# 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:

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:

# A neat counting problem from the 1995 AJHME

Stumbled on a neat counting problem tonight on an old math contest. Here’s a link to the 1995 AJHME hosted by Art of Problem Solving:

Problem 23 from the 1995 AJHSME

and here’s the problem:

How many four-digit whole numbers are there such that the leftmost digit is odd, the second digit is even, and all four digits are different?

I thought it would interesting to have both kids talk through the problem just to see if they would take different approaches. It turned out the way they approached the problem was similar, and both were left with the same multiplication problem as the last step. They way the multiplied out the final product was drastically different.

here’s my older son:

and my younger son:

Definitely a nice counting problem, and the difference in multiplying strategies was fascinating to me.