Choose a number on this clock face and, starting from 12, spell out that number’s English name as you advance clockwise around the face, one letter per numeral. For example, if you’ve chosen 3, count out T-H-R-E-E and you’ll land on the numeral 5. Adopt this new position as your next chosen number and proceed as before (in this case, counting F-I-V-E and landing on 9). After three or more moves you’ll reliably land on 1.
This recording presents four pairs of tones, each pair separated by three whole tones, or half an octave. Curiously, some listeners hear the interval as ascending, others as descending. (In fact the tones used are ambiguous as to octave, so there’s no objectively right answer.)
Even more curiously, sometimes a listener’s perception reverses when an interval is transposed, say from C-F# to G#-D, even though nothing else has changed.
A pleasing detail from Built for Speed, University of Idaho zoologist John A. Byers’ 2003 account of a year studying pronghorn antelope in western Montana:
“It can be really bizarre if, when I’ve been alone for several hours, I spot a human standing or walking in the distance. For an instant, my reaction is, ‘What the hell is that?’ A large mammal that walks on its hind legs seems very strange.”
This street scene, taken in Cork in 1927, seems to give a 3-D effect, but it’s only alternating between two photos taken from slightly different perspectives. The sense of depth comes from parallax and the occlusion of distant objects.
Small animals bob their heads to produce this effect as they plan a jump; it helps them to estimate distance.
Albert Einstein and Kurt Gödel used to walk home together from the Institute for Advanced Study in Princeton. In Incompleteness (2005), Rebecca Goldstein gives a sample of their conversation, broached by Gödel:
All of his thinking is governed by an ‘interesting axiom,’ as Ernst Gabor Straus, Einstein’s assistant from 1944 to 1947, once characterized it. For every fact, there exists an explanation as to why that fact is a fact; why it has to be a fact. This conviction amounts to the assertion that there is no brute contingency in the world, no givens that need not have been given. In other words, the world will never, not even once, speak to us in the way that an exasperated parent will speak to her fractious adolescent: ‘Why? I’ll tell you why. Because I said so!’ The world always has an explanation for itself, or as Einstein’s walking partner puts it, Die Welt is vernunftig, the world is intelligible. The conclusions that emanate from the rigorously consistent application of this ‘interesting axiom’ to every subject that crosses the logician’s mind — from the relationship between the body and the soul to global politics to the very local politics of the Institute for Avanced Study itself — often and radically diverge from the opinions of common sense. Such divergence, however, counts as nothing for him. It is as if one of the unwritten laws of his thought processes is: If reasoning and common sense should diverge, then… so much the worse for common sense! What, in the long run, is common sense, other than common?
A set of 4 elements can be partitioned in 15 ways.
Pleasingly, this is also the number of rhyme schemes that a 4-line poem can take: AAAA, AAAB, AABA, AABB, AABC, ABAA, ABAB, ABAC, ABBA, ABBB, ABBC, ABCA, ABCB, ABCC, ABCD.
A poem with 5 lines has 52 possible schemes, corresponding to the partitions of a 5-element set, and so on. These are called Bell numbers.
Two cannons are aimed directly at one another. One is on the floor of a valley, and the other is on a promontory. Neglecting air resistance, if the two fire simultaneously, what will happen?
The balls will collide. If there were no gravity, the balls would travel a straight line and meet between the cannons. When gravity is introduced, it imparts the same vertical acceleration to both balls, so they must still meet.
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.
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.”
