# Lessons from teaching gifted elementary school students (Part 2)

Every so often, I’ll informally teach a class of gifted elementary-school students. I greatly enjoy interacting with them, and I especially enjoy the questions they pose. Often these children pose questions that no one else will think about, and answering these questions requires a surprisingly depth of mathematical knowledge.

Here’s a question I once received:

Suppose

$A \times A = B$

$B \times B = C$

$C \times C = D$

If the pattern goes on, and if $A = 2$, what is $Z$?

I leave a thought bubble in case you’d like to think this. One way of answering this question appears after the bubble.

Let’s calculate the first few terms to try to find a pattern:

$B = 2 \times 2 = 2^2$

$C = 2^2 \times 2^2 = 2^4$

$D = 2^4 \times 2^4 = 2^8$

etc.

Written another way,

$A = 2^1 = 2^{2^0}$

$B = 2^{2^1}$

$C = 2^{2^2}$

$D = 2^{2^3}$

Continuing the pattern, we see that $Z = 2^{2^{25}}$, or $Z = 2^{33,554,432}$.

If you try plugging that number into your calculator, you’ll probably get an error. Fortuniately, we can use logarithms to approximate the answer. Since $2 = 10^{\log_{10} 2}$, we have

$Z = \left( 10^{\log_{10} 2} \right)^{33,554,432} = 10^{33,554,432 \log_{10} 2}$

Plugging into a calculator, we find that

$Z \approx 10^{10,100,890.5195}$

$\approx 10^{0.5195} 10^{10,100,890}$

$\approx 3.307 \times 10^{10,100,890}$

When this actually happened to me, it took me about 10 seconds to answer — without a calculator — “I’m not sure, but I do know that the answer has about 10 million digits.” Naturally, my class was amazed. How did I do this so quickly? I saw that the answer was going to be $Z = 2^{2^{25}}$, so I used the approximation $2^{10} \approx 1000$ to estimate

$2^{25} = 2^5 \times 2^{10} \times 2^{10} \approx 32 \times 1000 \times 1000 = 32,000,000$

Next, I had memorized the fact that that $\log_{10} 2 \approx 0.301 \approx 1/3$. So I multiplied $32,000,000$ by $1/3$ to get approximately 10 million. As it turned out, this approximation was a lot more accurate than I had any right to expect.

1. #### Tanya

/  September 8, 2016

John, can you fix the typo in the equation before the last paragraph? It says, or implies, (2^2)5 because the 5 didn’t get superscripted, when you really mean 2^(25). Threw me off for a bit! Thanks for a great post!