An attempt to explain Graham’s number to kids

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

http://blogs.scientificamerican.com/roots-of-unity/2014/04/01/grahams-number-is-too-big/

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:

Consider the sequence a, a^a, a^(a^a), etc., where x[1]=a and x[n+1]=a^(x[n]), n>1. For which a>0 does this sequence converge?

— Steven Strogatz (@stevenstrogatz) June 30, 2013

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.