Friday, February 14

What's so special about these numbers?

By which I mean the numbers in the first column. Each of those numbers is followed by the product of the number's digits, which is followed by the number's square, which is followed by the number's cube.

           0    0                        0                                    0
           1    1                        1                                    1
          93   27                     8649                               804357
        1941   36                  3767481                           7312680621
       22911   36                524913921                       12026302844031
       23313   54                543495969                       12670521525297
      111633   54              12461926689                     1391162262073137
      113163   54              12805864569                     1449150052221747
      115911   45              13435359921                     1557306003803031
      136311   54              18580688721                     2532752260248231
      192114   72              36907788996                     7090502975177544
      631122   72             398314978884                   251385346103227848
      631131   54             398326339161                   251396100761021091
      631212   72             398428588944                   251492906484520128
      912114   72             831951948996                   758835020006537544
     1111197   63            1234758772809                  1372060244069042373
     1111917   63            1236359414889                  1374729051525132213
     1312911   54            1723735293921                  2263111028477114031
     1334121   72            1779878842641                  2374573741423053561
     1831311   72            3353699978721                  6141667661731533231
     3111423   72            9680953084929                 30121540090369043967
     3113313   81            9692717835969                 30176464444054155297
     3114312   72            9698939233344                 30205522841674019328
     3121134   72            9741477445956                 30404456466806434104
     3143211   72            9879775390521                 31054218685014902931
     9211212   72           84846426508944                781538422016303080128
    11213232   72          125736571885824               1409913351440422023168
    13212231   72          174563047997361               2306367314205220922391
    16111341   72          259575308818281               4182106315551632224821
    22311312   72          497794643161344              11106451595501412323328
    23221311   72          539229284558721              12521610917045558103231
   111142911   72        12352746665553921            1372920223255206207404031
   111632121   72        12461730438958641            1391129400231214126107561
   131135121   90        17196419959684641            2255054612180060519376561
   132113511   90        17453979788747121            2305906550814420446451831
   132135111   90        17459687558982321            2307037753631448032372631
   151151212  100        22846688889068944            3453304715769704437160128
   211151125  100        44584797588765625            9414130168765149080078125
   211511313   90        44737035526983969            9462389124040026213141297
   311123151   90        96797615088168801           30115979015516220187011951
   311139111   81        96807546393870321           30120613923080067525224631
   311262111   72        96884101744176321           30156350031231103630673631
   311314113  108        96916476952976769           30171467057700905550840897
   312111513   90        97413596547149169           30403905005102302973282697
   312151131   90        97438328584579161           30415484470426014265181091
   313511211   90        98289279422686521           30814791020123832072086931
   313512111   90        98289843743676321           30815056401940106297423631
   314111313  108        98665916954583969           30992080722953331871341297
   353111121   90       124687463773876641           44028530107840471219224561
   521111215  100       271556898398776225          141511345266217833122863375
   921211131  108       848629947878299161          781767354085439020461161091
  1132112133  108      1281677881685809689         1451003080454243642845856637
  1132121313  108      1281698667348843969         1451038378149323463216411297
  1133123112  108      1283967986948564544         1454893801079532840030140928
  1231132113  108      1515686279659844769         1866010052122733611710966897
  1311331212  108      1719589547565388944         2254951445551453133186920128
  1321211313  108      1745599333599183969         2306305587516502867411041297
  1322111313  108      1747978323962583969         2311021916989711254127341297
  1651111131   90      2726167966912099161         4501206275144206623302861091
  1961111121  108      3845956828909876641         7542348708061053387403224561
  2121113313  108      4499121686585835969         9543146906224230191080155297
  2211311331  108      4889897802608991561        10813086418341264401322677691
  2211331131  108      4889985370929739161        10813376880871514620429121091
  2231131311  108      4977946926924578721        11106453253157656519907433231
  3111122133  108      9679080926442469689        30112802897353312400629526637
  3111123312  108      9679088262469849344        30112837132275522991246307328
  3111223113  108      9679709258865410769        30115735173302166140751903897
  3113131212  108      9691585943128588944        30171178693334067171084520128
  3311111322  108     10963458186676587684        36301230530178439032818158248
  6911111121  108     47763456926809876641       330098558344280221506253224561
 11112136131  108    123479569393875649161      1372121784502007371163027936091
 11116321311  108    123572599489392758721      1373672721159604442219333403231
 11133611121  108    123957296593654876641      1380092335884211352606120724561
 11213161131  108    125734982549769199161      1409886619134035257053123011091
 11311321611  108    127945996587475635321      1447238316240845405742382222131
 11312161131  108    127964989453707199161      1447560579827051502203940011091
 11613111117  126    134864349815788987689      1566194680132715995057302038613
 21111316311  108    445687676383094648721      9409053512038115542200462588231
 21113611311  108    445784582591987138721      9412122405283593350066231673231
 26311111131  108    692274568947832099161     18214513116751532102554082861091
 31111119141  108    967901734205496577881     30112506169647719011523576320221
 31111121163  108    967901860018866472569     30112512040940020294012574877747
 31111613211  108    967932476590869730521     30113940826060450950057153512931
 31111711116  126    967938568565437965456     30114225123242464523285859208896
 31114141113  144    968089777199676878769     30121281937843476384022059729897
 31141141113  144    969770669819778878769     30199765276106264342202348729897
 61112131311  108   3734692593372906578721    228235024172424195228533510433231
 71311161111  126   5085281698998998754321    362637342532137407597073138410631
111121142313  144  12347908268945998989969   1372113672021417832276299508458297
112119111117  126  12570695077666192987689   1409415158230780836621069204038613
112143211113  144  12576099798734886698769   1410324214707683209378187604219897
112191112311  108  12586845681579415760721   1412132217503301577476813213336231
119121111117  126  14189839113748660987689   1690309401801207047924464338038613
121131211911  108  14672770499027588271921   1777330472639180014401320942051031
129111112131  108  16669679275703655361161   2152240830153181511261437173344091
132291111111  108  17500938078982947654321   2315218543953464022012076230260631
134111221113  144  17985819628419976958769   2412100233085567028743834643289897
141211112133  144  19940578189838699809689   2815831222762166800008919138856637
161361111111  108  26037408178976487654321   4201425114210285198645410640260631
161613111111  108  26118797682976431654321   4221140152024599402144077316260631
211111312911  108  44567986439006155293921   9408806130938233066110015407114031
211132113114  144  44576769187982890776996   9411587474453873625025426821125544
213211111134  144  45458977910994898765956   9692359191419184521295324171754104
221111113911  108  48890124694963217715921  10810149930551006160340101469277031
221111191131  108  48890158843369613059161  10810161256441248405201661481501091
221131111413  144  48898968434748618856569  10813083236925167057019832205921997
221131131114  144  48898977147757058880996  10813086127001155938432544598909544
221311111911  108  48978608255283166071921  10839510252830001223537306705751031
223111411113  144  49778701768834099898769  11106196395017765153207562041619897
231211141113  144  53458591774775598878769  12360222006539902106268143838729897

Partial spoiler: The numbers are a subset of A117224.

Sunday, January 19

Spotted


I don't take a lot of photographs of people I don't know, but when I do, I prefer to do it on the sly. So it was somewhat unfortunate that Weston's "pigeon man" decided to break the fourth wall and acknowledge me across the intersection. The upraised hand, possibly a gesture of "peace"; the piercing stare, not so much.

Saturday, January 18

Anagrammatic sums

Éric Angelini asked about anagrammatic sums on MathFun on January 12: "Let a + b = c and a < b < c and a, b, c = anagrams of each other." Halfway down his sausage article, he lists the Gilles Esposito-Farèse calculation for 3- to 5-digit results: 1 @ 3-digit, 25 @ 4-digit, and 648 @ 5-digit. In that spirit, here are 17338 @ 6-digit results. I count 495014 @ 7-digit and 17565942 @ 8-digit.

The idea for these has been around a few years. Claudio Meller's A160851 appears to be a (currently) somewhat misguided attempt at enumeration, while Rajesh Bhowmick's A203024, fleshed out by Charles Greathouse, provides a seemingly complete listing of sums, including 9449 6-digit terms. My 17338 6-digit results yield only 9443 distinct sums. Why six fewer?

Apparently 6-digit sums are the first that allow the sums to be twice one of the addends (i.e., a = b). In A023086 we see that there are twelve such. It turns out that six of these are the six that are not in my 17338 sums (because I did not allow a = b):

251748 = 2 * 125874
257148 = 2 * 128574
285174 = 2 * 142587
285714 = 2 * 142857
517482 = 2 * 258741
825174 = 2 * 412587

The other six are included because they each had an alternate solution:

517428 = 2 * 258714 = 241587 + 275841
571428 = 2 * 285714 = 142857 + 428571
571482 = 2 * 285741 = 158724 + 412758
825714 = 2 * 412857 = 241587 + 584127
851742 = 2 * 425871 = 127584 + 724158
857142 = 2 * 428571 = 142857 + 714285 = 275418 + 581724 = 285714 + 571428

Addendum: Éric and Gilles had created A331468 for their triples.

Tuesday, January 14

Mysterious lights in St. John's Cemetery on the Humber


Photo taken from the middle of Denison Park, looking south, the evening darkness fast approaching, two bright lights appear in St. John's Cemetery on the Humber and, after a short while, go out. I suppose one hundred years ago when the park and surrounding area were still all fields, one might have thought such an occurrence — especially at night — to be mysterious, especially if the lights reappeared in the same location at irregular intervals — but never on for very long.

Of course I've seen this phenomenon innumerable times over the last forty years, as have many of the local residents. I've never given it much thought because the explanation was always a tad obvious, all the more so if one was in the cemetery when the lights came on. I was reminded of it only because I came across an article by Clark Kim in the 30 October 2013 edition of the York Guardian:

"There was a story that Denison Cemetery was haunted," said [Cherri] Hurst [of the Weston Historical Society]. It was also known as St. John’s Cemetery on the Humber where members of the Denison family are buried. 

Lights would mysteriously be seen in the cemetery at night. It turned out those lights were turned on and off by whiskey runners trying to hide their stashes of alcohol.

"Weston was a dry town for a long time. It was a small town. You couldn’t get away with stuff," she said.


Note the emphasis on the lights being turned on and off. Behind the south end of the cemetery is an almost 200-metre stretch of much lower ground before it rises again to where West Park Healthcare Centre has its presence. I don't know what the road layout was one hundred years ago, but today there is a curved West Park Receiving road off Emmett Ave. that presents a gated "employee entrance" a little along. Beyond the gate, one arrives at a 50-metre stretch of straight road that then abruptly turns right. That straight stretch is what allows the lights to "turn on" and the sharp right, to "turn off". In the following Google map, a yellow arrow extends the straight stretch of road over the low ground, through the cemetery, through Denison Park, stopping at the park's north end (at Lippincott St. W.):

Monday, January 13

Inundation

A lot of rain here on Saturday had the "Raymore" island on the east side of the Humber (just above the weir; photo taken on Friday) ...


... somewhat inundated with river water on Sunday:


By this morning, the torrent had subsided:

Tuesday, January 7

The Fu Yao fruit bag mystery

Yesterday morning, walking Bodie up the north side of Clouston Ave., I saw a bag by a tree, across the street at the southwest corner of Clouston Ave. and Centre Rd. Thinking that it was trash, I crossed the road, determined to pick it up so as to deposit it in the trash bin at Denison Park on my way back. It wasn't trash but, rather, a plastic bag full of fruit. Nestled against the tree were three Canadian quarters. I left the bag of fruit but decided to take the money home. This morning, I had another look:


The bag was from Fu Yao Supermarket which has two locations in Toronto, neither of them anywhere near the northwest part of the city where I reside. Inside the bag were nine apples, a few grapes, and some sort of flattish bread-like item:


I'm at a loss to imagine a scenario that could account for all this.

Thursday, December 19

True probabilities in self-referential statements

On December 8, Éric Angelini asked in an online MathFun forum for a solution to a self-referential sentence in which randomly picking four letters from the statement yielding the four letters F, O, U, and R, was a probability 𝜶/𝜷, where 𝜶 and 𝜷 were the English number names for the numeric quantities that allowed the sentence to be true. After determining that picking one letter depleted the letter availability of the next pick by one, I settled on my template being: "The odds of randomly picking four letters from this statement and having them be F, O, U, and R, are 𝜶 out of 𝜷."

Mathematica has a function for converting integers into English words so all I had to do was run through a bunch of them and test the resulting sentences for truthfulness. Here are nine solutions:

1. "The odds of randomly picking four letters from this statement and having them be F, O, U, and R, are two out of two hundred nineteen thousand six hundred eighty-seven."

2. "The odds of randomly picking four letters from this statement and having them be F, O, U, and R, are three out of two hundred ninety-two thousand nine hundred sixteen."

3. "The odds of randomly picking four letters from this statement and having them be F, O, U, and R, are three out of one hundred ninety thousand six hundred fifty."

4. "The odds of randomly picking four letters from this statement and having them be F, O, U, and R, are six out of seven hundred twenty-one thousand seven hundred ninety-one."

5. "The odds of randomly picking four letters from this statement and having them be F, O, U, and R, are twelve out of one million seven hundred sixty-six thousand six hundred twenty-two."

6. "The odds of randomly picking four letters from this statement and having them be F, O, U, and R, are eighteen out of one million six hundred thousand two hundred."

7. "The odds of randomly picking four letters from this statement and having them be F, O, U, and R, are twenty-one out of two million ninety-seven thousand twenty-four."

8. "The odds of randomly picking four letters from this statement and having them be F, O, U, and R, are thirty-five out of two million six hundred sixty-seven thousand four hundred twenty."

9. "The odds of randomly picking four letters from this statement and having them be F, O, U, and R, are seventy-one out of nine million nine hundred twenty thousand two hundred sixty-two."

I believe that this exhausts the possibilities for 𝜶 up to 100 and 𝜷 up to 10000000. But only for this particular template. One may easily alter the template by adding/subtracting/changing words without altering the essential thrust of what the sentence is saying. Each of these new statements would have its own set of solutions.

Of the above nine solutions, the first two are special: The ratio 𝜶/𝜷 is in its lowest terms. In the other seven the two numbers share a common factor, so they can be reduced. Of course we mustn't do that because that would rob the statements of their truthfulness.

If you are interested in verifying the nine statements, this will help:

 1.  {2,219687}  {5,11,6,8}  <132>   5*11*6*8/(132*131*130*129) = 2/219687
 2.  {3,292916}  {5,11,6,9}  <132>   5*11*6*9/(132*131*130*129) = 3/292916
 3.  {3,190650}  {7,10,6,9}  <126>   7*10*6*9/(126*125*124*123) = 1/63550
 4.  {6,721791}  {5,11,6,8}  <135>   5*11*6*8/(135*134*133*132) = 2/240597
 5. {12,1766622} {5,12,6,8}  <145>   5*12*6*8/(145*144*143*142) = 2/294437
 6. {18,1600200} {5,12,6,8}  <128>   5*12*6*8/(128*127*126*125) = 1/88900
 7. {21,2097024} {6,13,5,7}  <130>   6*13*5*7/(130*129*128*127) = 7/699008
 8. {35,2667420} {7,12,7,10} <147>  7*12*7*10/(147*146*145*144) = 1/76212
 9. {71,9920262} {5,13,6,8}  <146>   5*13*6*8/(146*145*144*143) = 1/139722

After {𝜶𝜷} are the letter counts of {F,O,U,R}, followed by the statement's <total-letter-count>, followed by the probability calculation and 𝜶/𝜷 in its lowest terms.

Friday, December 13

Sapphire


Today is the sixth Friday, December 13th anniversary of our marriage. The photograph of the somber newlyweds was taken by my friend, Alfy Marcuzzi.

Thursday, December 12

A330365

The idea for OEIS sequence A330365 came from Éric Angelini who asked me if I was willing to check and extend it. I was happy to. On the OEIS page for the sequence one can click "graph" and get (at the bottom) a logarithmic scatterplot. I have a much prettier version:


Click on the picture for a larger display of it. The blue points are the values at odd indices; the red points, values at even indices. Green lines connect adjacent values.