Friday, April 21

Aronson's sequence

I was made aware of Aronson's sequence by Greg Ross' Futility Closet article on it three weeks ago. A couple of things caught my eye. The first was his use of "nine billion one million second" to example the "few T-less ordinals" that "don’t arrange themselves to mop up all the incoming Ts". It would have been a little more compelling if 9001000002 was actually in the sequence — which it is not. The closest t-free ordinal that is is 9001000702.

The second thing was Greg's "We had supposed that the sentence would end with … letter in this sentence. But an infinite sentence has no end..." English number names have been well-defined only up to 10^66-1 — although I fully expect (once Mathematica debugs its IntegerName function) that that will go up to 10^306-1. Thus all strictly increasing, current English number-name sequences are necessarily finite, whether or not it is so recognized.

Sunday, April 16

Words from numbers

Last week I presented a "word" continuation puzzle. The algorithm used to create the list isn't too difficult to discover, applying English number words (one, two, three, ...) to the previous term (the zeroth assumed to be an empty string). Thus, one letter at a time, the second term from the first:

one +two
onet
onetw
netw

A letter gets added to the right if it doesn't already exist in the evolving string. It is deleted from the string if it does already exist. Thus the string will never contain more than one copy of any particular letter. If you noticed the double-comma near the end of my puzzle, that wasn't a typo: ourihten +onehundrednineteen results in an empty string, which +onehundredtwenty yields ohurweny, which +onehundredtwentyone yields one. Using Mathematica's built-in dictionary and ignoring already encountered words (such as one at index 121), here is a list of English found in a deep continuation:

         1 one
     21240 visaed
     45660 fads
     57242 ado
    155868 woad
    171524 aide
    271966 ad
    337664 waned
    347660 audit
    413700 and
    423066 roads
    507504 wained
    537056 goads
    557924 aid
    615808 wad
    619808 wade
    635830 wand
   1152766 mad
   1250766 moaned
   1272524 maid
   1298168 made
   2710904 maned
   3526644 mashed
  10984236 mawed
  16170624 maiden
  21730304 mated
  67092006 mead
 509056060 remands
 540798800 moated
1000080796 boards
1000146526 bards
1000152766 bad
1000298168 bade
1000530740 baud
1000558076 broads
1000562062 brands
1000748080 bandit
1000750040 band
1000816952 bandy
2000710904 baned
...

Why would all of our subsequent English dictionary words appear at even indices?

Sunday, April 9

What's next?

one, netw, nwhre, nwhefou, nwhouiv, nwhouvsx, whoux, wouxeigt, wouxgt, wouxgen, wouxglv, ouxgt, ouxghirtn, xghif, xghftn, ghfsi, ghfivt, fven, fvit, fviwenty, fvitone, fviy, fviwnthre, vihtyou, houwntf, houfetysix, houfixwtv, oufxvnyg, oufxvgwi, oufxvgwhry, ufxvgwine, ufxvgnehryto, ufxvgnoihre, xvgneyr, xgnhf, gnfrys, gfhiv, fvryeiht, fvti, viory, viftone, vinerytwo, vinwfthre, vinwheyfur, nwhuotf, nwhurysix, whuixfotv, wuxvryegt, wuxvgfoi, wuxvgofty, wuxvgifne, uxvgnefyo, uxvgnoifhre, xvgnhety, xgnhf, gnhftys, ghifv, vfyei, vfti, vfsxy, vfitone, vfnesxytw, vfnwithre, vnwhesxyou, nwhoutf, nwhoufy, whoufixtv, woufvsyeigt, woufvgxi, woufgxisnty, wufgxivne, ufgxiseyo, ufgxiovnhre, gxihsety, gxhnf, ghfvtyi, ghfiv, fsnye, fvti, fveghy, fvitone, fvnghytw, fvnwithre, vnwgyou, nwouhtf, nwoufegysx, woufxihtv, woufxvyiht, woufxvgi, woufxvgety, wufxvgine, ufxvgnyo, ufxvgoinhre, xvghnty, xghnf, ghfnetys, ghfiv, fvyi, fvti, fvtiohur, fvtione, fvihunrewo, fviwnthre, viwtner, wtohunf, wtfnrsix, wtfixohuv, wfxvregh, wfxvgoui, wfxvgihrten, wfxgitoul, fxgihrv, fxgvouhrn, xgvourt, xgvhin, gvourst, gh, ourihten, , ohurweny, ?

Friday, April 7

Counting t-free ordinals

Yesterday I introduced written-out English ordinals that lacked the letter "t" and I asked how many there are less than "one vigintillionth". I have come to conclude that the total number of t-free ordinals may be expressed by (c+1)^x*o, where 'c' is the number of t-free cardinals less than 1000, 'x' is the number of t-free -illions through which we traverse, and 'o' is the number of t-free ordinals less than 1000.

c = 55   (1, 4, 5, 6, 7, 9, 11, 100, 101, 104, 105, 106, 107, 109, 111, 400, 401, 404, 405, 406, 407, 409, 411, 500, 501, 504, 505, 506, 507, 509, 511, 600, 601, 604, 605, 606, 607, 609, 611, 700, 701, 704, 705, 706, 707, 709, 711, 900, 901, 904, 905, 906, 907, 909, 911)

x = 10   (10^6, 10^9, 10^15, 10^30, 10^33, 10^36, 10^39, 10^48, 10^51, 10^60)

o =  7   (2, 102, 402, 502, 602, 702, 902)

So we have 56^10*7 = 2123138423672799232.

The latest version of Mathematica has a built-in IntegerName function that does both cardinals and ordinals:

count1[ncard_] := 
 Length[Select[Range[10^(ncard - 1), 10^ncard - 1], 
   StringFreeQ[IntegerName[#, "Words"], "t"] &]]; 
m = {count1[1], count1[2], count1[3]}

{6, 1, 48}

... The number of t-free cardinals of 1-digit, 2-digit, and 3-digit base-ten numbers.

count2[nord_] := 
 Length[Select[Range[10^(nord - 1), 10^nord - 1], 
   StringFreeQ[IntegerName[#, "Ordinal"], "t"] &]]; 
s = {count2[1], count2[2], count2[3]}

{1, 0, 6}

... The number of t-free ordinals of 1-digit, 2-digit, and 3-digit base-ten numbers.

Do[If[StringFreeQ[IntegerName[10^(3*i), "Words"], "t"], 
  s = Join[s, m*Total[s]], s = Join[s, {0, 0, 0}]], {i, 21}]; s

{1, 0, 6, 0, 0, 0, 42, 7, 336, 2352, 392, 18816, 0, 0, 0, 131712, 21952, 1053696, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7375872, 1229312, 59006976, 413048832, 68841472, 3304390656, 23130734592, 3855122432, 185045876736, 1295321137152, 215886856192, 10362569097216, 0, 0, 0, 0, 0, 0, 72537983680512, 12089663946752, 580303869444096, 4062127086108672, 677021181018112, 32497016688869376, 0, 0, 0, 0, 0, 0, 227479116822085632, 37913186137014272, 1819832934576685056, 0, 0, 0}

... The number of t-free ordinals of 1-digit to 66-digit base-ten numbers. Finally:

Total[s]

2123138423672799232

IntegerName[%, "Words"]

two quintillion, one hundred twenty-three quadrillion, one hundred thirty-eight trillion, four hundred twenty-three billion, six hundred seventy-two million, seven hundred ninety-nine thousand, two hundred thirty-two

Interestingly, Mathematica has attempted to bridge the gap between the dictionary large-number names up to 10^63 (one vigintillion) and the next dictionary entry at 10^303 (one centillion):

Table[{i, IntegerName[10^i, "Words"]}, {i, 63, 306, 3}] // TableForm

 63 one vigintillion
 66 one unvigintillion
 69 one duovigintillion
 72 one trevigintillion
 75 one quattuorvigintillion
 78 one quinvigintillion
 81 one sexvigintillion
 84 one septenvigintillion
 87 one octovigintillion
 90 one novemvigintillion
 93 one trigintillion
 96 one untrigintillion
 99 one duotrigintillion
102 one trestrigintillions
105 one quattuortrigintillions
108 one quintrigintillions
111 one sextrigintillions
114 one septrigintillions
117 one octotrigintillions
120 one novemtrigintillions
123 one quadragintillions
126 one unquadragintillions
129 one duoquadragintillions
132 one tresquadragintillions
135 one quattuorquadragintillions
138 one quinquadragintillions
141 one sexquadragintillions
144 one septenquadragintillions
147 one octoquadragintillions
150 one novemquadragintillions
153 one quinquagintillions
156 one unquinquagintillions
159 one duoquinquagintillions
162 one tresquinquagintillions
165 one quattuorquinquagintillions
168 one quinquinquagintillions
171 one sexquinquagintillions
174 one septenquinquagintillions
177 one octoquinquagintillions
180 one novemquinquagintillions
183 one sexagintillions
186 one unsexagintillions
189 one duosexagintillions
192 one tresexagintillions
195 one quattuorsexagintillions
198 one quinsexagintillions
201 one sesexagintillions
204 one septensexagintillions
207 one octosexagintillions
210 one novemsexagintillions
213 one septuagintillions
216 one unseptuagintillions
219 one duoseptuagintillions
222 one treseptuagintillions
225 one quattuorseptuagintillions
228 one quinseptuagintillions
231 one seseptuagintillions
234 one septenseptuagintillions
237 one octoseptuagintillions
240 one novemseptuagintillions
243 one octogintillions
246 one unoctogintillions
249 one duooctogintillions
252 one tresoctogintillions
255 one quattuoroctogintillions
258 one quintoctogintillions
261 one sexoctogintillions
264 one septenoctogintillions
267 one octoctogintillions
270 one novoctogintillions
273 one nonagintillions
276 one unonagintillions
279 one duononagintillions
282 one trenonagintillions
285 one quattuornonagintillions
288 one quinonagintillions
291 one senonagintillions
294 one septenonagintillions
297 one octononagintillions
300 one novenonagintillions
303 one centillions
306 one billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion billion

Notice the terminal "s" for 10^102 to 10^303. I've alerted Wolfram to the bug. Then, starting at 10^306, all hell breaks loose!

Thursday, April 6

English t-free ordinals

second
one hundred second
four hundred second
five hundred second
six hundred second
seven hundred second
nine hundred second
one million second
one million one hundred second
one million four hundred second
one million five hundred second
one million six hundred second
one million seven hundred second
one million nine hundred second
four million second
four million one hundred second
four million four hundred second
four million five hundred second
four million six hundred second
four million seven hundred second
four million nine hundred second
five million second
five million one hundred second
five million four hundred second
five million five hundred second
five million six hundred second
five million seven hundred second
five million nine hundred second
six million second
six million one hundred second
six million four hundred second
six million five hundred second
six million six hundred second
six million seven hundred second
six million nine hundred second
seven million second
seven million one hundred second
seven million four hundred second
seven million five hundred second
seven million six hundred second
seven million seven hundred second
seven million nine hundred second
nine million second
nine million one hundred second
nine million four hundred second
nine million five hundred second
nine million six hundred second
nine million seven hundred second
nine million nine hundred second
eleven million second
eleven million one hundred second
eleven million four hundred second
eleven million five hundred second
eleven million six hundred second
eleven million seven hundred second
eleven million nine hundred second
one hundred million second
one hundred million one hundred second
one hundred million four hundred second
one hundred million five hundred second
one hundred million six hundred second
one hundred million seven hundred second
one hundred million nine hundred second
one hundred one million second
one hundred one million one hundred second
one hundred one million four hundred second
one hundred one million five hundred second
one hundred one million six hundred second
one hundred one million seven hundred second
one hundred one million nine hundred second
one hundred four million second
one hundred four million one hundred second
one hundred four million four hundred second
one hundred four million five hundred second
one hundred four million six hundred second
one hundred four million seven hundred second
one hundred four million nine hundred second
one hundred five million second
one hundred five million one hundred second
one hundred five million four hundred second
one hundred five million five hundred second
one hundred five million six hundred second
one hundred five million seven hundred second
one hundred five million nine hundred second
one hundred six million second
one hundred six million one hundred second
one hundred six million four hundred second
one hundred six million five hundred second
one hundred six million six hundred second
one hundred six million seven hundred second
one hundred six million nine hundred second
one hundred seven million second
one hundred seven million one hundred second
one hundred seven million four hundred second
one hundred seven million five hundred second
one hundred seven million six hundred second
one hundred seven million seven hundred second
one hundred seven million nine hundred second
one hundred nine million second
one hundred nine million one hundred second
...


There are 7 t-free ordinals less than 10^6, 392 less than 10^9, 21952 less than 10^15. How many are less than 10^63 (one vigintillion)?

Thursday, March 30

The sign of the four

In 1908, Matthew Burke was the head of one of twenty-two families living on the Conne River Mi'kmaq reservation in Newfoundland. Matthew's granddaughter, Margaret Burke Stewart, became the mother of sixteen children — the oldest (Catherine) ended up marrying my wife's now-deceased oldest brother (Larry). Catherine brought into that union two boys from her first marriage, Shawn and Jamie Beaupre.

Shawn (using his middle name) has been promoting himself as an aboriginal medium — Shawn Leonard. On Tuesday, Shawn teamed up with psychic/medium John Holland for a show in Moncton, New Brunswick, and they will do another tonight in Halifax, Nova Scotia. Yesterday, Holland did a Facebook interview with Shawn (click the "not now" in the pop-up sign-up if, like me, you don't do Facebook). It shows just how comfortable they are with each other in their overlapping, supportive roles.

Engineering coincidences into something that may be perceived to be meaningful is not of course everyone's cup of tea, least of all mine. This morning, Johnny Wills' Google+ photo-of-the-day theme was "four" and I quickly came up with an entry that I knew would be significantly different from the contributions of most other participants. Our brains exhibit a more-than-willing bent on assigning structure to the random bits and pieces in our lives!

Less than three hours after I posted the photo it was time for Bodie's morning walk. I have a habit of picking up any garbage that I encounter on the street so as to deposit it in a trash bin further along my route. A few houses away from my home I spotted (in light blue) just such a distraction lying in the middle of the road. Imagine my surprise as I approached to pick it up:

Tuesday, March 28

Rudolph Havermann genannt Draht

My great-great-grandfather Rudolph was born in 1800, likely in K├Ârbecke, and died in nearby Neheim in 1869. The use of genannt in my father's ancestors' names has always been bothersome to me and there are (German) explanations of its usage but I like to think that the Draht here is just an acknowledgement of Rudolph's mother's maiden name. I come to write about this man because of the mortality of his eight known-to-me children.

Rudolph's first marriage to Maria Christina Zentini (1811-1848) produced five offspring: Joseph (1837-1840), Heinrich (1839-1876), Friederich (1842-1842), Ferdinand (1843-1854), and Joseph (again, 1846-1864). Rudolph's second marriage to Maria Theresia Biermann (1813-1867) produced three more offspring: Anton (1850-1854), Heinrich Franz (1854-1854), and Maria Antonia (1858-1869). Wow, only one of these eight made it to age 18: my great-grandfather Heinrich — who died at age 36. Lucky man. Lucky me!

Monday, March 20

Fracture revisited

I noticed that the 2007 movie Fracture was on Netflix and so I thought I'd watch it again. I hadn't seen it since August 2009, a fact I was able to recover because I wrote a blog piece about a chess position in the film. I ended the item with a question about the six captured black pieces — which weren't visible in my movie-still grab. With the better-resolution Netflix version, I took it upon myself to retake a screen snippet of the board:


So the six captured black pieces were there all along, hidden in the poor contrast of my 2009 screen grab!

Tuesday, March 7

T cube


Designed by Yavuz Demirhan, realized by Brian Menold, this recently acquired puzzle more than tests my patience. The three identical pieces (two one-by-three bars attached to a one-by-five bar) fit — without any protuberances — within the (five-by-five-by-five) cubic cage. It arrived assembled and I was careful in taking it apart, but after I had reassembled it a second time it no longer came apart as I expected! And by the time I did get that disassembled I had forgotten how it was supposed to come together. Complicating things ever so slightly is that one of the frame's twenty-four inside edges is a touch less than three units long, presumably a production flaw and not part of the design. I'm getting too old for this type of toy.

Thursday, February 23

Movin' on up


When I bragged about my 100th Leyland prime find last August, I noted that I was moving up a leaderboard of probable-prime (PRP) discoverers. I currently have 131 Leyland primes under my belt and the last few days saw me take possession of position #44 on the PRP production score list, of which the above is a snippet.

Roughly, in the range where I am searching, every new find adds .01 to my production score. So to reach position #40 I have to come up with another 27 PRPs. My current rate of production is about five or six per month. So another five months.