The Pizza Theorem

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Images: Wikimedia Commons

If you’re sharing a pizza with another person, there’s no need to cut it into precisely equal slices. Make four cuts at equal angles through an arbitrary point and take alternate slices, and you’ll both get the same amount of pizza.

Larry Carter and Stan Wagon came up with this “proof without words”: Each piece in an odd-numbered sector corresponds to a congruent piece in an even-numbered sector, and vice versa.

Also: If a pizza has thickness a and radius z, then its volume is pi z z a.

(Larry Carter and Stan Wagon, “Proof Without Words: Fair Allocation of a Pizza,” Mathematics Magazine 67:4 [October 1994], 267-267.)

Math Notes

In a 1752 letter to Euler, Christian Goldbach suggested that every odd integer is the sum of a prime and twice a square. (At the time, 1 was considered a prime number.)

Only two exceptions, 5777 and 5993, have ever been found.

Illustration

A pleasing observation by W.V. Quine from 1988:

Fermat’s Last Theorem can be vividly stated in terms of sorting objects into a row of bins, some of which are red, some blue, and the rest unpainted. The theorem amounts to saying that when there are more than two objects, the following statement is never true:

Statement. The number of ways of sorting them that shun both colors is equal to the number of ways that shun neither.

He explains this, very concisely, here.

(W.V. Quine, “Fermat’s Last Theorem in Combinatorial Form,” American Mathematical Monthly 95:7 [September 1988], 636.)

A New Perspective

I find myself more than half convinced by the oddly repellent hypothesis that the peculiarity of the time dimension is not … primitive but is wholly a resultant of those differences in the mere de facto run and order of the world’s filling. It is then conceivable, though doubtless physically impossible, that one four-dimensional area of the time part of the manifold be slewed around at right angles to the rest, so that the time order of that area, as composed by its interior lines of strain and structure, run parallel with a spatial order in its environment. It is conceivable, indeed, that a single whole human life should lie thwartwise of the manifold, with its belly plump in time, its birth at the east and its death in the west, and its conscious stream running alongside somebody’s garden path.

— Donald C. Williams, “The Myth of Passage,” Journal of Philosophy 48:15 (1951), 457-472

A Near Miss

For a moment in the 1998 Simpsons episode “The Wizard of Evergreen Terrace,” it appears that Homer has found a solution to Fermat’s last theorem:

398712 + 436512 = 447212

If you check this on a calculator with a 10-digit display, it seems to work: Raise 3987 and 4365 each to the 12th power, take the 12th root of the sum, and you get 4472.

But that’s the fault of the display. The actual value for the third term is closer to 4472.000000007057617187512.

Simpsons writer David S. Cohen, who had studied physics at Harvard and contrived the ruse, told Simon Singh he was pleased at the consternation it caused online. “I feel great about it. It’s very easy working in television to not feel good about what you do on the grounds that you’re causing the collapse of society. So, when we get the opportunity to raise the level of discussion — particularly to glorify mathematics — it cancels out those days when I’ve been writing those bodily function jokes.”

(From Simon Singh, The Simpsons and Their Mathematical Secrets, 2013.)

Vanishing Act

There is often peculiar humour about self-frustration. Consider, for example, a train of events which started outside the old Clarendon Laboratory, Oxford. I came across a dirty beaker full of water just when I happened to have a pistol in my hand. Almost without thinking I fired, and was surprised at the spectacular way in which the beaker disappeared. I had, of course, fired at beakers before; but they had merely broken, and not shattered into small fragments. Following Rutherford’s precept I repeated the experiment and obtained the same result: it was the presence of the water which caused the difference in behavior. Years later, after the War, I found myself having to lecture to a large elementary class at Aberdeen, teaching hydrostatics ab initio. Right at the beginning came the definitions — a gas having little resistance to change of volume but a liquid having great resistance. I thought that I would drive the definitions home by repeating for the class my experiments with the pistol, for one can look at them from the point of view of the beaker, thus suddenly challenged to accommodate not only the liquid that it held before the bullet entered it, but also the bullet. It cannot accommodate the extra volume with the speed demanded, and so it shatters.

— R.V. Jones, “Impotence and Achievement in Physics and Technology,” Nature 207:4993 (1965), 120-125

(When the Royal Engineers tried to use this trick to demolish a tall chimney, filling its base with 6 feet of water and firing an explosive charge into the water, “it succeeded so well that it failed completely”: The incompressible water flung the surrounding ring of bricks outward, leaving a foreshortened chimney suspended above in midair. This dropped down neatly onto the old foundation, upright and intact, “presenting the Sappers with an exquisite problem.”)

A Special Year

Reader Alan Jackson points out that 2021 is the product of consecutive primes: 2021 = 43 × 47.

It’s the first time this has happened since 1763 (= 41 × 43), and it won’t happen again until 2491 (= 47 × 53).

A quick way to find the factors is to note that 2021 is the difference of two squares, 452 – 22, so 2021 = (45 – 2) × (45 + 2).

(Thanks, Alan.)

Small World

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In Antoine de Saint-Exupéry’s 1943 novella The Little Prince, the narrator encounters “a most extraordinary small person” whose planet is “scarcely any larger than a house.”

This led University of Ljubljana physicist Janez Strnad to consider the implications. If the radius of the prince’s planet were 64 meters and it had Earth’s density, then the weight of a prince with a mass of 30 kg would amount to 0.003 newtons, corresponding on Earth to the weight of a mass of 0.3 g. (If the planet had the density of an asteroid, his weight would be lower still.)

The planet cannot have an atmosphere, because the mean velocity of gas molecules is greater than the escape velocity.

If the prince moved faster than 80 millimeters per second he’d be sent into orbit around the planet; if faster than 11 centimeters per second he’d leave it altogether.

“He could overcome the limitations concerning his velocity by either binding himself with a rope to his planet or building a spherical shell around it,” Strnad concluded. “The human body adapts to weightlessness and astronauts have to perform special gymnastic exercises not to suffer on returning to the Earth. For the little prince, coming to Earth would be a serious adventure, were he not a fictitious character.”

(Janez Strnad, “The Planet of the Little Prince,” Physics Education 23:4 [1988], 224.)

Prime Circles

Is it always possible to arrange the numbers from 1 to 2n in a circle so that each adjacent pair sums to a prime?

  4  1
7      2
6      3
  5  8

As of 2016, solutions have been found for every case up to n = 106 — but no one has yet proven that it’s always possible.