The puzzling world of Raymond Smullyan
His work ranged from the academic study of mathematics to some of the world’s classic logic brainteasers.
RAYMOND SMULLYAN is a name that will be familiar to most mathematics enthusiasts. Born in 1919 in the small town of Far Rockaway, New York, Smullyan was fascinated by music, maths and magic from a very young age. Moving with his family to Manhattan at 13, he entered Theodore Roosevelt High School.
While the school provided an adequate music curriculum, Raymond was dissatisfied with its mathematics offerings and left soon after. He continued his studies on his own and eventually enrolled at Pacific College in Oregon. He moved around from one university to another, sometimes as a student and other times as a music or maths teacher for the next twenty-odd years. At one point, the University of Chicago even gave him credit for a calculus course that he had never actually taken, but that he had been teaching for some time. Eventually he received his PHD in mathematics from Princeton in 1959.
Smullyan published 30 books in his lifetime. While his academic works had intimidating titles such as Gödel’s Incompleteness Theorems and Recursive Theory for Metamathematics, his offerings to the public were more reflective of his true, whimsical nature. Alice in PuzzleLand, The Lady and The Tiger, and King Arthur in Search of His Dog are all typically Smullyan titles. Additionally they are considered classics in the field of logic puzzles – one of Smullyan’s specialties.
Two of my all-time favourite brainteasers come from his first book of logic puzzles, What is the Name of This Book?. The first puzzle introduces “Liars and Truth-tellers”. Liars can only tell lies and truth-tellers can only tell the truth. Liars and truth-tellers are identical in appearance. In some problems you must question them until you figure out which is which. In others, you must get them to reveal a location or specific pathway.
With that in mind, imagine you are on a deserted island whose only inhabitants are a single liar and a single truth-teller. You come to a fork in the road, guarded by these two men. You know one of them is a liar and the other a truth-teller, but you do not know which is which. You wish to know which way leads to the beach. You may turn to either of the two men but may ask only one question. The one you ask will only answer with “Yes” or “No.” Is there a question you can ask that will ensure you are directed down the appropriate fork? (Answer at the end of the article.)
A much simpler brainteaser wasn’t devised by Smullyan but was one of his favorites. A man is looking at a photograph and says: “Brothers and sisters I have none, but this man’s father is my father’s son.” Whose photograph is he holding? (Answer below.)
Smullyan had other interests apart from logic puzzles. He was a leading Taoist philosopher and also an amateur astronomer who even ground his own six-inch telescope mirror. His many adventures include appearing on The Tonight Show Starring Johnny Carson in 1982 and in 2001 being the subject of a biographical documentary entitled This Film Needs No Title: A Portrait of Raymond Smullyan. Seemingly indefatigable, he continued to give talks on mathematics and music until well into his 90s. Raymond Smullyan passed away on 7 February, 2017, at the age of 97.
I’ll close with a fiendishly difficult puzzle of Smullyan’s. It is preposterously hard. In 1979, two friends, Arthur and Robert, were curators at the Museum of American History. Both were born in the month of May, one in 1932 and the other a year later. Each was in charge of a beautiful antique clock. Both clocks worked pretty well, considering their ages, but one clock lost 10 seconds an hour while the other gained 10 seconds an hour. On one bright day in January, the two friends set both clocks right at exactly 12 noon. “You realise,” said Arthur, “that the clocks will start drifting apart, and they won’t be together again until – let’s see – why, on the very day you will be 47 years old. Am I right?” Robert then made a short calculation. “That’s right!” he said. Who is older, Arthur or Robert? Good luck.
ANSWER TO FIRST PUZZLE: The secret is to ask a question that forces both liars and truth-tellers to respond the same way. For instance, turn to either inhabitant while pointing down one of the roads and ask, “Would he tell me this road leads to the beach?” (“He” refers to the one you are not talking to). A liar will answer dishonestly “yes” if it’s the wrong road and “no” if it is the correct road. But a truth-teller would give exactly the same answers! He will say “yes” if you’re pointing down the wrong road and “no” if you’re pointing down the correct road. ANSWER TO SECOND PUZZLE: The man is looking at a picture of his son.
His public offerings were reflective of his true, whimsical nature.
There is nothing like cracking a tough mathematical nut.
YOU MIGHT BE FORGIVEN FOR supposing that a humdrum number like 4 would be of little interest to mathematicians or scientists. But 4 turns out to be deeply significant, cropping up across science and maths. Part of the number’s significance can be traced to antiquity. Greek philosophers associated various qualities with whole numbers, and 4 was chosen to signify justice. Even today we say that a fair bargain is “a square deal”. Other ancient uses include references to the four winds, the four corners of the Earth and the four horses of the Apocalypse.
In physics, 4 is the number of dimensions of spacetime – at least, those dimensions we perceive. The location of an object in space requires three numbers to specify it – longitude, latitude, altitude – and time requires just one to say when (e.g. 6 pm on Thursday).
The number of fundamental forces of nature is, so far as we know, also 4. They are gravitation, electromagnetism, the weak force responsible for some forms of radioactivity and the strong force that holds atomic nuclei together. Even the fundamental particles of matter the universe is made from cluster naturally into families of 4.
Professional mathematicians also get excited about the number 4. In 1852, an Englishman named Francis Guthrie was busy colouring a map showing the counties of England. Naturally he wanted to avoid having two adjacent counties the same colour, and he wondered how many different types of crayon that would require, for all possible arrangements of counties. ( Cosmos 65, p122) From common experience the minimum number seems to be four, but could that answer be rigorously proved?
In 1879 a mathematician named Alfred Kempe claimed to have solved the problem, but his proof was flawed. A subsequent attempt by Peter Tait in 1880 was also in error. Over time, the fourcolour problem became something of a cause célêbre; there is nothing like cracking a tough mathematical nut to motivate obsessive mathematicians. In 1900 the mathematical superstar David Hilbert listed solving the four-colour problem as one of the 23 great outstanding challenges in mathematics.
The issue remained unresolved when I was a child. I vividly remember drawing imaginary maps showing counties with incredibly convoluted shapes and arrangements in a futile attempt to disprove the four-colour conjecture. (I also tried to disprove Pythagoras’s theorem by analysing a complex mess of intersecting triangles, a lost cause if ever there was one.)
In 1976 Kenneth Appel and Wolfgang Haken at the University of Illinois UrbanaChampaign finally came up with a proof, but one with a difference. Rather than the usual pages of algebra, Appel and Haken resorted to 1,200 hours of computer time, resulting in thousands pages of output. A lively debate ensued over whether this construction constituted a real proof.
Most mathematicians who study the proofs of others in detail expect to arrive at an “Aha!” moment when they finally “get it”. Glimpsing the logical, and often elegant, reasoning of a preceding mathematician is what adds to the charm of the subject. But no human can “glimpse the mind of a computer” to appreciate how or even whether it has got the proof right.
This raises a profound issue. There is clearly a limit to the capabilities of even the cleverest human mathematician, yet there is no limit to the difficulty and subtlety of mathematical problems. As computers become more powerful, there is a growing temptation to do the mathematical equivalent of ‘asking Siri’ in order to settle outstanding conjectures. But this strikes at the very heart of what it is to know something. In many human affairs, to profess knowledge it is sufficient to be confident of the facts. But in the realm of pure mathematics, founded as it is on the ground of unshakeable logic, proofs are supposed to represent the most secure form of knowledge and understanding. To outsource this supreme intellectual responsibility to a machine represents a shattering demotion of human privilege.
Appel and Haken’s computations were the mathematical equivalent of Copernicus demoting Earth from the centre of the universe. It was a sign of things to come. Computers are now routinely used to explore the mathematical landscape.
In Douglas Adams’ A Hitchhiker’s Guide to the Galaxy, a computer is famously asked to solve the riddle of “life, the universe and everything”. The answer it delivers is 42 – at once banal yet profoundly meaningless to mere mortals. Why did Adams pick that number? Well, 42 = 43 – 42 – 4 – √4.
As the ancient Greeks might have commented: “Fair enough.”
There is clearly a limit to the capabilities of even the cleverest human mathematician.