Revisiting the last digits of Graham’s number

Several years ago we did a bunch of projects on Graham’s number.

An attempt to explain Graham’s number to kids, and

The last 4 digits of Graham’s number

These projects were inspired by this fantastic Evelyn Lamb article:

Graham’s numbers is too big for me to tell you how bit it is

Today I thought it would be fun to revisit the calculation of a few of the last digits of Graham’s number.

So, with no review, I asked my older son what he remembered about Graham’s number and then we talked about the surprising fact that you could calculate the last few digits even though you really couldn’t say much else about the number:

Next I asked my son about how he would approach calculating the last digit. He gravitated to the right idea -> modular arithmetic. The ideas were a little confusing to him, but I let him work mostly on his own.

We didn’t get to the end in this video, but you can see how the ideas start coming together.

In the last video he had made some progress on finding the last digit, but one piece of the argument kept giving him trouble.

BUT, he did have a correct argument – it just took him a minute to realize that he was on the right track.

Again, this is a nice example of how a kid works through some advanced mathematical ideas.

Next we went to the computer to begin looking at the last two digits of Graham’s number. The last two digits of powers of 3 repeat every 20 powers, so it was easier to use Mathematica to find the cycle than it was to do it by hand.

Here I just explain the short little computer program I wrote to him.

Finally, we tried to see if we could use the idea that the powers of 3 repeat their last two digits every 20 steps to see if we could find the last 2 digits of Graham’s number.

As we started down the path here, I didn’t know if we’d find those last two digits. But we did! It was a nice way to end the project.

Graham’s number and Skewes’ Number

Saw another great video from Numberphile today:

One thing that got my attention in the video was the comparison to Graham’s Number. Some of their previous videos about Graham’s number had inspired a few projects with the boys. For instance:

An attempt to explain Graham’s number to kids, and

The last 4 digits of Graham’s number

The fun (and sometimes frustrating) thing about talking about Graham’s number is that it is so large that it is nearly impossible to describe. In fact, the title of this Evelyn Lame piece on Graham’s number sums it up perfectly: Graham’s numbers is too big for me to tell you how bit it is!

Skewes’number doesn’t have this little problem – you can even write the number 🙂

Skewes’ number = 10^{10^{10^{34}}}

After seeing the video I thought these two questions might be interesting to kids learning about powers (or logarithms):

Question 1: Which is larger, Skewes’ number or this tower of 5 powers of 3 -> 3^{3^{3^{3^3}}}

Question 2: Which is larger, Skewes’ number or this tower of 6 powers of 3 -> 3^{3^{3^{3^{3^3}}}}

The reason I thought it would be interesting to compare to power of 3 is because of how Graham’s number is constructed.

More questions like these can be found in Richard Evan Schwartz’s book “Really Big Numbers.” A few projects that we did from that book are here (with the 3rd one being similar to the two questions above):

A few porjects for kids from Ricahrd Evan Schwartz’s “Really Big Numbers”

Oh, and by luck my older son came home from school as I was finishing this post, so I tried out question 1 with him:

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.

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:

Michael Pershan’s Exponential post (part 2 /3 )

Despite an extra day to try to think things through I remain confused about my own approach to teaching / talking about exponential functions.   I’m actually struggling to even understand what the struggle is.  After all, running across this fun little connection between \pi and e as a college freshman is what convinced me to major in math:


As I mentioned in the first post in this series, my approach to teaching exponentials has not been nearly as formal as my approach to teaching arithmetic.  We began by talking about powers where I introduced exponents as essentially a time saver.  Probably like just about everyone who has ever talked about exponents,  one of the early conversations was about the zeroth power and negative powers.  That talk about exponents with my younger son remains one of my favorite math  conversations he and I have ever had:

Feels as though you are almost forced to introduce integer exponents early one if you want to talk about place value or different bases or other similar topics, but the path to exponentials from here just isn’t that satisfying to me.  You’ll have to introduce fractional powers and then define non-rational powers by some sort of limit process (at least if you want to approach things formally).  If you are going to bed at night fearing a Grant Wiggins-like “conceptual understanding of exponential functions” exam, you probably won’t like this path at all.

There are fun topics, though, so I’m not suggesting that integer or rational exponents are a waste of time.  Two of my favorite topics here have been finding a formula for the Fibonacci numbers while we were studying quadratic equations:

and talking about Graham’s number, which is one of the most fun math activities that we’ve ever done, and probably as much fun and excitement with integer powers as you are ever going to have.  It took me a week to figure out how to put this one together (and I stopped after a week, because there was no way I was going to figure out how to do it anyway!)

With all this background I’m kind of surprised that I can’t really think of a nice, easy transition from exponents to exponential functions.   As I was riding home last night I tried to keep a look out for anything I saw that I naturally thought of as being associated with an exponential function – something / anything that kids might see occasionally in their life.  I couldn’t find a single thing which made me a little sad.  Maybe I’m just not being creative enough.

In yesterday’s post I mentioned a few things from finance and probability where exponential functions appear pretty naturally, but those are well outside the realm of things that kids see or worry about.   What I didn’t mention was a different field where exponentials play an incredibly important role – physics.  Representing waves in the form e^{i * \omega t} is pretty convenient, to say the least, but again is way outside of what might be reasonable examples for kids.

So, I’m lost.  A non-formal approach starting with integer exponents does let you talk about some really interesting problems, but doesn’t really seem to set you up too well to move to exponential functions in general.  I’m a little frustrated at my inability to find  any great (or even reasonable) natural exponential examples to share with kids.  And, to top it all off,  starting with a formal approach like defining e^x as the limit as n approaches infinity of (1 + x/n)^n just seems stupid.

The perplexing thing is that both e^x and ln(x) play such incredibly important roles in math.  You’d think that there would at least be a few easy examples you could talk through with kids to introduce / motivate these ideas.  I mentioned in yesterday’s post that I gave one formal approach a try.    That was in response to a question I saw about logs on twitter when I happened to be talking about them with my older son.  That question motivated me to throw together a fun overview of some of the areas in math where logs are part of important results.  I wasn’t expecting my son to get much of anything out of it other than to see some really amazing math involving prime numbers.  That blog post is here:

Tomorrow, or over the weekend, I’ll try to come out of the fog and write about what I’d like my kids to learn about logs and exponentials.  Hopefully I’ll have it all figured out by then.  Ha!

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.