What are some counterintuitive results that are really simple to explain?

Question

I am aware of other questions asking about counterintuitive mathematical results, all of which have many great answers. However, most examples rely on subtle concepts of probability (e.g., Monty Hall, birthday paradox) or infinity (e.g., Gabriel's Horn, Banach-Tarski).

Recently I came across a beautiful counterintuitive result that relies only on simple and finite Euclidean geometry. It should surprise anyone with a basic sense of scale, and yet it is easily explainable to someone with a middle-school understanding of geometry. Moreover, there's not much room for argument ("how do you define probability?", or "what is infinity, anyway?").

Back in Ancient Greece, Zeus commissions a blacksmith to forge him a ring that will go around the Earth. The blacksmith obliges, but he makes a mistake and forges the ring one meter too large in circumference. Zeus goes ahead and places the ring around the Earth, pressing it up against the South pole. How large is the gap left at the North pole, and what sort of animal can go through?

The counterintuitiveness comes from the fact that an error of one meter in circumference is negligible compared with the Earth's circumference. In fact, you might say the blacksmith did a great job in keeping the relative error so small. If you think like me, your intuition will say that the gap must also be negligible. However, the gap is actually around 32 cm: enough for a human to crawl through! Granted, you could argue it is still negligible compared with the Earth's diameter, but it is still much bigger than I would have guessed without calculating.

Perhaps even more surprisingly, the gap does not even depend at all on the circumference of the Earth or the ring - just on their difference. You could put a ring one meter too large in circumference around the sun or your favorite ginormous star, and the gap would still be 32 cm.

Do you know of any similar, counterintuitive facts that should surprise and be explainable to more or less anyone?

Answer 1

What about the book stacking problem: how far can a stack of $n$ books protrude over the edge of a table without the stack falling over? It turns out that you get a sequence which increases without bound.

Answer 2

The series $\sum_{n=1}^{\infty}\frac{1}{n}$ does not converge but $\sum_{n=1}^{\infty}\frac{1}{2^n}$ does.

It's easy to explain because in the latter series every term is only "half the remaining distance" to $1$, so there are not enough terms to reach $1$ itself. In the former, you are always above $M/2$ if you sum the first $2^M$ terms or more since:

$1 > \frac{1}{2}$

$\frac{1}{2} = \frac{1}{2}$

$\frac{1}{3} + \frac{1}{4} > \frac{1}{2}$

$\frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} > \frac{1}{2}$

$\frac{1}{9} + \frac{1}{10} + \frac{1}{11} + \frac{1}{12} + \frac{1}{13} + \frac{1}{14} + \frac{1}{15} + \frac{1}{16} > \frac{1}{2}$

And so on...

Answer 3

If $m$ and $n$ are positive integers chosen so that $(m-1)^2<n<(m+1)^2$ but $n\not=m^2$, then successive powers of $m+\sqrt{n}$ come successively closer to positive integers themselves. You prove by induction on $k$ that

$(m+\sqrt{n})^k+(m-\sqrt{n})^k$

is exactly a positive integer, and then the second term dies off when the absolute value of its base is smaller than unity.

Answer 4

A small trick for kids:

1) Think of a number between 1 and 10.
2) Multiply it by 2.
3) Add 8.
4) Divide it by 2.
5) Substract the number you thought.
6) The result is 4.

Why can I guess the result?

Let N be your number, and A the number I order to add. Then:

$(2N+A)/2 - N \;=\; A/2$