A hairy ball can’t be combed flat — it must always have a cowlick.
This result arose originally in algebraic topology, but it has intriguing applications elsewhere. For example, it can’t be windy everywhere at once on Earth’s surface — at any given moment, the horizontal wind speed somewhere must be zero.
In 1829 a correspondent to the Mechanic’s Magazine proposed this design for a “self-moving railway carriage.” Fill the car with passengers and cargo as shown and set it on two rails that undulate across the landscape:
In the descending sections (a, c, e) the two rails are parallel. In the ascending ones (b, d) they diverge so that the car, mounted on cones, will roll forward to settle more deeply between them, paradoxically “ascending” the slope. If the track circles the world the car will “assuredly continue to roll along in one undeviating course until time shall be no more.”
“How any one could ever imagine that such a contrivance would ever continue in motion for even a short time … must be a puzzle to every sane mechanic,” wrote John Phin in The Seven Follies of Science in 1911. But what does he know?
As a joke, Michael Collins submitted a travel voucher for his trip aboard Gemini 10. NASA reimbursed him $8 per day, a total of $24.
In his autobiography, Collins notes that he could instead have claimed 7 cents a mile, which would have yielded $80,000.
But one of the original Mercury astronauts had already tried this — and had received a bill for “a couple of million dollars” for the rocket he’d used.
Draw two circles of any size and bracket them with tangents, as shown.
The chords in green will always be equal.
Any group of six people must contain at least three mutual friends or three mutual strangers.
Represent the people with dots, and connect friends with blue lines and strangers with red. Will the completed diagram always contain a red or a blue triangle?
Because A has five relationships and we’re using two colors, at least three of A’s connections must be of the same color. Say they’re friends:
Already we’re perilously close to completing a triangle. We can avoid doing so only if B, C, and D are mutual strangers — in which case they themselves complete a triangle:
We can reverse the colors if B, C, and D are strangers to A, but then we’ll get the complementary result. The completed diagram must always contain at least one red or blue triangle.
I think this problem appeared originally in the William Lowell Putnam mathematics competition of 1953. Six is the smallest number that requires this result — a group of five people would form a pentagon in which the perimeter might be of one color and the internal connections of another.
(Update: In fact the more general version of this idea was adduced in 1930 by Cambridge mathematician F.P. Ramsey. It is very interesting.) (Thanks, Alex.)
Cut a notch in a stick and label the two parts p and q. Then draw the stick around the shore of a pond. The notch will describe a curve, and, remarkably, the area between the shore and this curve will be given by πpq.
“Two things immediately struck me as astonishing,” wrote British mathematician Mark Cooker in 1988. “First, the formula for the area is independent of the size of the given curve. Second, [the equation for the area] is the area of an ellipse of semi-axes p and q, but there are no ellipses in the theorem!”
For years Raymond Smullyan sought a “metaparadox,” a statement that is paradoxical if and only if it isn’t. He arrived at this:
Either this sentence is false, or (this sentence is paradoxical if and only if it isn’t).
He wrote, “I leave the proof to the reader.”
In 1976, CUNY mathematician Robert Feinerman showed that the game of dreidel is fundamentally unfair.
Each player contributes one unit to the pot, and then all take turns spinning the top. Each spin produces one of four outcomes: the player does nothing, collects the entire pot, collects half the pot, or puts one unit into the pot. When a player collects the entire pot, then each player contributes one unit to form a new pot and play continues.
Feinerman found that, if Xn is the payoff on the nth spin and p is the number of players, the expected value of Xn is
Thus if there are more than two players, “the first player has an unfair advantage over the second player, who in turn has an unfair advantage over the third player, etc.”
Dreidel is booming nonetheless. A Major League Dreidel tournament has been held in New York City every Hanukkah since 2007. The official playing surface is called the Spinagogue, and the tournament slogan is “no gelt, no glory.”
(Robert Feinerman, “An Ancient Unfair Game,” American Mathematical Monthly 83:623-625.)
The greater honeyguide of Africa eats beeswax but isn’t always able to invade a hive on its own. So it has forged a unique partnership with human beings: The bird attracts the attention of local honey hunters with a chattering call, flies toward a hive, then stops and calls again. When they arrive at the hive, the humans open it, subdue the bees with smoke, take the honey, and leave the wax for the bird.
This arrangement saves the humans an average of 5.7 hours in searching for hives, but it’s not foolproof. “We have been ‘guided’ to an abrupt precipice and to a bull elephant by greater honeyguides,” report biologists Lester Short and Jennifer Horne. “In these cases there were bee-hives below the cliff (in a valley) and beyond the elephant. Concern for the welfare of the guided person is beyond any reasonable expectation of a honeyguide.”
(Thanks, Tom.)
