Science & Math

The Hypergame Paradox

Call a game finite if it terminates in finitely many moves. Now consider Hypergame, which has two rules:

  1. The first player names a finite game.
  2. The two players play that game.

Is Hypergame a finite game? It seems so: It consists of a single game-naming move, followed by a subgame with a necessarily finite number of moves. But what if the first player names Hypergame itself as the subgame, and the second player names Hypergame as the sub-sub-game, and so on?

In presenting this question to students and colleagues at Union College, mathematician William Zwicker found that many saw the catch and quickly pointed out that it leads to infinite play, thinking that this settles the matter. But the proof that Hypergame is finite seems sound. “I … have to convince them that mathematicians cannot simply abandon a proof once a counter-example has been found, for if the internal flaw in such a proof cannot be identified then the counterexample threatens the entire edifice of mathematical proof.” What is the answer?

(William S. Zwicker, “Playing Games with Games: The Hypergame Paradox,” American Mathematical Monthly 94:6, 507-514)

Pleasure Gardening

The 1957 edition of Exotica: Pictorial Cyclopedia of Indoor Plants included a species called Rumandia cocacoliensis of the family Alcoholiaceae.

The description read “Cuba-libre tall, stemless, succulent, with brown-frosty bloom often with lemon flavor; good in summer, keep cool.”

It was indexed without a page number, and disappeared from subsequent editions.

Field Trip

In the early 1600s, Johannes Kepler wrote a fantasy in which he imagined a journey to the moon:

We congregate in force and seize a man of this sort; all together lifting him from beneath, we carry him aloft. The first getting into motion is very hard on him, for he is twisted and turned just as if, shot from a cannon, we were sailing across mountains and seas. Therefore, he must be put to sleep beforehand, with narcotics and opiates, and he must be arranged, limb by limb, so that the shock will be distributed over the individual members, lest the upper part of his body be carried away from the fundament, or his head be torn from his shoulders. Then comes a new difficulty: terrific cold and difficulty in breathing. The former we counter with our innate power, the latter by means of moistened sponges applied to the nostrils.

Somnium is largely a treatise on lunar astronomy, describing the motions of the planets as observed from the moon. But Kepler also considers the appearance of the moon’s inhabitants, who “wander in hordes over the whole globe in the space of one of their days, some on foot, whereby they far outstrip our camels, some by means of wings, some in boats pursue the fleeing waters, or if a pause of a good many days is necessary, then they creep into caves.” Carl Sagan and Isaac Asimov called it the first work of science fiction.


Full text of “The Unsuccessful Self-Treatment of a Case of ‘Writer’s Block’,” by Dennis Upper, from the Journal of Applied Behavior Analysis, Fall 1974:

“The Skeptic’s Horoscope”

For Capricorn, Aquarius, Pisces, Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, and Sagittarius:

The coming year is likely to present challenges; these trials are when your true character will show. Trusted friends can provide assistance in particularly pressing situations. Make use of the skills you have to compensate for ones you lack. Your reputation in the future depends on your honesty and integrity this year. Monetary investments will prove risky; inform yourself as much as possible. On the positive side, your chances of winning the lottery have never been greater!

(By Tim Harrod.)

Business as Usual

On July 1, 1858, the Linnean Society of London heard a joint presentation by Charles Darwin and Alfred Russel Wallace on the theory of evolution by natural selection.

In his annual report the following May, society president Thomas Bell wrote, “The year which has passed has not, indeed, been marked by any of those striking discoveries which at once revolutionize, so to speak, the department of science on which they bear.”

In Brief

In 1962, botanist Reid Moran published a note in the journal Madroño recounting his collection of a bush rue on a mountaintop in Baja California.

The note’s title was “Cneoridium dumosum (Nuttall) Hooker f. Collected March 26, 1960, at an Elevation of About 1450 Meters on Cerro Quemazón, 15 Miles South of Bahía de Los Angeles, Baja California, México, Apparently for a Southeastward Range Extension of Some 140 Miles.”

The text read, “I got it there then.”

This was followed by a 28-line acknowledgment section in which Moran thanked the person who had reviewed the text, his college professors, and the person who had mailed the manuscript.


  • SWARTHMORE is an anagram of EARTHWORMS.
  • The sum of the reciprocals of the divisors of any perfect number is 2.
  • We recite at a play and play at a recital.
  • Is sawhorse the past tense of seahorse?
  • “Things ’twas hard to bear ’tis pleasant to recall.” — Seneca

In Book II, Chapter 9, of H.G. Wells’ novel The War of the Worlds, a sentence begins “For a time I stood regarding …” These words contain 3, 1, 4, 1, 5, and 9 letters.

(Thanks, Dheeraj.)

Memorable Experiments

hanging boy experiment

In 1730 Stephen Gray found that an orphan suspended by insulating silk cords could hold an electrostatic charge and attract small objects.

In 1845, C.H.D. Buys Ballot tested the Doppler effect by arranging for an orchestra of trumpeters to play a single sustained note on an open railroad car passing through Utrecht.

In 1746 Jean-Antoine Nollet arranged 200 Carthusian monks in a circle, each linked to his neighbor with an iron wire. Then he connected the circuit to a rudimentary electric battery.

“It is singular,” he noted, “to see the multitude of different gestures, and to hear the instantaneous exclamation of those surprised by the shock.”

Project Management

How can six people be organized into four committees so that each committee has three members, each person belongs to two committees, and no two committees have more than one person in common?

It’s possible to work this out laboriously, but it yields immediately to a geometric insight:

project management diagram

If each line represents a committee and each intersection is a person, then the problem is solved.