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.
Freud observed that people who have much in common can still fight bitterly because they grow overly sensitive to the disagreements remaining between them. He called this the “narcissism of small differences”: “It is precisely the minor differences in people who are otherwise alike that form the basis of feelings of hostility between them.”
In Gulliver’s Travels, 11,000 people die in a war between the Big-endians, who break their eggs at the big end, and the Little-endians, who break them at the little end. A Lilliputian admiral attacks Gulliver because “he had good reasons to think you were a Big-endian in your heart; and, as treason begins in the heart, before it appears in overt acts, so he accused you as a traitor on that account, and therefore insisted you should be put to death.”
From Arthur Engel’s excellent Problem-Solving Strategies (1998):
On a cube, mark all the vertices and all the midpoints of the faces. Can an enterprising ant visit all these points by walking only along face diagonals and never retracing its steps?
For a 1752 essay, Samuel Johnson compiled these notes on how a man’s outlook changes as he grows older:
Hope predom. in youth. Mind not willingly indulges unpleasing thoughts. The world lies all enameld before him, as a distant prospect sun-gilt — inequalities only found by coming to it. Love is to be all joy — children excellent — Fame to be constant — caresses of the great — applauses of the learned — smiles of Beauty.
Fear of disgrace — Bashfulness — Finds things of less importance. Miscarriages forgot like excellencies; — if remembered, of no import. Danger of sinking into negligence of reputation. Lest the fear of disgrace destroy activity.
Confidence in himself. Long tract of life before him. — No thought of sickness. — Embarrasment of affairs. — Distraction of family. — Publick calamities. — No sense of the prevalence of bad habits. — Negligent of time — ready to undertake — careless to pursue — all changed by time.
Confident of others — unsuspecting as unexperienced — imagining himself secure against neglect, never imagines they will venture to treat him ill. Ready to trust; expecting to be trusted. Convinced by time of the selfishness, the meanness, the cowardice, the treachery of men.
“Such is the condition of life, that something is always wanting to happiness,” Johnson wrote in the finished essay. “In youth, we have warm hopes, which are soon blasted by rashness and negligence, and great designs, which are defeated by inexperience. In age, we have knowledge and prudence without spirit to exert, or motives to prompt them; we are able to plan schemes and regulate measures, but have not time remaining to bring them to completion.”
A puzzle from the 1975-1976 issue of Eureka, the journal of the Cambridge University Mathematical Society:
A jet fighter on the surface of the Earth is being chased by two identical fighters, all with ample fuel. Can the pursuers get within shooting range, no matter how they start out?
Yes. Start a pursuer at each pole and have them approach the quarry from either side, always matching its longitude. Eventually they must trap it between them.
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.