During one of their Brighton rendezvouses, after a round of oysters and a second bottle of champagne, Tom Haley asks Serena Frome the question every mathematician longs for her lover to utter:
I want you to tell me something…something interesting, no, counterintuitive, paradoxical. You owe me a good maths story.
Frome (“rhymes with plume”), a twenty-something blonde blessed with the looks of Scarlett Johansson might be the last person one would expect capable of satisfying Haley’s request. But readers of Booker-winning author Ian McEwan’s latest novel Sweet Tooth, a Cold War-era romance of multiple deception (semi-autobiographical, some have argued), should get use to mistrusting first readings. Even her newest lover Haley, a short story author and scholar of Spenser, doesn’t yet know her full story. When this conversation takes place Haley believes he is dating his liaison to the Cultural Foundation that is paying him to write the next great English novel. McEwan’s heroine is actually an agent for MI5 who is sent to lure Haley into a cultural war against communism (a la Encounter) as part of the project fittingly named Sweet Tooth. Though a fresh recruit, the service has decided that Frome’s beauty and steady diet of two to three paperbacks a week makes her ideal for the job.
What counterintuitive, paradoxical maths problem does Haley come up with? Why, The Monty Hall Problem, of course. You should read the rest of the article I linked to, if only to admire their impeccable choice of authorities.
I do love a good counterintuitive math problem. Tell people that
and just watch the mayhem that ensues! Which is funny, since no one balks at the idea that
But, somehow, accepting the results of adding these equations together is a bridge too far.
Here’s a little teaser for you. Would you rather receive $4000 for your first year of work, with an $800 raise every year, or $2000 for your first six months of work, with a $200 raise every six months? Don’t think too hard! Give me your first, knee-jerk reaction.
The first one sure sounds more appealing, but a quick calculation will show that you will do better with the second.
How about this classic? Imagine that you wrap a belt around the Earth at the equator, so that the belt hugs the earth tightly. You then cut the belt, straighten it out, and then extend its length by one mile. The lengthened belt is then wrapped around the Earth again, so that the distance between the belt and the ground is uniform. The problem is to estimate how high above the ground the belt will be. Will you be able to slip a pencil under it? Your arm? Could you crawl under it? Walk under it? That sort of thing.
Well, let’s see. The radius of the Earth is roughly 4000 miles. Since the circumference of a circle is given by the formula
we see that the initial length of the belt, which we shall call L, is
That’s in miles, incidentally. The stretched belt therefore has a length of
and we need to determine the radius that corresponds to this circumference. Let’s call that new radius R. Then we must solve the equation
From this we conclude that
For approximation purposes, let’s take pi to be three. Then the new radius is one sixth of a mile longer than the old radius, which is close to 900 feet. So, yes, I’d say you could walk under it.
Actually, it gets better. If you look at that calculation carefully, I think you’ll find that the radius of the earth was not actually relevant. Repeat the calculation with some variable, r say, to represent the original radius, and the conclusion will be exactly the same. The new radius will be a sixth of a mile longer than the old radius.
I think that’s cool!