A Rapid Sum

From Lewis Carroll’s diary, Feb. 5, 1856:

Varied the lesson at the school with a story, introducing a number of sums to be worked out. I also worked for them the puzzle of writing the answer to an addition sum, when only one of the five rows have been written: this … astonished them not a little.

He had started by writing an arbitrary number:


Then he asked the students to call out a second five-digit number. Carroll added a third, the students shouted a fourth, and Carroll added a fifth and immediately wrote the sum:

+ 17472


How did he do this?

Click for Answer

Podcast Episode 249: The Robbers Cave Experiment

robbers cave

In 1954 a social psychologist started a war between two teams of fifth graders at an Oklahoma summer camp. He wanted to investigate the sources of human conflict and how people might overcome them. In this week’s episode of the Futility Closet podcast we’ll review the Robbers Cave Experiment and examine its evolving reputation.

We’ll also dredge up a Dalek and puzzle over a hazardous job.

See full show notes …


Last August, researchers at Rome University produced tiny portraits of Albert Einstein and Charles Darwin by modifying E. coli cells to respond to light patterns. Bacteria that received more light would swim faster, so over time they tended to concentrate in the darker parts of a negative image.

Lead author Giacomo Frangipane said in a statement, “Much like pedestrians who slow down their walking speed when they encounter a crowd, or cars that are stuck in traffic, swimming bacteria will spend more time in slower regions than in faster ones.”

Using the same technique, they created a (tiny) version of the Mona Lisa.

One to Go

33 = 88661289752875283 + (-8778405442862239)3 + (-2736111468807040)3

That result was discovered by Andrew Booker of the University of Bristol just this year.

It leaves 42 as the only positive integer less than 100 that has not been represented as the sum of three cubes.

(We can omit numbers that give a remainder of 4 or 5 when divided by 9, since those are known to be ineligible. But can every other integer be expressed in this way? It’s an open problem.)

(Thanks, Kate.)

Getting Acquainted

Sheep can be trained to recognize human faces, even from photographs. In a 2017 study at Cambridge University, researchers trained sheep to recognize photographs of four celebrities (Fiona Bruce, Jake Gyllenhaal, Barack Obama, and Emma Watson). They learned to distinguish a celebrity’s face from another face 8 out of 10 times, and their performance dropped only about 15% when they were shown photographs taken at an angle. When an unfamiliar photograph of a handler was inserted randomly in place of a celebrity, they chose that photo 7 out of 10 times.

“During this final task the researchers observed an interesting behaviour. Upon seeing a photographic image of the handler for the first time — in other words, the sheep had never seen an image of this person before — the sheep did a ‘double take’. The sheep checked first the (unfamiliar) face, then the handler’s image, and then unfamiliar face again before making a decision to choose the familiar face, of the handler.”

Sheep are long-lived and have relatively large brains, so it’s hoped that studying them will shed light on illnesses such as Huntington’s disease.

Also: Pigeons can distinguish Monet from Picasso, and rats can distinguish spoken Japanese from spoken Dutch. “A previous study by Porter and Neuringer (1984), who reported discrimination by pigeons between music and Bach and Stravinsky, and the present study suggest that pigeons have abilities that enable them to identify both musical and visual artists.”



Many people know that you can form a pentagon by tying a strip of paper in a simple overhand knot.

Stephen Bleecker Luce’s seamanship manual of 1863 tells how to fold a blade of grass into an octagon (below):

It is first doubled short over itself, then 1 under 2, — leaving a space, then 2 over 1, and down through the centre of the triangle; next 1 over 2, and down through the centre, coming out on the opposite side, and so on until an octagonal figure is formed.

That’s pretty terse, but I think I’ve almost managed to do it tonight. Keep your eye on the drawing of the finished piece, and don’t form and flatten the finished shape until you’ve done all the weaving.


(Via The Ashley Book of Knots.)

Math Notes

The sum or difference of any pair of the numbers {150568, 420968, 434657} is a square:

420968 + 150568 = 7562
420968 – 150568 = 5202
434657 + 420968 = 9252
434657 – 420968 = 1172
434657 + 150568 = 7652
434657 – 150568 = 5332

Miracles and Agents

Jones tells a mountain to hop into the sea and it does so. Has he performed a miracle?

Well, no, writes University of Birmingham philosopher George Chryssides. If Jones repeats his feat, then he’s revealed an underlying causal principle that’s amenable to study just like the rest of the natural world. If he doesn’t repeat the feat, then there’s no support for the idea of a link between his command and the mountain’s movement — we know only that the two events coincided, not that one caused the other.

“In order … to determine the answer to the question, ‘Did Jones move the mountain?’ … we must ascertain whether similar effects would follow similar putative causes,” Chryssides writes. “Either an allegedly miraculous event is a violation of scientific law, in which case it could not be performed by an agent, or else it is performed by an agent, in which case it could not be a violation of scientific law.”

(George D. Chryssides, “Miracles and Agents,” Religious Studies 11:3 [September 1975], 319-327.)

Levine’s Sequence

In the summer of 1997 mathematician Lionel Levine discovered a sequence of numbers: 1, 2, 2, 3, 4, 7, 14, 42, 213, 2837, … . It’s made up of the final term in each row of this array:

1 1
1 2
1 1 2
1 1 2 3
1 1 1 2 2 3 4
1 1 1 1 2 2 2 3 3 4 4 5 6 7
1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 7 7 7 8 8 9 9 10 10 11 12 13 14

The array is built from a simple rule. Start with 1 1 and regard each line as a recipe for building the next one: Read it from right to left and think of it as an inventory of digits. The first line, 1 1, would be read “one 1 and one 2,” so that gives us 1 2 for the second line. The second line (again, reading from right to left) would be read “two 1s and one 2,” giving 1 1 2 for the third line.

And so on. That’s it. It’s excruciatingly simple, “yet it seems likely that the 20th term say is impossible to compute,” wrote N.J.A. Sloane that November. At that time only 15 terms were known, the last being 508009471379488821444261986503540. Two further terms have since been found, bringing us up to 5347426383812697233786139576220450142250373277499130252554080838158299886992660750432. And the 20th is still out of sight.

See the link below (page 15) for more.

(Neil J.A. Sloane, “My Favorite Integer Sequences,” in C. Ding, T. Helleseth, and H. Niederreiter, Sequences and Their Applications: Proceedings of SETA ’98, 2012, 103-130.)