Science & Math

Math Notes

multigrade equation

Never Mind

“It has been asserted (by C.S. Lewis, for instance) that no determinist rationally can believe in determinism, for if determinism is true, his beliefs were caused, including his belief in determinism. The idea seems to be that the causes of belief, perhaps chemical happenings in the brain, might be unconnected with any reasons for thinking determinism true. They might be, but they need not be. The causes might ‘go through’ reasons and be effective only to the extent that they are good reasons.”

— Robert Nozick, “Reflections on Newcomb’s Paradox,” 1974

“If … [determinism] is true, then the intellectual or cognitive operations of its upholders, including their choice or decision to maintain the thesis, … are themselves only the effects of inexorable forces. But if this is so, why should the thesis … be accepted as valid or true?”

— Alan Gewirth, Reason and Morality, 1978

Pandigital Approximations

The digits 1-9 can work some impressive tricks:
The first formula, found by B. Ziv in 2004, produces the first 10 digits of pi.

The second, astonishingly, reproduces e to 18,457,734,525,360,901,453,873,570 decimal places. It was discovered by Richard Sabey, also in 2004.

(Thanks, Robin.)

Party of One

“A man wrote to say that he accepted nothing but Solipsism, and added that he had often wondered it was not a more common philosophy. Now Solipsism simply means that a man believes in his own existence, but not in anybody or anything else. And it never struck this simple sophist, that if his philosophy was true, there obviously were no other philosophers to profess it.”

— G.K. Chesterton, St. Thomas Aquinas, 1933

Digit Acrobatics

214358976 = (3 + 6)2 + (4 + 7)8 + (5 + 9)1

“Triangle Rhyme”

Although the altitudes are three,
Remarks my daughter Rachel,
One point’ll lie on all of them:
The orthocenter H’ll.

By mathematician Dwight Paine of Messiah College, 1983.

(Further recalcitrant rhymes: month, orange. W.S. Gilbert weighs in.)


  • River Phoenix was born River Bottom.
  • Every natural number is the sum of four squares.
  • What happens if Pinocchio says, “My nose will grow now”?
  • Shakespeare has no living descendants.
  • “All generalizations are dangerous — even this one.” — Dumas

Math Notes

000002569 is prime.

You Are Here

In “Partial Magic in the Quixote,” Borges writes:

Let us imagine that a portion of the soil of England has been levelled off perfectly and that on it a cartographer traces a map of England. The job is perfect; there is no detail of the soil of England, no matter how minute, that is not registered on the map; everything has there its correspondence. This map, in such a case, should contain a map of the map, which should contain a map of the map of the map, and so on to infinity.

This sequence tends to a single point, the point on the map that corresponds directly to the point it represents in the territory.

Cover England entirely with a 1:1 map of itself, then crumple the map into a ball. So long as it remains in England, the balled map will always contain at least one point that lies directly above the corresponding point in England.

See Garganta and Papered Over.



Draw a semicircle and surmount it with two smaller semicircles.

A line drawn through A, at any angle, will divide the perimeter precisely in half.

This probably has some romantic symbolism, but I’m not very good at that stuff.

Squares Into Squares

squares into squares


Imagine you have a little robot that holds a pencil. Set it down on a sheet of paper and give it these instructions:

  1. Move forward 3 units and turn right.
  2. Move forward 1 unit and turn right.
  3. Move forward 2 units and turn left.
  4. Move forward 1 unit and turn left.
  5. Move forward 2 units and turn right.
  6. Repeat.

If the robot makes its turns at 90° angles, it will produce this figure:


But, remarkably, if it turns at 120° it will draw this:

star of david

The Grid Flower

ellipses and hyperbolas

Any pair of points define an infinity of ellipses and an infinity of hyperbolas.

The ellipses do not touch one another, nor do the hyperbolas.

But every ellipse meets every hyperbola at a right angle.

The Black Hole

unilluminable room

If Satan plays miniature golf, this is his favorite hole. A ball struck at A, in any direction, will never find the hole at B — even if it bounces forever.

The idea arose in the 1950s, when Ernst Straus wondered whether a room lined with mirrors would always be illuminated completely by a single match.

Straus’ question went unanswered until 1995, when George Tokarsky found a 26-sided room with a “dark” spot; two years later D. Castro offered the 24-sided improvement above. If a candle is placed at A, and you’re standing at B, you won’t see its reflection anywhere around you — even though you’re surrounded by mirrors.

The Franklin Square

franklin magic square

In a 1769 letter, Ben Franklin describes a magic square he devised in his youth. The magic total of 260 can be reached by adding the numbers in each row or column, as in a normal magic square. But “bent rows” (shaded) produce the same total, even when “wrapped across” the border of the table. This works in all four directions.

Further: Half of each row or column sums to half of 260, as does any 2×2 subsquare. And the four corners and the four center squares sum to 260. (Alas, the main diagonals don’t, so this doesn’t strictly qualify as a magic square by the modern definition.)

Interestingly, no one knows how Franklin created the square. Many methods have been devised, but none apparently as quick as his, which he claimed could generate them “as fast as he could write.”

The Perfect Family

Take any Platonic solid, join the centers of its faces, and, charmingly, you get another Platonic solid. The cube and the octahedron produce one another, as do the dodecahedron and the icosahedron, and the tetrahedron produces another tetrahedron.

Bonus factoid: If you inscribe a dodecahedron and an icosahedron in the same sphere, the dodecahedron will occupy more of the sphere’s volume. It has fewer faces than the icosahedron, but its faces are more nearly circular, so it fits the sphere more snugly.

See The Pup Tent Problem.

Mathematicians’ Graves

Archimedes wanted no other epitaph than a sphere inscribed within a cylinder — he had determined the sphere’s relative volume and considered this his greatest achievement.

Henry Perigal’s tomb in Essex displays his graphic proof of the Pythagorean theorem (left).

Gauss wanted to be buried under a heptadecagon, which he’d shown can be constructed with compass and straightedge. (The stonemason demurred, fearing he’d produce only a circle.)

And Jakob Bernoulli opted for a logarithmic spiral and the words Eadem mutata resurgo—the motto means “I shall arise the same though changed.”

Benardete’s String Paradox

benardete's string paradox

Let us take a piece of string. In the first half minute we shall form an equilateral triangle with the string; in the next quarter minute we shall employ the string to form a square; in the next eighth minute we shall form a regular pentagon; etc. ad infinitum. At the end of the minute what figure or shape will our piece of string be found to have assumed? Surely it can only be a circle. And yet how intelligible is that process? Each and every one of the polygons in our infinite series contains only a finite number of sides. There is thus a serious conceptual gap separating the circle, as in the limiting case, from each and every polygon in the infinite series.

— Jose Amado Benardete, Infinity: An Essay in Metaphysics, 1964

Math Notes

Turn each of these palindromes “inside out” and their sum remains the same:

13031 + 42024 + 53035 + 57075 + 68086 + 97079 = 31013 + 24042 + 35053 + 75057 + 86068 + 79097

Remarkably, this holds true even if you square or cube them:

130312 + 420242 + 530352 + 570752 + 680862 + 970792 = 310132 + 240422 + 350532 + 750572 + 860682 + 790972

130313 + 420243 + 530353 + 570753 + 680863 + 970793 = 310133 + 240423 + 350533 + 750573 + 860683 + 790973

From Albert Beiler, Recreations in the Theory of Numbers, 1964.


  • Douglas Adams claimed that the funniest three-digit number is 359.
  • Romeo has more lines than Juliet, Iago than Othello, and Portia than Shylock.
  • Friday the 13th occurs at least once a year.
  • “By nature, men love newfangledness.” — Chaucer
  • John was the only apostle to die a natural death.

The Paradox of Dives and Lazarus

The Gospel of Luke contains a parable about a rich man and a beggar. Both men die, and the rich man is consigned to hell while the beggar is received into the bosom of Abraham. The rich man pleads for mercy, but Abraham tells him that in his lifetime he received good things and the beggar evil things: “now he is comforted and thou art tormented.” The rich man then begs that his brothers be warned of what lies in store for them, but Abraham rejects this plea as well, saying, “If they hear not Moses and the prophets, neither will they be persuaded though one rose from the dead.”

Now, writes E.V. Milner:

Suppose … that this last request of Dives had been granted; suppose, in fact, that some means were found to convince the living, whether rich men or beggars, that ‘justice would be done’ in a future life, then, it seems to me, an interesting paradox would emerge. For if I knew that the unhappiness which I suffer in this world would be recompensed by eternal bliss in the next world, then I should be happy in this world. But being happy in this world I should fail to qualify, so to speak, for happiness in the next world. Therefore, if there were such a recompense awaiting me, its existence would seem to entail that I should at least be not wholly convinced of its existence.

“Put epigrammatically, it would appear that the proposition ‘Justice will be done’ can only be true for one who believes it to be false. For one who believes it to be true justice is being done already.”


Write the numbers 82 to 1 in descending order and string them together:


The resulting 155-digit number is prime.

Benardete’s Book Paradox

Here is a book lying on a table. Open it. Look at the first page. Measure its thickness. It is very thick indeed for a single sheet of paper — one half inch thick. Now turn to the second page of the book. How thick is this second sheet of paper? One fourth inch thick. And the third page of the book, how thick is this third sheet of paper? One eighth inch thick, etc. ad infinitum. We are to posit not only that each page of the book is followed by an immediate successor the thickness of which is one half that of the immediately preceding page but also (and this is not unimportant) that each page is separated from page 1 by a finite number of pages. These two conditions are logically compatible: there is no certifiable contradiction in their joint assertion. But they mutually entail that there is no last page in the book. Close the book. Turn it over so that the front cover of the book is now lying face down upon the table. Now, slowly lift the back cover of the book with the aim of exposing to view the stack of pages lying beneath it. There is nothing to see. For there is no last page in the book to meet our gaze.

— Patrick Hughes and George Brecht, Vicious Circles and Infinity, 1978

The Answer Wheel

answer wheel

Multiply 212765957446808510638297872340425531914893617 by any number from 2 to 46 and you’ll find the product on the ring above.

Einstein wrote, “Pure mathematics is, in its way, the poetry of logical ideas.”