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

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

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).

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.)

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

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.)

Effects of lightning on Mrs. T.T. Boddington, struck as her post-chariot departed Tenbury on April 13, 1832, as reported in the Lancet:

The … electric fluid … struck the umbrella she had in her hand; it was an old one made of cotton, and had lost the ferule that is usually placed at the end of the stick, so that there was no point to attract the spark. It was literally shivered to pieces, both the springs in the handle forced out, the wires that extended the whalebones broken, and the cotton covering rent into a thousand shreds. From the wires of the umbrella the fluid passed to the wire that was attached to the edge of her bonnet, the cotton thread that was twisted round that wire marking the place of entrance, over the left eye, by its being burnt off from that spot all round the right side, crossing the back of the head and down into the neck above the left shoulder. The hair that came in contact with it was also singed; it here made a hole through the handkerchief that was round the throat, and zigzagged along the skin of the neck to the steel busk of her stays, leaving a painful but not deep wound, and also affecting the hearing of the left ear. … There were marks of burning on the gown and petticoat above the steel; and the inside of the stays, and all the garments under the stays, were pierced by the passage of the fluid to her thighs, where it made wounds on both; but that on the left so deep, and so near the femoral artery, that the astonishment is, that she escaped with life;–even as it was, the hemorrhage was very great.

It also magnetized her corset:

“Both ends attract strongly the south pole of the needle, the upper part for some considerable way down; it then begins to lose power over the south pole, and the point of northern attraction is at about one third of the length of the busk from the bottom; so that by far the greatest portion of the whole has acquired southern attraction.”