A long run

Continuing with Eric Angelini's digit-difference/add-subtract/iterate procedure, I have now come across a particularly long run that does not repeat a previous element (i.e., confirm that a loop has been entered) until step 175128. The starting number (linking to a 32 MB evolution) is 293613.

Update (8 Nov 2012): An even longer run that does not repeat a previous element until step 337550: The starting number (linking to a 55 MB evolution) is 294066. I've made a graph of the run.

A long loop

Eric Angelini's digit-difference/add-subtract/iterate procedure, to which I drew attention two weeks ago, continues to delight. I knew there was a length-25 loop starting with 20971:

 0 20971
 1 50232
 2 102345
 3 223459
 4 212312
 5 101102
 6 211223
 7 110212
 8 98101
 9 80983
10 170138
11 841395
12 408752
13 889984 *maximum
14 879840
15 758392
16 522717
17 217055
18 49552
19 103584
20 235927
21 111172
22 110521
23 95211
24 52103
25 20971

Now I have discovered a significantly longer loop starting with 204099163.

Update (3 Nov 2012): There exists an intermediate length-85 loop starting with 17175432.

13094

Start with 13094. This is p.
From left to right, determine the absolute differences between p's adjacent digits:

1-3 = 2
3-0 = 3
0-9 = 9
9-4 = 5
4-1 = 3     (the last digit of p minus the first one)

Collecting the resulting digits, we get q = 23953.

If q > p, compute p + q. If q < p, compute p - q. Either way, this is our new p.
Repeat:

0     13094 + 23953
1     37047 + 47434
2     84481 - 40477
3     44004 -  4040
4     39964 + 60321
5    100285 ...

The procedure was created by Eric Angelini, who presented it to MathFun on October 9. I chose 13094 to illustrate the method because this is a number for which I do not have an eventual outcome. Lesser starting numbers end up in small loops, although it may take a while (for example 199, 10853, 10886) to get there. In my graph of ten million iterations of 13094's evolution, the upward climb is relentless.

Topsy-turvy

On September 24, Apple announced that it had sold 5 million iPhone 5s in the three days since the phone's introduction. This represents a profit of one to two billion American dollars. In three days! One might have expected AAPL to do better than drop $10, that day, from its $700 previous close — and another $17 the day after.

On September 27, Research in Motion announced that it had lost $235 million in its second quarter. How did NASDAQ react to this news? RIMM went from $7.14 per share to $7.50. I guess the fundamentals were sound, in the fundamentally flawed illogic of what constitutes the American stock market system.

Disclosure: I do not use a mobile phone nor have I ever put money into the stock market.

56 by 2's

two nonillion
nine trillion thirteen
one hundred thousand three hundred
thirteen billion forty-three million forty-three
?
one hundred twenty-one million two hundred thirty-six thousand six hundred thirty-six

One, two, three, four

What is the smallest English number expression that has one letter occurring once, two letters occurring twice, three letters occurring three times, and four letters occurring four times?

wordthreepeating

1, 21, 1000013, 4000414, 131256, 21445566, ________________.

Coverage

BBC news anchor this morning: "Give us the background to it. What's at the centre of it?"

Flagpole

oneseventhreesixfiveeleveneightninetwothirteenfourtentwelvesixteenfifteentwentysixfourteeneighteenseventeennineteentwentyonehundredeleventwentyonetwentyfivetwentytwotwentythreetwentyfourtwentyeightthirtyeightthirtytwentyseventwentyninethirtyonethirtyseventhirtythreethirtytwofortytwothirtyfourthirtyfivethirtysixthirtyninefortyfortyonefortythreefortyfivefortysevenfortyfourfortysixfiftyfourfortyeightfortyninefiftyonefiftysevenfiftyfiftytwofiftythreeonehundredtwelvefiftyfivefiftyeightfiftysixfiftyninesixtysixtyonesixtythreesixtyfivesixtysevensixtyfoursixtytwoseventyfoursixtysixsixtyeightsixtynineseventyseventyoneseventytwoseventythreeseventyfiveseventysixseventysevenseventyeightseventynineeightysixeightyfoureightyoneninetyfoureightythreeeightyeightytwoeightyfiveeightyseveneightyeighteightynineninetyeightninetythreeninetyninetyoneninetytwoninetyfiveninetysixninetysevenonehundredfiveninetynineonehundredonehundredtwoonehundredoneonehundredthreeonehundredfouronehundredsixonehundredeightonehundredsevenonehundrednineonehundredtenonehundredthirteen...

This is the top of the flagpole of Eric Angelini's English flagpole sequence. If we subtract the indices from the sequence, we get: 0, 5, 0, 2, 0, 5, 1, 1, -7, 3, -7, -2, -1, 2, 0, 10, -3, 0, -2, -1, -1, 89, -2, 1, -3, -3, -3, 0, 9, 0, -4, -3, -2, 3, -2, -4, 5, -4, -4, -4, -2, -2, -2, -1, 0, 1, -3, -2, 5, -2, -2, -1, 4, -4, -3, -3, 55, -3, -1, -4, -2, -2, -2, -1, 0, 1, -3, -6, 5, -4, -3, -3, -3, -3, -3, -3, -2, -2, -2, -2, -2, 4, 1, -3, 9, -3, -7, -6, -4, -3, -3, -3, 5, -1, -5, -5, -5, -3, -3, -3, 4, -3, -3, -2, -4, -3, -3, -2, -1, -3, -2, -2, 0, ... Herein, the positions of the zeros are: 1, 3, 5, 15, 18, 28, 30, 45, 65, 113, ... In effect, these are the indices of the flagpole sequence where a(n) = n. It turns out that the majority of flagpole sequence numbers lie on that line — of the first 1000000 terms, 688153. Within this region, the largest spacing between consecutive on-the-line terms is between term #321947 (=321947) and term #411273 (=411273). This large gap is due to the demand for the letter L exceeding the local supply. A much larger gap will begin ~10^9 when demand for the letter M exceeds its local supply.

Bye bye Birdo

Just a few hours shy of a 492-day stay with us, Harry Vance attempts to put a reluctant Birdo into her carrier. I will miss her anticipation of the microwave beep, half-way (or so) through my one-minute milk heat-ups, and the imitation squeaks as the window blind by her cage came up in the morning. Birdo's vocal repertoire included coughing, sneezing, and a hearty nose-blowing whenever I brought a hankie up to my face. She arrived with a couple of well-worn, two-note whistles but Catherine eventually taught her the theme to The Andy Griffith Show, which fell into disuse after I taught her I Was Kaiser Bill's Batman — phrases of which became our back-and-forth contact calling. She could do a sparrow, robin, or gull, a cat or a dog, and would not hesitate to join in at the first hint of laughter. Birdo's "English" included yeap (guessing correctly, occasionally, the answer to a query being asked of someone else), naow (said with a snarl), c'mere, c'mon, good girl, and of course Har-ree followed by — in a different voice — Birdo. She had a mock people-speak but the monologue was largely unintelligible. Catherine tried once to get her to do a happy wheee but it came out incredibly cranky when she finally succumbed to it. Birdo would greet you with a cluck and thank you with a coo. When she really wanted your attention, she'd let out an ear-piercing smoke-alarm. She had an impossible-to-describe, funny bit that we called the 'dying-parrot'. Watching her grapple with falling-asleep had me conjecture that birds evolved the behaviour to rest on one leg, with the head tucked into the back, so as to make the mathematics/mechanics of balancing that much easier. (I'm aware of the heat-loss theory but I'm not buying it as anything more than a modest advantage in limited circumstances. On the other hand, trying to maintain one's balance with a minimum of thought/correction is, high up — where a bird is likely to find itself at night, a matter of some gravity.) As for having a parrot as a room-mate, I'm now aware of how much dander and feather fluff (such) a bird generates each and every day and this detritus becomes a cleaning-burden far beyond the confines of the cage.