Science & Math

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


Cast of Thought

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

A Penny Saved

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.

“Plane Geometry”

‘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

Reservation Trouble

Socrates likes company. He wants to eat only if Plato wants to eat.

But Plato is angry at Socrates. He wants to eat only if Socrates does not want to eat.

Does Socrates want to eat?

(From Buridan’s Sophismata.)

À La Carte

Suppose there were an experience machine that would give you any experience you desired. Superduper neuropsychologists could stimulate your brain so that you would think and feel you were writing a great novel, or making a friend, or reading an interesting book. All the time you would be floating in a tank, with electrodes attached to your brain. Should you plug into this machine for life, preprogramming your life’s experiences? If you are worried about missing out on desirable experiences, we can suppose that business enterprises have researched thoroughly the lives of many others. You can pick and choose from their large library or smorgasbord of such experiences, selecting your life’s experiences for, say, the next two years. After two years have passed, you will have ten minutes or ten hours out of the tank, to select the experiences of your next two years. Of course, while in the tank you won’t know that you’re there; you’ll think it’s all actually happening. … Would you plug in?

– Robert Nozick, Anarchy, State, and Utopia, 1974


10102323454577 is the smallest 14-digit prime number that follows the rhyme scheme of a Shakespearean sonnet (ababcdcdefefgg).

(Discovered by Jud McCranie.)


  • Richard Gere’s middle name is Tiffany.
  • Where does the hinterland begin?
  • WORLD CUP TEAM is an anagram of TALCUM POWDER.
  • log 237.5812087593 = 2.375812087593
  • “Why is it that something can be transparent green but not transparent white?” — Wittgenstein

Round Trip

chandler perpetual motion

Eric Chandler offered this perpetual-motion scheme for Edward Barbeau’s “Fallacies, Flaws and Flimflam” column in the College Mathematical Journal. Points A and B are at the same height, and C is halfway between them. The ramp AC is a segment of a cycloid, a curve designed to produce the fastest descent under gravity.

A ball released at A rolls down the ramp AC to C covering a greater distance in a shorter time than it would have had it rolled down BC to C. The relation Velocity = Distance/Time thus implies that the ball arrives at C with greater velocity than it would have had it rolled down BC. This added velocity enables the ball to roll from C up to and past B to a point D a little farther along. It then returns to A along the inclined ramp DA to repeat the cycle endlessly.

Where is the error?

Math Notes

99999999999999999999999999999999999999999999999999999999999999 is prime.