Dan Pearcy’s Twitter feed highlights a pretty link between Pascal’s triangle and the tangent angle formulae:

The pattern is described more fully here.

In 1988, Florida International University mathematician T.I. Ramsamujh offered a proof that all positive integers are equal. “The proof is of course fallacious but the error is so nicely hidden that the task of locating it becomes an interesting exercise.”

Let p(n) be the proposition, ‘If the maximum of two positive integers is n then the integers are equal.’ We will first show that p(n) is true for each positive integer. Observe that p(1) is true, because if the maximum of two positive integers is 1 then both integers must be 1, and so they are equal. Now assume that p(n) is true and let u and v be positive integers with maximum n + 1. Then the maximum of u – 1 and v – 1 is n. Since p(n) is true it follows that u – 1 = v – 1. Thus u = v and so p(n + 1) is true. Hence p(n) implies p(n+ 1) for each positive integer n. By the principle of mathematical induction it now follows that p(n) is true for each positive integer n.

Now let x and y be any two positive integers. Take n to be the maximum of x and y. Since p(n) is true it follows that x = y.

“We have thus shown that any two positive integers are equal. Where is the error?”

(T.I. Ramsamujh, “72.14 A Paradox: (1) All Positive Integers Are Equal,” Mathematical Gazette 72:460 [June 1988], 113.)

(The error is explained here and here.)

When a limb is paralyzed and then amputated, the patient may perceive a “phantom limb” in its place that is itself paralyzed — the brain has “learned” that the limb is paralyzed and has not received any feedback to the contrary.

University of California neuroscientist V.S. Ramachandran found a simple solution: The patient holds the intact limb next to a mirror, looks at the reflected image, and makes symmetric movements with both the good and the phantom limb. In the reflected image, the brain is now able to “see” the phantom limb moving. The impression of paralysis lifts, and the patient can now move the phantom limb out of painful positions.

A 2018 review called the technique “a valid, simple, and inexpensive treatment for [phantom-limb pain].”

Discovered by Archimedes: The volume of a cone, sphere, and cylinder of the same height and radius fall in the ratio 1:2:3.

A cone plus a sphere is a cylinder.

The surface of a standard soccer ball is covered with 20 hexagons and 12 pentagons. Interestingly, while we might vary the number of hexagons, the number of pentagons must always be 12.

That’s because the Euler characteristic of a sphere is 2, so V – E + F = 2, where V is the number of vertices, or corners, E is the number of edges, and F is the number of faces. If P is the number of pentagons and H is the number of hexagons, then the total number of faces is F = P + H; the total number of vertices is V = (5P + 6H) / 3 (we divide by 3 because three faces meet at each vertex); and the total number of edges is E = (5P + 6H) / 2 (dividing by 2 because two faces meet at each edge). Putting those together gives

and since the Euler characteristic is 2, this means P must always be 12.

If you arrange the first 12 primes into exponentiated summands, you get a prime number:

2³ + 5⁷ + 11¹³ + 17¹⁹ + 23²⁹ + 31³⁷ = 15148954872646850196557152427604893685308877022260348791 is prime.

(From Fermat’s Library.)

There’s only one normal magic hexagon — only one hexagonal arrangement of cells like the one shown here that can be filled with the consecutive integers 1 to n such that every row, in all three directions, sums to the same total. (This excludes the trivial example of a single cell on its own, as well as rotations and reflections of the hexagon shown here.)

Amazingly, it appears that this lonely example may not have been discovered until 1888! In a 1988 letter to the Mathematical Gazette, Martin Gardner reported that the magic hexagon was given as a problem in the German magazine Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht (Volume 19, page 429) in 1888. The proposer was identified only as von Haselberg, of Stralsund; his solution was published in the next volume.

Gardner writes, “The structure could easily have been discovered by mathematicians in ancient times, but as of now, this is the earliest known publication.”

(Martin Gardner, “The History of the Magic Hexagon,” Mathematical Gazette 72:460 [June 1988], 133.)