# Impossible? Surprising Solutions to Counterintuitive Conundrums

## Excerpt

We've quips and quibbles heard in flocks,
But none to beat this paradox!

The Pirates of Penzance

We begin with a classic puzzle.

Imagine a rope just long enough to wrap tightly around the
equator of a perfectly spherical Earth. Now imagine that the
length of the rope is increased by 1 metre and again wrapped
around the Earth, supported in a regular way so that it forms
an annulus. What is the size of the gap formed between the
Earth and the extended rope?

The vast Earth, the tiny 1 metre—surely the rope will be in effect as tight after its extension as before it? Yet, let us perform a small calculation: in standard notation, if C = 2πr for the Earth and the original length of the rope and C + 1 = 2πR for the rope lengthened, we require the size of

The Cs have cancelled, leaving 1/2π = 0.159 … m ≈ 16 cm as the gap. The Earth could have been replaced by any other planet, an orange or a ping-pong ball and the result would have been the same: a fact which is hard to accept even though the reasoning is irrefutable.

In Nonplussed! we gathered together a variety of counterintuitive situations and in this sequel we chronicle eighteen more mathematical phenomena which, if allowed to do so, confound one's reason. The criterion for inclusion has been, as it was in Nonplussed!, that, in the recent or distant past, the matter . . .

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