If you touch a gold ball, you touch its surface and you touch gold. It seems reasonable to conclude that the surface is made of gold. But University of Exeter computer scientist Antony Galton points out that the surface is two-dimensional; it can’t contain any quantity of gold.

What then is it? We can’t say it’s the outermost layer of gold atoms, for that’s a film with two surfaces. And we can’t say it’s an abstract boundary with no physical existence, for we can see it and touch it. So what is it?

J.L. Austin asked, “Where and what exactly is the surface of a cat?”

[Lewis Carroll] told me of a simple, too simple, rule by which, he thought, one could be almost sure of making something at a horse-race. He had on various occasions noted down the fractions which represented the supposed chances of the competing horses, and had observed that the sum of these chances amounted to more than unity. Hence he inferred that, even in the case of such hard-headed men as the backers, the wish is often father to the thought; so that they are apt to overrate the chances of their favourites. His plan, therefore, was to bet against all the horses, keeping his own stake the same in each case. He did not pretend to know much about horse-racing, and I probably know even less; but I understand that it would be impossible to adjust the hedging with sufficient exactitude — in fact, to get bets of the right amount taken by the backers.

— Lionel Arthur Tollemache, Old and Odd Memories, 1908

The number 360 is centered across the 360th decimal place of π:

6998970 = 3⁶ + 1⁹ + 4⁹ + 1⁸ + 5⁹ + 9⁷ + 2⁰

6998971 = 3⁶ + 1⁹ + 4⁹ + 1⁸ + 5⁹ + 9⁷ + 2¹

(Thanks, Pablo.)

The 22nd, 7th, 355th, 113th, and 52163rd digits of π (counting from the 3) are 2s.

The 16604th digit, alas, is a 1 — but it’s flanked by 2s.

At the end of your visit to an elderly, infirm relative who lives alone, the relative says, ‘I’m sorry but my arthritis won’t let me get up from this chair today. You’ll have to show yourself out.’ How can you show yourself out of someone’s house? If you know the way out, you can act as a guide to someone else. But how can you act as your own guide?

— T.S. Champlin, Reflexive Paradoxes, 1988

If three hula hoops cover a common point, then a fourth hoop of the same size will cover their remaining intersections.

Mathematician G.H. Hardy had an ongoing feud with God. Once, after spending a summer vacation in Denmark with Harald Bohr, he found he’d have to take a small boat across the tempestuous North Sea to return to England. Before boarding, he sent Bohr a postcard that said “I have proved the Riemann hypothesis. — G.H. Hardy.”

When Bohr excitedly asked about this later, “Oh, that!” Hardy said. “That was just insurance. God would never let me drown if it meant I’d get undue credit.”

From Samuel Isaac Jones, Mathematical Wrinkles (1929), a magic square with a twist:

“It will be observed that this square when turned upside down is still magic.”

Discovered by J.A.H. Hunter.

A line that bisects the right angle in a right triangle also bisects a square erected on the hypotenuse.

Proof: