Lessons from teaching gifted elementary students (Part 6b)

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 surprising depth of mathematical knowledge.

Here’s a question I once received:

255/256 to what power is equal to 1/2? And please don’t use a calculator.

Here’s how I answered this question without using a calculator… in fact, I answered it without writing anything down at all. I thought of the question as $\displaystyle \left( 1 - \epsilon \right)^x = \displaystyle \frac{1}{2}$. $\displaystyle x \ln (1 - \epsilon) = \ln \displaystyle \frac{1}{2}$ $\displaystyle x \ln (1 - \epsilon) = -\ln 2$

I was fortunate that my class chose 1/2, as I had memorized (from reading and re-reading Surely You’re Joking, Mr. Feynman! when I was young) that $\ln 2 \approx 0.693$. Therefore, we have $x \ln (1 - \epsilon) \approx -0.693$.

Next, I used the Taylor series expansion $\ln(1+t) = t - \displaystyle \frac{t^2}{2} + \frac{t^3}{3} \dots$

to reduce this to $-x \epsilon \approx -0.693$,

or $x \approx \displaystyle \frac{0.693}{\epsilon}$.

For my students’ problem, I had $\epsilon = \frac{1}{256}$, and so $x \approx 256(0.693)$.

So all I had left was the small matter of multiplying these two numbers. I thought of this as $x \approx 256(0.7 - 0.007)$.

Multiplying $256$ and $7$ in my head took a minute or two: $256 \times 7 = 250 \times 7 + 6 \times 7$ $= 250 \times (8-1) + 42$ $= 250 \times 8 - 250 + 42$ $= 2000 - 250 + 42$ $= 1750 + 42$ $= 1792$.

Therefore, $256 \times 0.7 = 179.2$ and $256 \times 0.007 = 1.792 \approx 1.8$. Therefore, I had the answer of $x \approx 179.2 - 1.8 = 177.4 \approx 177$.

So, after a couple minutes’ thought, I gave the answer of 177. I knew this would be close, but I had no idea it would be so close to the right answer, as $x = \displaystyle \frac\ln \frac{1}{2} }\ln \frac{255}{256}} \approx 177.0988786\dot$