My fun interaction with prime numbers this week

One of the strange patterns that I noticed was that each of the 64 last digit triples seems to have a partner that occurs with the same frequency. I did not know why this paring was happening and wrote to Lemke Oliver and Soundararajan to see if there was an explanation that I could understand.

They wrote back (yay!) and told me that the pairing related to something that occurs in the Sieve of Eratosthenes. The idea goes something like this:

(1) The pairing I was seeing was that the last digit triple (A,B,C) occurs with about the same frequency as the triple (-C,-B,-A). All of these last digits are being represented mod 10, so 3 = -7, 1 = -9, 7 = -3, and 9 = -1.

(2) Lemke Oliver and Soundararajan told me that you can see a similar groping in the Sieve of Erathosthenes at every step. So, if you stop after crossing out all the multiples of 2, 3, and 5 you’ll see the same number of consecutive last digits of the form, say,(7,1,3) as you’ll see of the form (7,9,3).

The reason has to do with properties of modular arithmetic – I think. I have to confess that I didn’t understand the full explanation (even though it only took 1 paragraph!), but it did seem like a fun idea to try to explore.

So, this morning I decided to explore the idea a little bit with my kids. I started by explaining the ideas about last digits of consecutive primes that caught my eye last week.

Then we worked out the first three steps of the Sieve of Eratosthenes and I asked the kids to talk about some of the patterns they saw:

Next we looked at some of the last digits in consecutive numbers which remained on the board. I also explained the (A,B,C) and (-C,-B,-A) pattern which led to a brief diversion into what -X means in mod 10.

Another fun question for kids in this part was – how many different triples of last digits of prime numbers are there?

The last thing we did was look to see if there were any (A,B,C) and (-C,-B,-A) pairs on the board. The e-mail from Lemke Oliver and Soundararajan said that there would be the same number of pairs in the numbers. Of course we only had the integers up to 60 on the board so I had no idea if we’d see any pairs. Turns out that we did – exciting!

It is fun to be able to find new math reserach that you can (partially) explain to kids ðŸ™‚

So, a fun little impromptu project for today. Sorry this one was a little rushed – the kids were heading out to go hiking in NH and I wanted to get in a short project before they left.

# My fun interaction with prime numbers this week

Last week I saw a amazing new result about primes by two mathematicians at Stanford – Robert J. Lemke Oliver and Kanna Soundararajan – via an Evelyn Lamb article:

Peculiar Pattern found in “Random” Prime Numbers by Evelyn Lamb

Erica Klarreich at Quanta magazine also wrote a fantastic article about the result:

Mathematicians Discover Prime Conspiracy by Erica Klarreich

and there’s also a neat discussion of the result on Terry Tao’s blog:

Terry Tao’s blog post about the new result

After seeing the two articles (I only saw Tao’s blog post today) I thought it would be fun to play around with some similar ideas and chose to look at the last digits of triples of consecutive primes. Over the course of the week I was able to use a simple program in Mathematica to count how often the different triples of last digits occur in consecutive primes in the first 10 billion primes. Right from the start I found something I didn’t expect – counting the occurrence of the triples of last digits seemed to pair the sets of last digits quite naturally into groups of 2.

For example, for 3 consecutive primes in the first 10 billion prime numbers the last digits (3,7,1) occur 178,500,881 times and the last digits (9,3,7) occur 178,500,928 times. Another example of the strange grouping is that the triple (1,1,3) occurs 147,750,170 times and the triple (7,9,9) occurs 147,761,746 times. Weird – what’s causing this clustering?

All of my data is in the google doc linked below. I’m sorry that the data in the google doc isn’t organized very well – I was just playing around for myself, but thought that it might be fun to share anyway:

My google doc with all of the data I collected this week

I didn’t really study any number theory in college or graduate school, so I have essentially no way to know if something like the counts for the last digits of consecutive prime triples pairing up is an easy to prove fact or an impossible to prove fact. After thinking about the strange groups of two for a few days without having any decent ideas I sent an e-mail to authors of the new paper and asked them for help. They wrote back last night – which was super cool! – and provided a (possibly) easy way to think about it. I sort of can’t believe that they wrote back, but I’m really excited to spend a bit more time trying to understand their explanation.

Receiving their e-mail got me even more interested in / excited about their paper, so I spent several hours today going through it one more time. The results and conjectures are general enough to apply to the problem of consecutive triples and that led me to try to see if the paper could help me get a better understanding of the data I’d collected. Happily, I was able to understand a bit more of the paper the 2nd time through,

With sort of an “I know enough to be dangerous” understanding I attempted to predict the number of various prime triples in the next set of 1 billion primes (so, last digits of three consecutive primes from the 10 billionth prime number to the 11 billionth prime number). My guesses are in column R and column U of the “Approximations” tab in my google doc. The results should be in tomorrow morning ðŸ™‚

One fun thing about the two sets of guesses is that the sum of the guesses for all of the triples adds up to almost exactly 1 billion! Since I’m looking at 1 billion primes the sum be 1 billion, but I didn’t take that constraint into account (not directly anyway) when I was playing with the numbers.

One other bit of structure I was able to notice in the data after re-reading the paper today was a different set of clustering. The triples with three of the same numbers have the lowest counts, triples with two of the same number in a row have (generally) the next lowest counts, triples with two numbers that are the same, but not in a row have (generally) the next lowest counts, and triples with three different numbers have (generally) the highest counts. *I think* their paper predicts this ordering.

So, a really fun week of playing around with prime numbers. There are still a few things to think about – the e-mail from the paper’s authors, and seeing if there’s any way to improve the predictions – but I’m extremely happy with how this little side project went this week. Haven’t had that much fun learning new math in a long time ðŸ™‚

# Carl Sagan on the 4th dimension

I decided to have the boys watch the video and then talk to start our Family Math project for today. After they watched it we discussed the parts they found interesting.

My younger son thought these two things were interesting:

(1) How the apple appeared as it fell through “Flatland”, and also how the creatures in Flatland interacted with the apple.

(2) The other 2D world that was curved.

My older son found these parts interesting:

(1) The discussion of projecting images from one dimension to a lower dimension, and

(2) The idea that lower dimensional things would have a hard time, but not an impossible time, noticing higher dimensions.

Right at the end we talked about the similarity between Carl Sagan walking around the sphere and (i) Vi Hart’s “Wind and Mr. Ug” video and (ii) a bicycle trip around a Klein Bottle. These are two of my all time favorite videos – sadly, though, the Klein Bottle video is no longer on youtube ðŸ˜¦ Luckily Wind and Mr. Ug remains:

Definitely a fun little project today – always fun to hear the boys’ ideas about complicated math topics ðŸ™‚

# 4 dimensional spheres

Last night my older son asked me what a 4-dimensional sphere looked like. We talked about it a little bit last night and I decided to turn the question into our 400th Family Math project (and, sorry for writing 399 in the videos, I sometimes lose track of where we are!).

We started the project today by talking about squares and cubes. Getting up to the hypercube by “sliding” the each shape in a new dimension is the typical way that we’ve discussed 4 dimensions in the past.

Next we talked about circles in 1, 2, and 3 dimensions. The main idea of this part of the project was to have the kids understand what it looks like when an “n+1” dimensional circle is projected down to “n” dimensions. The boys also had some great ideas about how circles in different dimensions related to each other.

In the final part of the project we used the ideas from the last video to understand 4-dimensional spheres. It was fun to hear my younger son describe what he thought the shape would look like.

I’m happy to have had our 400th project come from a question my son, and also happy that it was a topic that both kids could enjoy.

Can’t believe we’ve made it to 400. #1 feels like it was a looooooong time ago ðŸ™‚

# Patrick Honner’s Pi Day exercise in 4d part 5: The 120- and 600-cells

This is the 5th (of 5) in series of 4 dimensional explorations inspired by Patrick Honner’s Pi Day exercise:

The first four parts in the series are here:

Playing with 4 dimensional shapes using Zometool

Introducing Patrick Honner’s Pi day idea in 4 dimensions

Patrick Honner’s Pi day exercise in 4d: part 3

Patrick Honner’s Pi day exercise in 4d part 4: They Hyperdiamond

Also, since I didn’t want to really dive into the “surface volume” and “hyper volume” calculations, this website was critical for today’s project:

Regular Convex Four-Dimensional Polytopes

So, today we are looking at the 120-cell and the 600-cell. We have studied the 120-cell previously:

A Stellated 120-cell made from our Zometool Set

We’ve also played around with the game Hypernom and experienced the 120-cell first hand!

Using Hypernom to get kids talking about math

However, since we do not currently have a model of the 120-cell anywhere around the house, before starting the project tonight we took a look at two of Henry Segerman’s movies:

(1) Half of a 120-cell

and

(2) Toroidal Half 120-cell

With Segerman’s videos as background, we now calculated what “$\pi$” would be for the 120-cell:

We approached the 600-cell the same way – starting with a video from Henry Segerman:

After that quick introduction we discussed the shape and calculated the value of “$\pi$” for it. Turned out that it was the most 4-d spherical of the shapes that we looked at. That was a fun fact, and thinking about that fact caused my son to ask what a 4-d sphere actually looked like!

Well, I couldn’t end this week-long project with the question about the 4-dimensional sphere hanging in the air. We talked about the shape for about 5 min and then took the dog for a walk ðŸ™‚

As we walked up the street at the end of the walk my son turned to me and said:

“Wait a minute, all of the spheres would have to differ by infinitesimal amounts . . . . but, oh, there are infinitely many of them so I guess that’s ok.”

The project couldn’t have ended on a better note! Thanks to Patrick Honner for the great Pi day exercise which inspired this project. Thanks also to Henry Segerman for his videos about the 120- and 600- cells. I hope to own a few more of his 3d prints soon!

# Weird clustering with last digits of 3 consecutive primes

[note: as I count a few more of the last digit triples, the results are updated on this google-doc spreadsheet:

As of the morning of March 25th I’m up to the first 6 billion primes]

I found the latest news about patterns in last digits of consecutive primes to be really interesting. Here’s Evelyn Lamb’s piece about the new paper:

Peculiar Pattern Found in “Random” Prime Numbers

Although the main results in the paper are way over my head, I thought it would be fun to try to understand the results a bit more. I decided to look at the last digits (in base 10) of triples of consecutive primes. Mathematica makes this task pretty simple since there is a function Prime[i] which tells you the $i^{th}$ prime number.

So, I meant to study the patterns in the fist 10 billion primes, but when that program finished running I accidentally deleted the data – oops. Now I’m running 1 billion at a time. The results so far are in the table below. The first column is the pattern of 3 digits, the second column is how many times that pattern occurs in the first 100 million primes, the 3rd column is how often that pattern appears in the first billion primes, and the last column is how often it appears in the 2nd billion primes. The clustering in the patterns is strange. For example, why are the numbers for the patter (3,7,7) so similar to the ones for (3,3,7)? I’ll update this as more results pop out of the computer.

First First 1,000,000,000 to
100,000,000 1,000,000,000 2,000,000,000

(7,7,7) 735,435 8,006,387 8,351,773
(3,3,3) 737,172 8,013,553 8,356,530
(9,9,9) 752,906 8,139,168 8,471,313
(1,1,1) 752,991 8,143,311 8,472,066
(1,9,9) 1,057,410 10,978,711 11,198,641
(1,1,9) 1,057,849 10,981,805 11,199,329
(7,7,1) 1,089,299 11,397,105 11,659,414
(9,3,3) 1,090,380 11,398,336 11,661,478
(9,7,7) 1,120,365 11,634,379 11,854,429
(3,3,1) 1,121,750 11,639,504 11,854,065
(9,9,3) 1,123,151 11,654,003 11,880,474
(7,1,1) 1,123,346 11,652,871 11,877,020
(3,1,1) 1,152,478 11,848,595 12,031,756
(9,9,7) 1,153,949 11,854,029 12,028,070
(7,3,3) 1,190,879 12,315,608 12,517,157
(7,7,3) 1,191,047 12,310,589 12,518,011
(3,7,7) 1,253,361 12,896,307 13,090,240
(3,3,7) 1,253,949 12,898,170 13,087,114
(1,9,7) 1,304,421 13,561,347 13,799,747
(3,1,9) 1,305,280 13,556,507 13,800,610
(3,3,9) 1,329,690 13,563,836 13,703,504
(1,7,7) 1,330,194 13,565,657 13,699,576
(3,9,9) 1,337,398 13,624,284 13,756,109
(1,1,7) 1,338,238 13,626,051 13,752,462
(7,7,9) 1,423,574 14,388,649 14,466,820
(1,3,3) 1,424,130 14,387,566 14,466,048
(1,1,3) 1,473,963 14,743,098 14,762,437
(1,9,3) 1,474,440 14,895,602 14,973,534
(7,1,9) 1,475,115 14,902,832 14,973,667
(7,9,9) 1,475,202 14,749,205 14,762,144
(9,9,1) 1,592,910 15,844,168 15,808,349
(9,1,1) 1,594,226 15,849,488 15,805,452
(9,1,9) 1,604,100 16,251,163 16,325,968
(1,3,1) 1,605,399 16,249,250 16,330,014
(1,9,1) 1,606,073 16,256,647 16,327,652
(9,7,9) 1,606,645 16,251,438 16,329,446
(9,3,1) 1,623,274 16,186,003 16,171,463
(9,7,1) 1,624,894 16,197,827 16,178,227
(3,1,3) 1,637,164 16,478,634 16,516,505
(7,9,7) 1,638,318 16,483,693 16,519,549
(7,3,1) 1,660,559 16,560,034 16,531,550
(9,7,3) 1,660,835 16,560,577 16,533,455
(3,7,3) 1,817,402 17,915,772 17,771,206
(7,3,7) 1,818,700 17,918,059 17,766,848
(3,9,3) 1,819,555 17,915,593 17,758,829
(7,1,7) 1,821,161 17,911,268 17,768,419
(9,3,9) 1,827,515 17,942,425 17,766,508
(1,7,1) 1,829,358 17,936,238 17,769,169
(3,7,1) 1,830,430 18,059,352 17,921,956
(9,3,7) 1,831,771 18,057,340 17,926,542
(3,9,7) 1,916,051 18,745,152 18,548,191
(3,1,7) 1,916,059 18,751,055 18,538,221
(7,1,3) 1,954,359 19,123,551 18,909,660
(7,9,3) 1,955,794 19,118,907 18,913,154
(7,3,9) 2,085,057 20,272,160 19,973,655
(1,7,3) 2,085,911 20,278,923 19,966,538
(1,3,7) 2,139,275 20,808,168 20,500,739
(3,7,9) 2,142,502 20,810,306 20,497,841
(1,7,9) 2,259,149 21,792,998 21,392,025
(1,3,9) 2,260,634 21,792,207 21,386,368
(7,9,1) 2,362,556 22,891,586 22,491,285
(9,1,3) 2,363,951 22,891,907 22,494,566
(9,1,7) 2,429,154 23,285,442 22,768,206
(3,9,1) 2,429,892 23,285,599 22,766,906