Home, sweet home

Yesterday, the replacement for my nine-year-old dual 2-GHz Power-PC G5 arrived and — as suggested in the final entry of my old blog — in the wee hours of this morning I purchased and subsequently downloaded the home edition of Mathematica 9.

The 27" iMac (2012) had been maxed out with what I could add at the Apple store, with the exception of 32 GB RAM which I had ordered from a third party and installed myself.

For the most part the setup went smoothly. I had to figure out how to do my web-sharing, resolve a will-not-fetch-mail issue, and resign myself to not using Java for now. I booted up Mathematica and asked for $MaxNumber. It suggested: 8.768126706828697*10^2711437152599256. So I knew I could do a lot better than the 180 million terms of the pi continued-fraction generated back in 2002. Requesting one billion terms, the calculation required just over one hour!

Time wasted

Since November 19, I've been trying to find the eighth term of A219325 and have only just managed to get up to five billion. Giovanni Resta has now beat me to it with 34505916416. The binary circulant that results from this number is:

1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 1 1 0 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 1 1 0 1
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 1 1 0
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 1 1
1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 1
1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0
0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1
1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1
1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0
0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1
1 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0
0 1 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0
0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0
0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
1 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
0 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
0 0 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

Keyboards

Design — as Apple (of all companies) should know — is how it works. When Tim Cook was asked recently how adept he was at using Apple's virtual keyboard, he replied: "Pretty good. I think if you stick with it a little while you get quite good at it. And the auto correction is quite good." Ouch!

Apple's virtual keyboard suffers of course from a fundamental design flaw. It fails to adequately mimic an actual keyboard by splitting it into two discrete entities, the second of which sports the numbers and some much-used punctuation (such as the apostrophe and the hyphen). This becomes a difficult-to-ignore annoyance in message chats on my iPad and would be relatively easy to fix with a properly designed, replacement virtual keyboard. I'm aware of course that space/size concerns might limit what is reasonably possible but intelligent designers would figure out a way. The reason they have not done so (unless perhaps there is a design disconnect at Apple these days) surely is to steer folk into Apple's $70 wireless keyboard accessory solution!

So, in order to better enjoy my iPad Messages chats, I purchased an Apple wireless keyboard. The first thing I grappled with was why there wasn't an obvious on/off indicator light on the keyboard. This design flaw may be overcome by noting that the caps-lock light will not come on when the keyboard is off (an extra and surely unnecessary step). The second thing I dealt with is that there is no way to send my wireless-keyboard typed blurbs without touching the Send button on the iPad. This is of course an easily fixed software limitation — except that Apple hasn't addressed the issue in the several years that people have been complaining about it! Maybe Apple really has lost its design moxie.

What's so special about 798 and 1182?

I've been working on A219357 for nine days now. I'm especially proud of my graph, where the two wayward points (798, 2164818573) and (1182, 5394998141) are easily picked out.

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?