Math Notes

2015 = 4 + 8 + 4 + 9 + 3 + 3 + 1 + 9 + 6 + 7 + 7 + 7 + 1 + 1 + 4 + 0 + 7 + 4 + 1 + 4 + 3 + 9 + 5 + 8 + 6 + 0 + 1 + 2 + 9 + 6 + 0 + 4 + 7 + 1 + 9 + 1 + 3 + 0 + 3 + 2 + 3 + 8 + 8 + 8 + 8 + 3 + 2 + 8 + 4 + 9 + 3 + 7 + 8 + 8 + 8 + 9 + 3 + 4 + 1 + 7 + 1 + 2 + 4 + 6 + 3 + 6 + 4 + 6 + 1 + 2 + 8 + 2 + 5 + 3 + 5 + 5 + 3 + 5 + 1 + 9 + 7 + 0 + 3 + 0 + 8 + 1 + 3 + 9 + 9 + 2 + 8 + 9 + 5 + 7 + 8 + 3 + 3 + 0 + 8 + 5 + 9 + 5 + 6 + 8 + 3 + 2 + 6 + 3 + 1 + 8 + 8 + 3 + 5 + 1 + 0 + 9 + 0 + 1 + 9 + 6 + 3 + 2 + 2 + 5 + 1 + 3 + 8 + 3 + 9 + 4 + 4 + 5 + 9 + 5 + 9 + 2 + 7 + 1 + 1 + 5 + 7 + 6 + 4 + 9 + 5 + 1 + 8 + 0 + 4 + 1 + 5 + 4 + 0 + 1 + 8 + 3 + 1 + 4 + 6 + 5 + 9 + 6 + 1 + 7 + 1 + 1 + 0 + 2 + 0 + 2 + 2 + 8 + 3 + 0 + 0 + 8 + 9 + 6 + 5 + 0 + 2 + 9 + 2 + 1 + 7 + 7 + 5 + 8 + 7 + 7 + 9 + 7 + 2 + 9 + 0 + 1 + 8 + 8 + 6 + 9 + 1 + 2 + 8 + 2 + 7 + 7 + 5 + 4 + 3 + 6 + 0 + 9 + 5 + 4 + 1 + 0 + 1 + 0 + 0 + 3 + 8 + 4 + 0 + 4 + 2 + 1 + 8 + 1 + 2 + 8 + 9 + 4 + 6 + 0 + 8 + 3 + 7 + 6 + 3 + 8 + 8 + 8 + 5 + 5 + 9 + 2 + 4 + 7 + 7 + 5 + 3 + 0 + 7 + 4 + 6 + 1 + 6 + 8 + 6 + 4 + 3 + 7 + 0 + 2 + 5 + 8 + 1 + 3 + 7 + 9 + 8 + 7 + 3 + 3 + 6 + 3 + 4 + 7 + 5 + 8 + 8 + 5 + 2 + 6 + 5 + 6 + 6 + 6 + 6 + 0 + 8 + 6 + 0 + 6 + 6 + 9 + 0 + 8 + 2 + 5 + 6 + 1 + 1 + 3 + 2 + 6 + 9 + 6 + 0 + 9 + 4 + 5 + 2 + 4 + 2 + 5 + 1 + 2 + 5 + 7 + 6 + 8 + 8 + 0 + 6 + 9 + 9 + 8 + 7 + 1 + 3 + 4 + 0 + 9 + 4 + 6 + 1 + 5 + 0 + 8 + 4 + 3 + 6 + 8 + 3 + 1 + 5 + 2 + 3 + 4 + 0 + 1 + 0 + 3 + 5 + 3 + 4 + 1 + 8 + 1 + 0 + 3 + 2 + 6 + 6 + 3 + 0 + 1 + 4 + 1 + 9 + 3 + 7 + 7 + 6 + 8 + 2 + 0 + 6 + 8 + 2 + 7 + 2 + 3 + 6 + 4 + 4 + 3 + 4 + 2 + 2 + 2 + 3 + 9 + 8 + 5 + 3 + 6 + 2 + 0 + 2 + 1 + 6 + 5 + 9 + 5 + 6 + 6 + 3 + 0 + 8 + 0 + 7 + 2 + 4 + 7 + 5 + 4 + 0 + 4 + 6 + 0 + 0 + 2 + 5 + 4 + 4 + 7 + 8 + 2 + 8 + 2 + 2 + 9 + 5 + 1 + 6 + 7 + 4 + 4 + 6 + 1 + 3 + 6 + 4 + 7 + 4 + 6 + 0 + 9 + 3 + 7 + 5

2015137 = 484933196777114074143958601296047191303238888328493788893417124636461282
535535197030813992895783308595683263188351090196322513839445959271157649518041540
183146596171102022830089650292177587797290188691282775436095410100384042181289460
837638885592477530746168643702581379873363475885265666608606690825611326960945242
512576880699871340946150843683152340103534181032663014193776820682723644342223985
362021659566308072475404600254478282295167446136474609375

(Thanks, Pablo.)

| Science & Math

Misc

  • Most Muppets are left-handed.
  • The largest prime number in the Bible is 22273 (Numbers 3:43).
  • SEE, HE, and IS are spelled identically in Morse code (ignoring spaces).
  • Maine is the only one-syllable state name.
  • “More things grow in the garden than the gardener sows.” — Spanish proverb

Fritz Zwicky referred to his colleagues at the Mount Wilson Observatory as “spherical bastards” because they were bastards whichever way one looked at them.

| Language · Science & Math · Trivia

In a Word

http://commons.wikimedia.org/wiki/File:MazurGes.jpg

anserine
adj. of or resembling a goose

In 1936, Polish mathematician Stanislaw Mazur offered a live goose to the first person who could determine whether every Banach space has a Schauder basis. Thirty-seven years later, his Swedish colleague Per Enflo claimed the prize. The ceremony was broadcast throughout Poland. (Thanks, Jeremy.)

In the public gardens at Halifax, there is an eccentric goose that seems to manifest a genuine affection. Whenever a certain old gentleman, whose name we do not know, approaches the pond and calls ‘Bobby,’ the goose will leave the pond and sit beside him, and when he leaves to go home, will follow close at his feet, like a dog, to the gate, and sometimes into the street, when it has to be forcibly put back, to its manifest disgust, for it goes off to its native element twisting its tail with indignation, and giving vent to sundry discordant squeaks. The old gentleman says he has never fed it, or petted it in any way, which makes it more remarkable; but we are told by a frequenter of the gardens that about two or three years ago a man used to come there and feed this identical goose regularly, so we are inclined to think that it is a case of mistaken identity on the part of his gooseship. Anyway, it is an interesting question for ornithologists to solve, whether geese (supposed to be the most stupid of birds) have memory and can experience the sensation of gratitude.

— James Baird McClure, ed., Entertaining Anecdotes From Every Available Source, 1879

| Language · Oddities · Science & Math

Math Notes

14 + 50 + 54 = 15 + 40 + 63 = 18 + 30 + 70 = 21 + 25 + 72
14 × 50 × 54 = 15 × 40 × 63 = 18 × 30 × 70 = 21 × 25 × 72

6 + 56 + 75 = 7 + 40 + 90 = 9 + 28 + 100 = 12 + 20 + 105
6 × 56 × 75 = 7 × 40 × 90 = 9 × 28 × 100 = 12 × 20 × 105

| Science & Math

Narcissus Prime

Repeat the string 1808010808 1560 times, and tack on a 1 the end:

tetradic prime

The resulting 15601-digit number is prime, and because it’s a palindrome made up of the digits 1, 8, and 0, it remains prime when read backward, upside down, or in a mirror.

Unrelated mirror curiosity: Outline the reflection of your head on the glass of a mirror, then back away. At any distance, the image continues to fill the outline (which is half the size of your head).

| Science & Math

A Study in Oils

http://commons.wikimedia.org/wiki/File:2008_inside_the_National_Portrait_Gallery,_London.jpg
Image: Wikimedia Commons

In seeking to understand how a person’s ability might vary with his complexion, Havelock Ellis chose an unusual data set: the National Portrait Gallery. Ellis spent two years examining paintings of notable Britons in various fields and established an “index of pigmentation” in each group by multiplying the number of fair people by 100 and dividing by the number of dark people. Results:

http://books.google.com/books?id=jCo4AAAAMAAJ&source=gbs_navlinks_s

An index greater than 100 means that fair people predominate in the group; one less than 100 means that dark people predominate. The list includes both men and women.

In general, Ellis concluded, the fair man tends to be “bold, energetic, restless, and domineering,” while the dark man is “resigned and religious and imitative, yet highly intelligent.” “While the men of action thus tend to be fair, the men of thought, it seems to me, show some tendency to be dark.”

Ellis speculated that the British aristocracy tended to be dark because peers could choose the most beautiful women, and British women with the greatest reputation for beauty tended to be dark: a group of 15 English women of letters had an index of 100, while 13 famous beauties rated 44.

(“The Comparative Abilities of the Fair and the Dark,” Monthly Review, August 1901.)

| Art · Science & Math

Math Notes

1/473684210526315789 = 0.0000000000000000021111111111111111132222222222222
22224333333333333333335444444444444444446555555555555555557666666666666666
66877777777777777777988888888888888889100000000000000000211111111111111111
32222222222222222243333333333333333354444444444444444465555555555555555576
66666666666666668777777777777777779888888888888888891000000000000000002111
11111111111111322222222222222222433333333333333333544444444444444444655555
55555555555576666666666666666687777777777777777798888888888888888910000000
00000000002111111111111111113222222222222222224333333333333333335444444444
44444444655555555555555555766666666666666666877777777777777777988888888888
88888910000000000000000021111111111111111132222222222222222243333333333333
33335444444444444444446555555555555555557666666666666666668777777777777777
77988888888888888889100000000000000000211111111111111111322222222222222222
43333333333333333354444444444444444465555555555555555576666666666666666687
77777777777777779888888888888888891000000000000000002111111111111111113222
22222222222222433333333333333333544444444444444444655555555555555555766666
66666666666687777777777777777798888888888888888910000000000000000021111111
11111111113222222222222222224333333333333333335444444444444444446555555555
55555555766666666666666666877777777777777777988888888888888889100000000000
00000021111111111111111132222222222222222243333333333333333354444444444444
44446555555555555555557666666666666666668777777777777777779888888888888888
89100000000000000000211111111111111111322222222222222222433333333333333333
54444444444444444465555555555555555576666666666666666687777777777777777798
88888888888888891000000000000000002111111111111111113222222222222222224 ...

(Thanks, William.)

| Science & Math

A Niente

I was seriously tormented by the thought of the exhaustibility of musical combinations. The octave consists only of five tones and two semitones, which can be put together in only a limited number of ways, of which but a small proportion are beautiful: most of these, it seemed to me, must have been already discovered, and there could not be room for a long succession of Mozarts and Webers, to strike out, as these had done, entirely new and surpassingly rich veins of musical beauty. This source of anxiety may, perhaps, be thought to resemble that of the philosophers of Laputa, who feared lest the sun should be burnt out.

— John Stuart Mill, Autobiography, 1873

| Science & Math

“Scooping the Loop Snooper”

Given a particular input, will a computer program eventually finish running, or will it continue forever?

That sounds straightforward, but in 1936 Alan Turing showed that it’s undecidable: It’s impossible to devise a general algorithm that can answer this question for every possible program and input.

The most charming proof of this was published in 2000 by University of Edinburgh linguist Geoffrey Pullum — he did it in the style of Dr. Seuss:

No program can say what another will do.
Now, I won’t just assert that, I’ll prove it to you:
I will prove that although you might work til you drop,
You can’t predict whether a program will stop.

Imagine we have a procedure called P
That will snoop in the source code of programs to see
There aren’t infinite loops that go round and around;
And P prints the word “Fine!” if no looping is found.

You feed in your code, and the input it needs,
And then P takes them both and it studies and reads
And computes whether things will all end as they should
(As opposed to going loopy the way that they could).

Well, the truth is that P cannot possibly be,
Because if you wrote it and gave it to me,
I could use it to set up a logical bind
That would shatter your reason and scramble your mind.

Here’s the trick I would use — and it’s simple to do.
I’d define a procedure — we’ll name the thing Q —
That would take any program and call P (of course!)
To tell if it looped, by reading the source;

And if so, Q would simply print “Loop!” and then stop;
But if no, Q would go right back to the top,
And start off again, looping endlessly back,
Til the universe dies and is frozen and black.

And this program called Q wouldn’t stay on the shelf;
I would run it, and (fiendishly) feed it itself.
What behaviour results when I do this with Q?
When it reads its own source, just what will it do?

If P warns of loops, Q will print “Loop!” and quit;
Yet P is supposed to speak truly of it.
So if Q’s going to quit, then P should say, “Fine!” —
Which will make Q go back to its very first line!

No matter what P would have done, Q will scoop it:
Q uses P’s output to make P look stupid.
If P gets things right then it lies in its tooth;
And if it speaks falsely, it’s telling the truth!

I’ve created a paradox, neat as can be —
And simply by using your putative P.
When you assumed P you stepped into a snare;
Your assumptions have led you right into my lair.

So, how to escape from this logical mess?
I don’t have to tell you; I’m sure you can guess.
By reductio, there cannot possibly be
A procedure that acts like the mythical P.

You can never discover mechanical means
For predicting the acts of computing machines.
It’s something that cannot be done. So we users
Must find our own bugs; our computers are losers!

Pullum, Geoffrey K. (2000) “Scooping the loop snooper: An elementary proof of the undecidability of the halting problem.” Mathematics Magazine 73.4 (October 2000), 319-320.

(Thanks, Pål.)

| Science & Math

Misc

  • There’s no “u” in solipsism.
  • Wagner said the saxophone “sounds like the word Reckankreuzungsklankewerkzeuge.”
  • FDR was related by blood or marriage to 11 other presidents.
  • 3909511 = 53 + 59 + 50 + 59 + 55 + 51 + 51
  • “If you can’t stand the heat, stay out of the chicken.” — Ted Giannoulas, San Diego Chicken

(Thanks, Eric.)

| Language · Quotations · Science & Math