# The False Position Method

In David Hayes and Tatiana Shubin’s Mathematical Adventures (2004), University of California-Davis mathematician Don Chakerian describes a method used in antiquity for solving an equation in one unknown. He illustrates it with a problem from Daboll’s Schoolmaster’s Assistant (1800):

A, B, and C built a house which cost $500, of which A paid a certain sum, B paid 10 dollars more than A, and C paid as much as A and B both; how much did each man pay? We’ll make two guesses as to how much A paid, check them, and plug the “errors” into a formula to get the right answer. First, suppose A pays$80. That means that B pays $90 and C pays$170, giving a total of $340. That’s 500 – 340 =$160 short of the goal, so our guess of $80 yields an “error” of$160. As a second guess, suppose that A pays $150. In that case B pays$160, C pays $310, and the total is now$620. This time the “error” is 500 – 620 = -$120. The false position method (technically here the double false position method) offers this formula for finding the right answer: $\displaystyle \frac{\left ( \textup{first guess} \right ) \left ( \textup{second error} \right ) - \left ( \textup{second guess} \right ) \left ( \textup{first error} \right )}{\left ( \textup{second error} \right ) - \left ( \textup{first error} \right )}$ In this case it gives $\displaystyle \frac{\left ( 80 \right ) \left ( -120 \right ) - \left ( 150 \right ) \left ( 160 \right )}{ -120 -160 } = \frac{-9600 - 24000}{ -280 } = 120.$ When A pays$120 then B pays $130, C pays 250, and together they pay$500, so this solution works.

This is hardly the most efficient way to solve a simple linear equation given the tools we have today, but it served for centuries. In his Ground of Artes of 1542, Robert Recorde offered a rule:

Gesse at this woorke as happe doth leade.
By chaunce to truthe you may procede.
And firste woorke by the question,
Although no truthe therein be don.
Suche falsehode is so good a grounde,
That truth by it will soone be founde.
From many bate to many mo,
From to fewe take to fewe also.
With to much ioyne to fewe againe,
To to fewe adde to manye plaine.
In crossewaies multiplye contrary kinde,
All truthe by falsehode for to fynde.

# Many-Sided Story

A 55-sided figure is a pentacontapentagon; one with 79 sides is a heptacontaenneagon. A system exists to go even higher: A figure with 9,000 sides is an enakischiliagon, and one with a million is a megagon.

René Descartes suggested the 1,000-sided chiliagon as an example of a thing that can be considered without being explicitly imagined; one “does not imagine the thousand sides or see them as if they were present.” So the intellect is not dependent on imagination.

# 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:

21879

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:

  21879
62593
37406
82527
+ 17472

221877


How did he do this?

# Podcast Episode 249: The Robbers Cave Experiment

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 …

# Miniatures

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.

# Math Notes

Suppress each of the digits in 1729404 and add the resulting numbers, and you get 1729404 again:

  729404
129404
179404
172404
172904
172944
+ 172940
1729404


Another example:

  3758846
1758846
1358846
1378846
1375846
1375846
1375886
+ 1375884
13758846


There are 41 such numbers.

# 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.)

09/06/2019 UPDATE: The case of 42 has now been solved, by Andrew Booker at Bristol and Andrew Sutherland at MIT:

42 = (-80538738812075974)3 + 804357581458175153 + 126021232973356313

The lowest unsolved case is now 114.

# 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.”

# Cultivation

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