In 1962, a burnt golf ball arrived at the botanic gardens at Kew, in southwest London. The head of mycology, R.W.G. Dennis, may have rolled his eyes: The office had received another burnt golf ball 10 years earlier, which the submitter had claimed to be a “rare fungal species.” In that case the staff had got as far as trying to collect spores before they’d realized the hoax.

Twice provoked, Dennis responded in good humor. He published an article titled “A Remarkable New Genus of Phalloid in Lancashire and East Africa,” formally nominating it as a new species of fungus, “Golfballia ambusta,” and describing the specimens as “small, hard but elastic balls used in certain tribal rites of the Caledonians, which take place all season in enclosed paddocks with partially mown grass.” When a third burnt golf ball arrived in 1971, it was accepted into the collection, where all three balls now reside.

That creates a sort of Dadaist dilemma in mycology. By accepting the specimens and publishing a description, Dennis had arguably honored them as a genuine species. The precise definition of a fungus has varied somewhat over time; in publishing his article, Dennis may have been satirically questioning criteria that could accept a nonliving golf ball as a species. But what’s the solution? Some specialists have argued that fungi should be defined as “microorganisms studied by mycologists.” But in that case, points out mycologist Nathan Smith, we should be asking, “Who is a mycologist?”

Thinking Big

Called on to give an after-dinner talk at a 1961 meeting, Los Alamos physicist Darol Froman proposed a new project “without much attention to some of the practical aspects”: What if we turned Earth into a giant spaceship and drove it around? If we built a fusion rocket and fed it the moon and some damp sand, it could (in principle) carry us out of orbit. Our atmosphere would protect us from interstellar radiation, and we’d be free of the worry of our sun’s impending death.

It might take 100 million years to make our way out of the solar system, but by accelerating smoothly and sacrificing 4 percent of the planet’s mass we might have enough fuel to travel for 8 billion years. We could harvest deuterium from the oceans to generate heat and light during the journey, which might cover 1,300 light-years, and if we can refuel (“i.e., fill an ocean or two”) at hospitable planets then we might continue in this way indefinitely, exploiting the pull of stars along our path to help with navigation.

“I predict a pleasant existence in space — we will get away from the daily grind,” he said. “Perhaps we shall not wish to join another star — life in space may be more desirable.”

(He added, “I haven’t yet figured out a good way to leave certain people behind. It has been suggested that we use them for propellant. The human body is not constituted with quite as good proportions of the elements for our propellant as is damp sand. Nevertheless, the proportions are not too bad and we can in this way take care of the problem of overpopulation.”)

(Darol Froman, “The Earth as a Man-Controlled Space Ship,” Physics Today 15:7 [July 1962], 19-23.)


Hinchliffe’s rule, named after physicist Ian Hinchliffe, states that if the title of a scholarly article takes the form of a yes-no question, the answer to that question will be no.

In 1988 Boris Peon tested this proposition by writing a paper titled “Is Hinchliffe’s Rule True?”:

Hinchliffe has asserted that whenever the title of a paper is a question with a yes/no answer, the answer is always no. This paper demonstrates that Hinchliffe’s assertion is false, but only if it is true.

This seems to threaten the integrity of the universe. Happily, Harvard computer scientist Stuart Shieber pointed out that Hinchliffe’s rule might simply be false, in which case Peon’s title presents no problem.

Unfortunately Shieber also managed to resurrect the paradox by titling his article “Is This Article Consistent With Hinchliffe’s Rule?”

We await developments.

The Denomination Effect

In 2009, marketing professors Priya Raghubir and Joydeep Srivastava gave $1 to each of 89 undergraduates and told them they could keep the money or spend it on candy. The students received the money in different denominations — 43 students were given four quarters, and 46 were given a dollar bill. About 63 percent of the students who’d received quarters chose to buy candy, but only 26 percent of those who’d received a dollar bill did so.

In related studies, Raghubir and Srivastava found that subjects who foresaw a need to exert self-control in spending chose to receive money in large denominations. And “[t]ightwads choose to receive money in a large denomination as a precommitment device when the need for self-control is high.”

The lesson seems to be that people are less likely to spend money when they receive it in large denominations. “[L]arge denominations are psychologically less fungible than smaller ones, allowing them to be used as a strategic device to control and regulate spending.”

(Priya Raghubir and Joydeep Srivastava. “The Denomination Effect,” Journal of Consumer Research 36:4 [December 2009], 701-713.)


Dry Falls, in central Washington, has a stunningly dramatic history: At the end of the last glaciation, when ice dams to the east gave way, torrents of water roared through the landscape from flooded Montana, pouring over a 400-foot rock face at 65 mph in a waterfall five times as wide as Niagara and carrying 10 times the flow of all the world’s rivers combined.

The scale of the cataclysm is hard to imagine, so geologist Nick Zentner of Central Washington University commissioned these animations from Newlands & Company of Portland to help to convey its magnitude.

Twice True

Each of these sums is valid in two ways, once when the words at taken at their face value and again when each letter is interpreted as a particular digit:

   THREE    79322           ONE       483       ZERO   4206      TRECE  69858
    NINE     6562          FIVE      7293        SEI    827      CINCO  57354
     TEN      726           TEN       138      SETTE  82112       OCHO   4504
FOURTEEN 40837226        ELEVEN    363938       OTTO   6116       ----   ----
 FIFTEEN  4547226      NINETEEN  82831338       NOVE   9652     QUINCE 127358
 -------  -------     FORTYFIVE 745107293       ----   ----       ONCE   4358
FIFTYONE 45471062     --------- ---------     TRENTA 102913
                      NINETYONE 828310483

All are from the Journal of Recreational Mathematics, collected by Leonard Gordon in “Doubly-True Alphametics,” Word Ways 27:1 [February 1994], 10-12. More alphametics.

Small World

Physicist Bruce Sherwood has created a 3-D computer model of a flat earth so that conspiracy theorists can examine the implications of their own theory.

For example, if Antarctica is a mountain range surrounding a flat disk, what should we expect to see in the sky? “Walk inside the model and look up,” Sherwood told philosopher Lee McIntyre. “If you’re standing at the North Pole, then Polaris should be directly overhead. Fair enough. But if you’re standing at the ‘edge of the Earth’ — and Polaris is only a few thousand miles overhead — shouldn’t you at best see it at an angle? But if you’re actually in Antarctica, you won’t be able to see it at all. Their model is inconsistent with physical observation. And they can see that for themselves.”

(Bruce Sherwood, “A Flat Earth?”, via Lee McIntyre, et al., How to Talk to a Science Denier, 2021.)

In a Word

n. a leisurely walk

In the ancient world, distances were sometimes measured by pacing. Specialists known as bematists were employed for this purpose in both Egypt and Greece, and their accuracy could be startling: In his Naturalis Historia, Pliny the Elder notes that two bematists employed by Alexander the Great had measured the distance from Hecatompylos to Alexandria Areion on the Silk Road at 851 kilometers. The actual distance is 855 kilometers, a deviation of just 0.4 percent. In general, according to Pliny’s records, Alexander’s bematists showed a median deviation of just 2.8 percent from the true distances; a separate account by Strabo shows a median deviation of only 1.9 percent.

This accuracy suggests that the bematists may have been using an early odometer, such as one described by Heron of Alexandria, though the records don’t mention this.

12/30/2023 UPDATE: Reader Charlotte Fare has made a data visualization. (Thanks, Charlotte.)

Easy Street

The following was rather widely quoted a few years ago. It bothered one banker so much that he made a hasty trip to consult his neighbor, a college professor of mathematics. Assume we make a deposit of $50 in a bank.

    withdraw $20.00 leaving $30.00
now withdraw  15.00 leaving  15.00
now withdraw   9.00 leaving   6.00
now withdraw   6.00 leaving   0.00
             ------         ------
             $50.00         $51.00

We now present our figures to the bank, showing the discrepancy, and demand the extra dollar. Repeat ten thousand times, and retire for a while.

— Cecil B. Read, “Mathematical Fallacies,” School Science and Mathematics, June 1933