Navigators from the Poluwat atoll of Micronesia find their way among islands by appealing to parallax — a reference island at one side of their course will appear to pass beneath a succession of stars:

The star bearings of the reference island from both the starting and ending points of the trip are known, since on another occasion the reference island may itself become a destination. In between there are other navigation star positions under which the reference island will pass as it ‘moves’ backwards. Its passage under each of these stars marks the end of one etak and the beginning of another. Thus the number of star positions which lie between the bearing of the reference island as seen from the island of origin and its bearing as seen from the island of destination determine the number of etak, which can here be called segments, into which the voyage is conceptually divided. When the navigator envisions in his mind’s eye that the reference island is passing under a particular star he notes that a certain number of segments have completed and a certain proportion of the voyage has therefore been accomplished.

This is a dynamic model: Where Western navigators think of a vessel moving among stationary islands, the Poluwatese find it more natural to think of the canoe as stationary and the islands as moving around it. “Etak is perfectly adapted for its use by navigators who have no instruments, charts, or even a dry place in which to spread a chart if they had one,” writes Stephen D. Thomas in The Last Navigator. “Etak allows the Micronesian navigator to process all his information — course, speed, current drift, and so on — through a single, sea-level perspective.”

(Thomas Gladwin, East Is a Big Bird: Navigation and Logic on Puluwat Atoll, 1970, quoted in Lorenzo Magnani, Philosophy and Geometry, 2001.)

A Modest Proposal
Image: Flickr

While a visiting fellow at All Souls College, Oxford, in 1978, Claude Shannon pondered a personal challenge he faced there:

An American driving in England is confronted with a wild and dangerous world. The cars have the driver on the right and he is supposed to drive on the left side of the road. It is as though English driving is a left-handed version of the right-handed American system.

I can personally attest to the seriousness of this problem. Recently my wife and I, together with another couple on an extended visit to England, decided to jointly rent a car. … With our long-ingrained driving habits the world seemed totally mad. Cars, bicycles and pedestrians would dart out from nowhere and we would always be looking in the wrong direction. The car was usually filled with curses from the men and with screams and hysterical laughter from the women as we careened from one narrow escape to another.

His solution was “grandiose and utterly impractical — the idle dream of a mathematician”:

How will we do this? In a word, with mirrors. If you hold your right hand in front of a mirror, the image appears as a left hand. If you view it in a second mirror, after two reflections it appears now as a right hand, and after three reflections again as a left hand, and so on.

Our general plan is to encompass our American driver with mirror systems which reflect his view of England an odd number of times. Thus he sees the world about him not as it is but as it would be after a l80° fourth-dimensional rotation.

A corresponding adjustment to the steering system will turn the car left when the driver steers right, and vice versa. And filling the cabin with a high-density liquid will reverse the feeling of centrifugal force as well. “A snorkel provides for his breathing and altogether, with our various devices, he feels very much as though he were at home in America!”

(Claude E. Shannon, “The Fourth-Dimensional Twist, or a Modest Proposal in Aid of the American Driver in England,” typescript, All Souls College, Oxford, Trinity term, 1978; via Jimmy Soni and Rob Goodman, A Mind at Play: How Claude Shannon Invented the Information Age, 2017.)

The IKEA Effect
Image: Flickr

In 2011, Michael I. Norton of Harvard Business School, Daniel Mochon of Yale, and Dan Ariely of Duke found that test subjects were willing to pay 63% more for IKEA furniture that they had assembled than for identical units that came preassembled. In a separate study, they found that subjects who had finished building items were willing to pay more for their creations than subjects who had only partially completed assembly. The lesson seems to be that consumers place a disproportionately high value on products they’ve had a hand in creating.

The principle had been understood, though not named, as early as the 1950s, when homemakers initially disdained instant cake mixes, which they felt made cooking too easy and inspired no investment in the outcome. When the recipe was adjusted to require the cook to crack an egg, sales went up.


John Conway’s author biography in Melvin Fitting and Brian Rayman’s 2017 book Raymond Smullyan on Self Reference:

John H. Conway is the John von Neumann Distinguished Professor Emeritus of Applied and Computational Mathematics at Princeton. He has four doctorate degrees: PhD (Cambridge), DSc (Liverpool), D.h.c. (Iasi), PhD (Bremen). He has written or co-written 11 books, one of which has appeared in 11 languages. He has both daughters and sons, each daughter has as many sisters as brothers and each son twice as many sisters as brothers. He has met Raymond Smullyan repeatedly at many Gatherings for Gardner and Andrew Buchanan in Cambridge, New York and Princeton.

The size of his family is left as an exercise.

The Beal Conjecture

In 1993, banker and amateur mathematician Andrew Beal proposed that if Ax + By = Cz, where A, B, C, x, y, and z are positive integers and x, y, and z are all greater than 2, then A, B, and C must have a common prime factor.

Is it true? No one knows, but Beal is offering $1 million for a peer-reviewed proof or a counterexample.

The Kolakoski Sequence

Write down the digit 1:


This can be seen as describing itself: It might denote the length of the string of identical digits at this point in the sequence. Well, in that case, if the length of this run is only one digit, then the next digit in the sequence can’t be another 1. So write 2:

1 2

Seen in the same light, the 2 would indicate that this second run of digits has length 2. So add a second 2 to the list to fulfill that description:

1 2 2

We can continue in this way, adding 1s and 2s so that the sequence becomes a recipe for writing itself:
Animation: Wikimedia Commons

This is a fractal, a mathematical object that encodes its own representation. It was described by William Kolakoski in 1965, and before him by Rufus Oldenburger in 1939. University of Evansville mathematician Clark Kimberling is offering a reward of $200 for the solution to five problems associated with the sequence:

  1. Is there a formula for the nth term?
  2. If a string occurs in the sequence, must it occur again?
  3. If a string occurs, must its reversal also occur?
  4. If a string occurs, and all its 1s and 2s are swapped, must the new string occur?
  5. Does the limiting frequency of 1s exist, and is it 1/2?

So far, no one has found the answers.

The Reminiscence Bump

If you seem to recall your adolescence and early adulthood years more clearly than your later life, that’s normal. Most of us can recall a disproportionate number of autobiographical memories made between ages 10 and 30, perhaps because of the important changes in identity, goals, attitudes, and beliefs that most of us went through in those years. (Also, that’s the span in which many of us have novel experiences such as graduation, marriage, and the birth of a child.)

Interestingly, this phenomenon extends to favorite books, movies, and records. In a 2007 study, psychologist Steve M.J. Janssen and his colleagues at the University of Amsterdam found that subjects best recorded memories of these things between 11 and 25. This is particularly true of music: Items that aren’t revisited frequently, such as books, are more likely to be forgotten, but records have a strong “reminiscence bump.”

“Books are read two or three times, movies are watched more frequently, whereas records are listened to numerous times. The results suggest that differential encoding initially causes the reminiscence bump and that re-sampling increases the bump further.” See the appendices for lists of favorite books, movies, and records and the average ages at which subjects first encountered them.

(Steve M.J. Janssen, Antonio G. Chessa, and Jaap M.J. Murre, “Temporal Distribution of Favourite Books, Movies, and Records: Differential Encoding and Re-Sampling,” Memory 15:7 [2007], 755-767.)

The Dining Cryptographers Problem
Image: Wikimedia Commons

Three cryptographers are having dinner at their favorite restaurant. The waiter informs them that arrangements have been made for their bill to be paid anonymously. It may be that the National Security Agency has picked up the tab, or it may be that one of the cryptographers himself has done so. The cryptographers respect each other’s right to pay the bill anonymously, but they want to know whether the NSA is paying. Happily, there is a way to determine this without forcing a generous cryptographer to reveal himself.

Each cryptographer flips a fair coin behind a menu between himself and his right-hand neighbor, so that only the two of them can see the outcome. Then each cryptographer announces aloud whether the two coins he can see — one to his right and one to his left — had the same outcome or different outcomes. If one of the cryptographers is the payer, he states the opposite of what he sees. If an even number of cryptographers say that they saw different outcomes, then the NSA paid; if an odd number say so, then one of the cryptographers paid the bill, but his anonymity is protected.

Computer scientist David Chaum offered this example in 1988 as the basis for an anonymous communication network; these networks are often referred to as DC-nets (for “dining cryptographers”).

(David Chaum, “The Dining Cryptographers Problem: Unconditional Sender and Recipient Untraceability,” Journal of Cryptology 1:1 [1988], 65-75.)

Bases Into Gold

You have 12 coins that appear identical. Eleven have the same weight, but one is either heavier or lighter than the others. How can you identify it, and determine whether it’s heavy or light, in just three weighings in a balance scale?

This is a classic puzzle, but in 1992 Washington State University mathematician Calvin T. Long found a solution “that appears little short of magic.” Number the coins 1 to 12 and make three weighings:

First weighing: 1 3 5 7 vs. 2 4 6 8
Second weighing: 1 6 8 11 vs. 2 7 9 10
Third weighing: 2 3 8 12 vs. 5 6 9 11

To solve the problem, note the result of each weighing and assemble a three-digit numeral in base 3 as follows:

Left pan sinks: 2
Right pan sinks: 0
Balance: 1

For example, if coin 7 is light, that produces the number 021 in base 3. Now converting that to base 10 gives 7, the number of the odd coin, and an examination of the weighings shows that it must be light. Another example: If coin 2 is heavy, then we get 002 in base 3, which is 2 in base 10. Note that it’s possible to get an answer that’s higher than 12, e.g. when coin 7 is heavy — in that case subtract the base-10 answer you get from 26.

Another curious method to solve the classic puzzle, this one involving verbal mnemonics, appeared in Eureka in 1950.

(Calvin T. Long, “Magic in Base 3,” Mathematical Gazette 76:477 [November 1992], 371-376.)

09/30/2018 UPDATE: Due to an error in the original paper, the weighings I originally specified don’t work in every case — in the third weighing, the left pan should contain 2 3 8 12, not 1 2 8 12. I’ve amended this in the post above; everything should work now. Sorry for the error; thanks to everyone who wrote in.