A warden offers a challenge to two prisoners. The first prisoner will enter a room that contains a chessboard. On each of the board’s 64 squares is a coin that’s either heads up or tails up. The guard will identify one square as the “target.”
The first prisoner must turn over exactly one coin and then leave the room. The second prisoner must then enter and, solely by viewing the board, determine which square is the target.
If they succeed, both prisoners will go free. They can confer beforehand on a strategy, but they may not communicate after that. Can they establish a plan that will always work?
Surprisingly, the answer is yes. Number the squares 0 to 63 and express these numbers in 6-bit binary (000000 to 111111). List the numbers of the squares that bear coins that are heads-up. Below each column in that list, write a 1 if the column contains an odd number of 1s or a 0 if it contains an even number of 1s. For example, suppose that the only heads-up coins are on squares 17, 21, 38, 44, and 55. Listing these in binary and condensing the list as described, we’d get:
010001
010101
100110
101100
110111
------
111001
111001 is 57 in decimal. Call that the encoded square; the second prisoner, on just viewing the board, could conduct the same calculations and identify this square independently.
Now how can we communicate the identity of the target square? It turns out that, by flipping a single coin, we can move the encoded square to any square we choose.
Suppose the target square is square 11. List this below the number of the encoded square, both in binary:
111001
001011
Below each column, write a 0 if the two numbers match and a 1 if they don’t (this is an application of the binary logical operator XOR, “exclusive OR”):
111001
001011
------
110010
This result, 110010, is 50 in decimal. This tells us that flipping the coin on square 50 will move the encoded square from square 57 to square 11, the target square. Now we can leave the room, and prisoner 2 can enter and, following the steps above, independently calculate the encoded square, which we’ve now arranged to coincide with the target square. So we both go free.
In Pascal’s triangle, each number is the sum of the two above it. Obviously, the infinite pyramid contains an infinite number of 1s, but most numbers appear surprisingly seldom:
2 appears just once.
3, 4, 5, and all odd primes appear exactly twice.
6 appears three times.
Infinitely many numbers appear exactly six times, but we don’t know whether any appear exactly five or seven times.
3003 appears eight times, possibly the only such specimen.
In 1971, Berkeley mathematician David Singmaster suggested that there may be a finite upper bound on the number of times that any number can appear (apart from 1). But that remains an unsolved problem.
In one copy of a 1942 edition of German historian Gert Buchheit’s biography of Rainer Maria Rilke, someone has glued a typewritten and hectographed alphanumeric text. The text fills 33 pages with 18,760 characters in groups of four. Analysis shows that it’s less ordered than English or German but more ordered than random text. To date, no one has been able to make sense of it.
I’d missed this: In 2006 a geneticist, a philosopher, and a chicken farmer all agreed that the egg came before the chicken.
Nottingham University geneticist John Brookfield pointed out that the first chicken (the first creature bearing chicken DNA) must have begun as an embryo in an egg. “The first living thing which we could say unequivocally was a member of the species would be this first egg, so I would conclude that the egg came first.”
David Papineau, philosopher of science at King’s College, London, agreed. “I would argue it is a chicken egg if it has a chicken in it. … If a kangaroo laid an egg from which an ostrich hatched, that would surely be an ostrich egg, not a kangaroo egg.”
And Charles Bournes, chair of trade body Great British Chicken, said, “Eggs were around long before the first chicken arrived. Of course they may not have been chicken eggs as we see them today, but they were eggs.”
According to the BBC, “Professor Brooke added the debate could finally be laid to rest.”
According to legend, French highwayman Claude Duval agreed not to rob one gentleman if his wife would dance the courante with him by the wayside.
He was hanged at Tyburn in 1670 “to the great grief of the women.” A memorial in Covent Garden reads, “Here lies DuVall: Reder, if male thou art, Look to thy purse; if female, to thy heart.”
A problem from the October 1961 issue of Eureka, the journal of the Cambridge University Mathematical Society:
When A was three times as old as B was the year before A was a half of B’s present age, B was 3 years younger than A was when B was two thirds of A’s present age. A’s and B’s ages now total 73. How old are A and B?
An “easy and curious method of foretelling rainy or fine weather,” from an 1860 book on conjuring, of all places:
“[T]he best instrument of all, is a good pair of scales, in one of which let there be a brass weight of a pound, and in the other a pound of salt, or of saltpetre, well dried; a stand being placed under the scale, so as to hinder it falling too low. When it is inclined to rain, the salt will swell, and sink the scale: when the weather is growing fair, the brass weight will regain its ascendancy.”
A puzzle in chess logic from The Batsford Chess Puzzle Book. Who made the last move, and what was it? (There’s no trick — everything is just as it seems.)
Black cannot have moved last, as there’s no square from which the king can legally have moved. The white pawns have not moved, the rook is entombed, and the knight would have had to move from b3, in which which case Black had no legal preceding move. So White just moved his king to c1. What was Black’s move before that? He can’t have moved his king, so he must have moved a black piece to c1, where it was captured by the white king. What piece? Only a knight could legally arrive on that square to be captured. So the white king captured a black knight on c1.
Have Angleworms attractive homes?
Do Bumblebees have brains?
Do Caterpillars carry combs?
Do Ducks dismantle drains?
Can Eels elude elastic earls?
Do Flatfish fish for flats?
Are Grigs agreeable to girls?
Do Hares have hunting hats?
Do Ices make an Ibex ill?
Do Jackdaws jug their jam?
Do Kites kiss all the kids they kill?
Do Llamas live on lamb?
Will Moles molest a mounted mink?
Do Newts deny the news?
Are Oysters boisterous when they drink?
Do Parrots prowl in pews?
Do Quakers get their quills from quails?
Do Rabbits rob on roads?
Are Snakes supposed to sneer at snails?
Do Tortoises tease toads?
Can Unicorns perform on horns?
Do Vipers value veal?
Do Weasels weep when fast asleep?
Can Xylophagans squeal?
Do Yaks in packs invite attacks?
Are Zebras full of zeal?
“P.S. Shake well and recite every morning in a shady place.”
The second star in the Big Dipper’s handle is actually two stars, Mizar and Alcor. Distinguishing the two with the naked eye has been used as a test of vision for hundreds of years. Arabic tradition held that only those with the sharpest eyesight could see Mizar’s companion, and the 13th-century Persian astronomy writer Zakariya al-Qazwini wrote that “people tested their eyesight by this star.” In Japan, it was said that being unable to see Alcor with the naked eye foretold an impending death of old age, and Alexander von Humboldt and François Arago both noted that Alcor can be seen only with difficulty.
In The Herschels and Modern Astronomy (1901), Irish astronomer Agnes Mary Clerke wrote, “The Arabs in the desert regarded it as a test of penetrating vision; … Vidit Alcor, at non lunam plenam (Latin for ‘he saw Alcor, but not the full moon’), came to be a proverbial description of one keenly alive to trifles, but dull of apprehension for broad facts.”
