Science & Math

Perpetual Locomotion

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?

Business Trip

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.

Eye to Eye

equal chord theorem

Draw two circles of any size and bracket them with tangents, as shown.

The chords in green will always be equal.

The Elevator Problem

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:

elevator problem 1

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:

elevator problem 2

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.)

Holditch’s Theorem

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!”

Knot Mirage

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.”

Top Secret

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

dreidel expected value

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.)

Cancel That

freak cancellations

A Feathered Maître d’

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.)

Reality Unperceived

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

Math Notes

math notes - equal sums

The Sleeping Beauty Problem

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?

A Pretty Fact

three squares theorem

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.”

A Square Surprise

3x3 magic square

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.

Ladies’ Night

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+

18 queens

This may be the shortest possible such game.

Numerius Negidius

Only 43 numbers have names that lack the letter N.

One of them, fittingly, is forty-three.

Varignon’s Theorem

Draw any quadrilateral and connect the midpoints of its sides.

You’ll always get a parallelogram.

Triangles, too, have perfection at heart.

Actual Size

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.

Yes or No?

Will you either answer no to this question or pay me a million dollars?

(Raymond Smullyan)


  • 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

Gilbreath’s Conjecture

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:

gilbreath's conjecture

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

Person to Person

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

Math Notes