After Archimedes’ death in 212 B.C., his tomb in Sicily fell into obscurity and was eventually lost. It was rediscovered by, of all people, Cicero, who had been sent to the island in 75 B.C. to administer corn production:

When I was Quaestor, I tracked down his grave; the Syracusans not only had no idea where it was, they denied it even existed. I found it surrounded and covered by brambles and thickets. I remembered that some lines of doggerel I had heard were inscribed on his tomb to the effect that a sphere and a cylinder had been placed on its top. So I took a good look around (for there are a lot of graves at the Agrigentine Gate cemetery) and noticed a small column rising a little way above some bushes, on which stood a sphere and a cylinder. I immediately told the Syracusans (some of their leading men were with me) that I thought I had found what I was looking for. Slaves were sent in with scythes to clear the ground and once a path had been opened up we approached the pedestal. About half the lines of the epigram were still legible although the rest had worn away.

“So, you see, one of the most celebrated cities of Greece, once upon a time a great seat of learning too, would have been ignorant of the grave of one of its most intellectually gifted citizens — had it not been for a man from Arpinum who pointed it out to them.”

(From Anthony Everitt, Cicero, 2003.)

True and False

Since its launch in 1991, arXiv, Cornell’s open-access repository of electronic preprints, has cataloged more than 2 million scientific papers.

In 2010, Caltech physicist David Simmons-Duffin created snarXiv, a random generator that produces titles and abstracts of imaginary articles in theoretical high-energy physics. Then he challenged visitors to distinguish real titles from fake ones.

After 6 months and 750,000 guesses in more than 50,000 games played in 67 countries, “the results are clear,” he concluded. “Science sounds like gobbledygook.” On average, players had guessed right only 59 percent of the time. Real papers most often judged to be fake:

  • “Highlights of the Theory,” by B.Z. Kopeliovich and R. Peschanski
  • “Heterotic on Half-Flat,” by Sebastien Gurrieri, Andre Lukas, and Andrei Micu
  • “Relativistic Confinement of Neutral Fermions With a Trigonometric Tangent Potential,” by Luis B. Castro and Antonio S. de Castro
  • “Toric Kahler Metrics and AdS_5 in Ring-Like Co-ordinates,” by Bobby S. Acharya, Suresh Govindarajan, and Chethan N. Gowdigere
  • “Aspects of U_A(1) Breaking in the Nambu and Jona-Lasinio Model,” by Alexander A. Osipov, Brigitte Hiller, Veronique Bernard, and Alex H. Blin
  • “Energy’s and Amplitudes’ Positivity,” by Alberto Nicolis, Riccardo Rattazzi, and Enrico Trincherini

Try it yourself.

While we’re at it: SCIgen randomly generates research papers in computer science, complete with graphs, figures, and citations; and Mathgen generates professional-looking mathematics papers, with theorems, proofs, equations, discussion, and references.

(Via Andrew May, Fake Physics: Spoofs, Hoaxes and Fictitious Science, 2019.)

A Parabolic Calculator

mathematikum calculator

Mathematikum, the science museum in Giessen, Germany, contains this clever device for multiplying pairs of numbers. The parabola presents the curve y = x2. Now suppose we want to multiply 10 by 9. We find the points -10 and 9 on the x axis, follow them up to the curve, and hang a weighted string over the pegs that we find there. The string crosses the y axis at (0, 90), so 10 × 9 = 90.

A simulator, and an explanation of the principle, are here.

A New Dawn

In July 1054 Chinese astronomers saw a reddish-white star appear in the eastern sky, its “rays stemming in all directions.” Yang Weide wrote:

I humbly observe that a guest star has appeared; above the star there is a feeble yellow glimmer. If one examines the divination regarding the Emperor, the interpretation is the following: The fact that the star has not overrun Bi and that its brightness must represent a person of great value. I demand that the Office of Historiography is informed of this.

It’s now believed they were witnessing SN 1054 — the supernova that gave birth to the Crab Nebula.

No Sale

If Chicken McNuggets come in packs of 6, 9, and 20, what’s the largest number of McNuggets that you can’t buy?

Steve Omohundro and Peter Blicher posed this question in MIT Technology Review in May 2002, and Ken Rosato contributed a neat solution.

The answer is 43. To start, notice that we can use the 6-packs and 9-packs to piece together any multiple of 3 other than 3 itself. 43 itself is not divisible by 3, so 6-packs and 9-packs alone won’t get us there, and adding some 20-packs won’t help, since we’d have to add them to a quantity of either 23 or 3, neither of which can be assembled from packs of other sizes. So that shows that 43 itself can’t be reached.

But we still need to show that every larger number can be. Well, we can create all the larger even numbers by adding some quantity of 6-packs to either 36, 38, or 40, and each of those foundations can be assembled from the packs we have (36 = 9 + 9 + 9 + 9, 38 = 20 + 9 + 9, and 40 = 20 + 20). So that takes care of the even numbers. And adding 9 to any of these even numbers will give us any desired odd number above 43, starting with 36 + 9 = 45.

So 43 is the largest number of Chicken McNuggets that can’t be formed by combining 6-packs, 9-packs, and 20-packs.

(I think Henri Picciotto was the first to broach this arresting question, in Games magazine in 1987. Since then, McNuggets have found their way into Happy Meals in 4-piece servings, reducing the largest non–McNugget number to 11. In some countries, though, the 9-piece allotment has been increased to 10 — and in that case there is no largest such number, as no odd quantity can ever be assembled.)


In their 1981 book Facts and Fallacies, Chris Morgan and David Langford note that the biblical reference to the “four corners of the earth” would apply equally well if the world were a tetrahedron.

In a similar spirit, as American airman Joseph Portney was flying over the North Pole in 1968 he wondered, “What if the Earth were … ?” He made sketches of 12 fanciful alternate Earths and gave them to Litton’s Guidance & Control Systems graphic arts group, which created models that were featured in the company’s Pilots and Navigators Calendar of 1969. This made an international sensation, and Portney’s creations were subsequently published for use in classrooms worldwide, inviting students to ponder what life would be like on a cone or a dodecahedron.

Portney graduated from the U.S. Naval Academy and went on to work for Litton on high-altitude navigation problems — for example, designing control systems that could guide an aircraft around one of these strange worlds.

The Internet Archive has the whole complement.


Consider the set (2, 5, 9, 13). Which of these numbers can be tossed out, and for what reason?

We might choose:

  • 2 because it’s the only even number.
  • 9 because it’s the only non-prime.
  • 13 because it doesn’t fit in the sequence AnAn-1 = 1 + (An-1An-2).

“Hence one could toss out either 2, 9 or 13,” observes Marquette University mathematician George R. Sell. “Therefore one should toss out 5 because it is the only number that cannot be tossed out.”

(George R. Sell, “A Paradox,” Pi Mu Epsilon Journal 2:6 [Spring 1957], 278.)

A Panmagic Geomagic Square

sallows panmagic geomagic square

Another amazing contribution by Lee Sallows:

“The picture above shows a 4×4 geomagic square, which is to say a magic square using geometrical shapes that can be fitted together so as to form an identical target shape, in this case a 4×6 rectangle, rather than numbers adding to a constant sum. In addition, the square is also panmagic, meaning that besides the usual 4 rows, 4 columns, and both main diagonals, the shapes occupying each of the so-called ‘broken’ diagonals, afkn, dejo, cfip, bglm, chin, belo, are also able to tile the rectangle. Lastly, the 4 shapes contained in the corner cells of the four embedded 3×3 sub-squares, acik, bdjl, fhnp, egmo, are also ‘magic’, bringing the total number of target-tiling shape sets to 20, a small improvement over the 16 achieved by a panmagic-only square. With that said, it is worth noting that 4×4 geomagic squares have been found achieving target-tiling scores as high as 48.”

Click the image to expand it. Thanks, Lee!

A Fateful Choice

A disease is spreading rapidly across the country. Half the people who have contracted it have died, and half have recovered on their own. A crash program to ward off the epidemic has produced two serums, A and B, but there’s been little time to test them. All three of the patients who were given serum A recovered, and so did 7 of the 8 patients who were given serum B. Unfortunately, you’ve just learned that you have the disease. If you get no treatment, your chances of surviving are 50-50. Both serums have a better record than that, but which one should you take?

“There doesn’t seem to be anything we can do other than appeal to our intuitive feelings on the matter,” writes University of Waterloo mathematician Ross Honsberger. “However, a very ingenious notion, the so-called ‘null hypothesis,’ permits a measure of analysis which, in this case, yields a definite preference.”

The key is to ask how likely it is that 3 out of 3 patients would have recovered if serum A were neither helping nor hindering them. An untreated patient has a 50-50 chance of recovery, so the answer is

\displaystyle \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}.

On the other hand, if serum B had no effect, then the chance of 7 recoveries out of 8 is

\displaystyle 8 \left ( \frac{1}{2} \times \frac{1}{2} \cdots \frac{1}{2}  \right ) = 8 \left ( \frac{1}{2} \right )^{8} = \frac{1}{32}.

(Here the factor 8 reflects the fact that there are 8 different possible victims, and again the probability of dying is 1/2.)

So the available evidence suggests that it’s 4 times as likely that serum A has no effect as that serum B has no effect. Your best course is to take serum B.

(Ross Honsberger, “Some Surprises in Probability,” in his Mathematical Plums, 1979.)