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.”
But, say you, surely there is nothing easier than for me to imagine trees, for instance, in a park, or books existing in a closet, and nobody by to perceive them. I answer, you may so, there is no difficulty in it; but what is all this, I beseech you, more than framing in your mind certain ideas which you call books and trees, and the same time omitting to frame the idea of any one that may perceive them? But do not you yourself perceive or think of them all the while? This therefore is nothing to the purpose; it only shews you have the power of imagining or forming ideas in your mind: but it does not shew that you can conceive it possible the objects of your thought may exist without the mind.
— George Berkeley, A Treatise Concerning the Principles of Human Knowledge, 1710
Prince Charming tells Sleeping Beauty, “I’m going to put you to sleep with this potion, and then I’ll flip a coin. Today is Sunday. If the coin lands heads, I’ll wake you again on Monday. If it lands tails, then I’ll wake you on Monday, put you to sleep again, and wake you on Tuesday. The potion induces a mild amnesia, so you won’t remember the intermediate awakening if it happens, but otherwise it won’t hurt you.”
When Sleeping Beauty awakes, what probability should she assign that the coin landed heads?
There seem to be two contradictory answers to this. From one perspective, the coin was fair, so it would seem the chance is 1/2. But from another, Beauty finds herself in one of three equally likely situations (heads/Monday, tails/Monday, and tails/Tuesday), so the chance of heads appears to be 1/3. Which is correct?
Given three adjacent squares,
a + b = c.
“Beauty is the first test,” wrote G.H. Hardy. “There is no permanent place in the world for ugly mathematics.”
The lowly 3×3 magic square has modest pretensions — each row, column, and diagonal produces the same sum.
But perhaps it’s magicker than we suppose:
6182 + 7532 + 2942 = 8162 + 3572 + 4922 (rows)
6722 + 1592 + 8342 = 2762 + 9512 + 4382 (columns)
6542 + 1322 + 8792 = 4562 + 2312 + 9782 (diagonals)
6392 + 1742 + 8522 = 9362 + 4712 + 2582 (counter-diagonals)
6542 + 7982 + 2132 = 4562 + 8972 + 3122 (diagonals)
6932 + 7142 + 2582 = 3962 + 4172 + 8522 (counter-diagonals)
(R. Holmes, “The Magic Magic Square,” The Mathematical Gazette, December 1970)
More: Any of the equations above will still hold if you remove the middle digit or any two corresponding digits in each of the six addends.
Yet more: (6 × 1 × 8) + (7 × 5 × 3) + (2 × 9 × 4) = (6 × 7 × 2) + (1 × 5 × 9) + (8 × 3 × 4)
I think everything above will work for any rotation or reflection of the square (that is, for any normal 3×3 magic square). I haven’t checked, though.
Is a legal chess game possible in which all the pawns promote and each player has nine queens?
Yes — Freidrich Burchard of Germany and Friedrich Hariuc of Romania reached nearly identical solutions in 1980:
1. e4 f5 2. e5 Nf6 3. exf6 e5 4. g4 e4 5. Ne2 e3 6. Ng3 e2 7. h4 f4 8. h5 fxg3 9. h6 g5 10. Rh4 gxh4 11. g5 g2 12. g6 Bg7 13. hxg7 g1=Q 14. f4 h3 15. f5 h2 16. b4 a5 17. b5 a4 18. b6 a3 19. Bb2 Ra7 20. bxa7 axb2 21. a4 b5 22. a5 b4 23. a6 b3 24. c4 h1=Q 25. c5 h5 26. c6 Bb7 27. cxb7 c5 28. d4 c4 29. d5 Nc6 30. dxc6 c3 31. c7 c2 32. c8=Q c1=Q 33. b8=Q Qc7 34. a8=Q d5 35. a7 d4 36. Nc3 dxc3 37. Qa6 c2 38. Qa8b7 c1=Q 39. a8=Q Qd5 40. gxh8=Q+ Kd7 41. g7 bxa1=Q 42. g8=Q b2 43. f7 b1=Q 44. f8=Q h4 45. f6 h3 46. f7 h2 47. Qfa3 h1=Q 48. f8=Q exf1=Q+
This may be the shortest possible such game.
Only 43 numbers have names that lack the letter N.
One of them, fittingly, is forty-three.
Draw any quadrilateral and connect the midpoints of its sides.
You’ll always get a parallelogram.
Triangles, too, have perfection at heart.
The familiar Mercator projection is useful for navigation, but it exaggerates the size of regions at high latitudes. Greenland, for example, appears to be the same size as South America, when in fact it’s only one eighth as large.
An equal-area projection such as the Mollweide, below, distorts the shapes of regions but preserves their relative size. This reveals some surprising facts: Russia is larger than Antarctica, Mexico is larger than Alaska, and Africa is just mind-bogglingly huge — larger than the former Soviet Union, larger than China, India, Australia, and the United States put together.
Will you either answer no to this question or pay me a million dollars?
- The telephone number 266-8687 spells both AMOUNTS and CONTOUR.
- 38856 = (38 – 85) × 6
- CARTHORSE is an anagram of ORCHESTRA.
- The French for paper clip is trombone.
- “The oldest books are only just out to those who have not read them.” — Samuel Butler
Doodling on a napkin in 1958, mathematician Norman L. Gilbreath noticed something odd. First he wrote down the first few prime numbers in a row. Then, on each succeeding row, he recorded the (unsigned) difference between each pair of numbers in the row above:
The first digit in each row (except the first) is 1. Will this always be true, no matter how many prime numbers we start with? It’s been borne out in computer searches extending to hundreds of billions of rows. But no one knows for sure.
- Tarzan’s yell is an aural palindrome.
- CONTAMINATED is an anagram of NO ADMITTANCE.
- The Swiss Family Robinson have no surname (“Robinson” refers to Robinson Crusoe).
- x2 – 2999x + 2248541 produces 80 primes from x = 1460 to 1539.
- “A great fortune is a great slavery.” — Seneca
Suppose that you enter a cubicle in which, when you press a button, a scanner records the states of all the cells in your brain and body, destroying both while doing so. This information is then transmitted at the speed of light to some other planet, where a replicator produces a perfect organic copy of you. Since the brain of your Replica is exactly like yours, it will seem to remember living your life up to the moment when you pressed the button, its character will be just like yours, and it will be in every other way psychologically continuous with you. Is it you?
— Derek Parfit, “Divided Minds and the Nature of Persons,” in Mindwaves, 1987
Suppose, therefore, a person to have enjoyed his sight for thirty years, and to have become perfectly acquainted with colours of all kinds, except one particular shade of blue, for instance, which it never has been his fortune to meet with. Let all the different shades of that colour, except that single one, be placed before him, descending gradually from the deepest to the lightest; it is plain, that he will perceive a blank, where that shade is wanting, and will be sensible, that there is a greater distance in that place between the contiguous colours than in any other. Now I ask, whether it be possible for him, from his own imagination, to supply this deficiency, and raise up to himself the idea of that particular shade, though it had never been conveyed to him by his senses?
— David Hume, An Enquiry Concerning Human Understanding, 1748
When German physicist Walther Nernst learned that his cowshed was warm because of the cows’ metabolic activity, he resolved to sell them and invest in carp.
A thinking man, he said, cultivates animals that are in thermodynamic equilibrium with their surroundings and does not waste his money in heating the universe.
‘Twas Euclid, and the theorem pi
Did plane and solid in the text,
All parallel were the radii,
And the ang-gulls convex’d.
“Beware the Wentworth-Smith, my son,
And the Loci that vacillate;
Beware the Axiom, and shun
The faithless Postulate.”
He took his Waterman in hand;
Long time the proper proof he sought;
Then rested he by the XYZ
And sat awhile in thought.
And as in inverse thought he sat
A brilliant proof, in lines of flame,
All neat and trim, it came to him,
Tangenting as it came.
“AB, CD,” reflected he–
The Waterman went snicker-snack–
He Q.E.D.-ed, and, proud indeed,
He trapezoided back.
“And hast thou proved the 29th?
Come to my arms, my radius boy!
O good for you! O one point two!”
He rhombused in his joy.
‘Twas Euclid, and the theorem pi
Did plane and solid in the text;
All parallel were the radii,
And the ang-gulls convex’d.
— Emma Rounds