# Taking the Fun Out of Puzzles & Games

## Wednesday, November 2, 2016

### How big is 10^100^100?

Elliot Temple posed the following math puzzle: How many square roots does it take to get 10100100 down to a small number? (And what number is it?)
Note that abc = a(bc).
Answer: 665 square roots takes you down from 10100100 to around 4.5.
Proof:
Here, lg means the base 10 logarithm, and dot (·) means multiplication. I will use the following facts:
• (ab)c = abc
• xaxb = xa+b
• x = 10lg x
First, note that 10100100= 1010200 . We next prove the following lemma:
abc = 1010c lg b + lg lg a
Example: 223 = 28 = 256 = 10103 lg 2 + lg lg 2
Proof:

 abc = (10lg a)(10lg b)c = (10lg a)10c lg b = 10lg a · 10c lg b = 1010lg lg a · 10c lg b = 1010c lg b + lg lg a ▢

We can now use the lemma to show that 4.52665 = 1010665 lg 2 + lg lg 4.5. And 665 lg 2 + lg lg 4.5 ≈ 200.▢
You can also check the answer with Robert Munafo's Hypercalc, an online calculator that works with very large integers. To do this, enter 4.5^(2^665) in Hypercalc. It evaluates it to something very close to 10^10^200.