A simple and surprisingly effective illusion by Lisbon anamorphosis specialists Sonhos com Dimensão.

More on their YouTube channel.

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.)

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.

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.)

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

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.”)

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.)