Tuesday, February 18

The hidden palindromes


I submitted today, on behalf of Éric Angelini, OEIS A332661. The sequence (1, 2, 3, 11, 4, 22, 101, 5, 111, 6, 1001, 7, 88, 77, 8, 1111, 9, 10001, 33, 121, 131, 202, ...) is composed entirely of palindromic (base ten) integers with an added proviso that the products of adjacent terms are also palindromes. We don't see those products there, so I thought I'd show them here:

1. 1 * 2 = 2
2. 2 * 3 = 6
3. 3 * 11 = 33
4. 11 * 4 = 44
5. 4 * 22 = 88
6. 22 * 101 = 2222
7. 101 * 5 = 505
8. 5 * 111 = 555
9. 111 * 6 = 666
10. 6 * 1001 = 6006
11. 1001 * 7 = 7007
12. 7 * 88 = 616
13. 88 * 77 = 6776
14. 77 * 8 = 616
15. 8 * 1111 = 8888
16. 1111 * 9 = 9999
17. 9 * 10001 = 90009
18. 10001 * 33 = 330033
19. 33 * 121 = 3993
20. 121 * 131 = 15851
21. 131 * 202 = 26462
22. 202 * 44 = 8888
23. 44 * 2002 = 88088
24. 2002 * 141 = 282282
25. 141 * 212 = 29892
26. 212 * 1221 = 258852
27. 1221 * 303 = 369963
28. 303 * 2112 = 639936
29. 2112 * 222 = 468864
30. 222 * 3003 = 666666
31. 3003 * 232 = 696696
32. 232 * 10101 = 2343432
33. 10101 * 55 = 555555
34. 55 * 99 = 5445
35. 99 * 555 = 54945
36. 555 * 979 = 543345
37. 979 * 5555 = 5438345
38. 5555 * 9779 = 54322345
39. 9779 * 55555 = 543272345
40. 55555 * 97779 = 5432112345
41. 97779 * 100001 = 9777997779
42. 100001 * 66 = 6600066
43. 66 * 1000001 = 66000066
44. 1000001 * 151 = 151000151
45. 151 * 11011 = 1662661
46. 11011 * 161 = 1772771
47. 161 * 11111 = 1788871
48. 11111 * 171 = 1899981
49. 171 * 101101 = 17288271
50. 101101 * 181 = 18299281
51. 181 * 110011 = 19911991
52. 110011 * 242 = 26622662
53. 242 * 10201 = 2468642
54. 10201 * 313 = 3192913
55. 313 * 20002 = 6260626
56. 20002 * 323 = 6460646
57. 323 * 20102 = 6492946
58. 20102 * 333 = 6693966
59. 333 * 21012 = 6996996
60. 21012 * 404 = 8488848
61. 404 * 111111 = 44888844
62. 111111 * 252 = 27999972
63. 252 * 1001001 = 252252252
64. 1001001 * 191 = 191191191
65. 191 * 1010101 = 192929291
66. 1010101 * 262 = 264646462
67. 262 * 1100011 = 288202882
68. 1100011 * 272 = 299202992
69. 272 * 1101011 = 299474992
70. 1101011 * 343 = 377646773
71. 343 * 200002 = 68600686
72. 200002 * 414 = 82800828
73. 414 * 120021 = 49688694
74. 120021 * 1331 = 159747951
75. 1331 * 12021 = 15999951
76. 12021 * 10301 = 123828321
77. 10301 * 2222 = 22888822
78. 2222 * 20202 = 44888844
79. 20202 * 3113 = 62888826
80. 3113 * 11211 = 34899843
81. 11211 * 4004 = 44888844
82. 4004 * 210012 = 840888048
83. 210012 * 2332 = 489747984
84. 2332 * 30003 = 69966996
85. 30003 * 3223 = 96699669
86. 3223 * 300003 = 966909669
87. 300003 * 3333 = 999909999
88. 3333 * 1002001 = 3339669333
89. 1002001 * 424 = 424848424
90. 424 * 2000002 = 848000848
91. 2000002 * 434 = 868000868
92. 434 * 2001002 = 868434868
93. 2001002 * 444 = 888444888
94. 444 * 2002002 = 888888888
95. 2002002 * 1441 = 2884884882
96. 1441 * 201102 = 289787982
97. 201102 * 10401 = 2091661902
98. 10401 * 21112 = 219585912
99. 21112 * 11311 = 238797832
100. 11311 * 102201 = 1155995511
101. 102201 * 12121 = 1238778321
102. 12121 * 30103 = 364878463
103. 30103 * 12221 = 367888763
104. 12221 * 40004 = 488888884
105. 40004 * 112211 = 4488888844
106. 112211 * 31013 = 3479999743
107. 31013 * 1011101 = 31357275313
108. 1011101 * 4114 = 4159669514
109. 4114 * 1110111 = 4566996654
110. 1110111 * 10501 = 11657275611
111. 10501 * 1111111 = 11667776611
112. 1111111 * 10601 = 11778887711
113. 10601 * 10000001 = 106010010601
114. 10000001 * 282 = 2820000282
115. 282 * 10100101 = 2848228482
116. 10100101 * 292 = 2949229492
117. 292 * 100000001 = 29200000292
118. 100000001 * 353 = 35300000353
119. 353 * 10011001 = 3533883353
120. 10011001 * 363 = 3633993363
121. 363 * 11000011 = 3993003993
122. 11000011 * 454 = 4994004994
123. 454 * 11011011 = 4998998994
124. 11011011 * 2442 = 26888888862
125. 2442 * 20000002 = 48840004884
126. 20000002 * 3443 = 68860006886
127. 3443 * 20011002 = 68897879886
128. 20011002 * 11411 = 228345543822
129. 11411 * 1200021 = 13693439631
130. 1200021 * 12321 = 14785458741
131. 12321 * 1003001 = 12357975321
132. 1003001 * 20302 = 20362926302
133. 20302 * 121121 = 2458998542
134. 121121 * 301103 = 36469896463
135. 301103 * 21212 = 6386996836
136. 21212 * 310013 = 6575995756
137. 310013 * 1012101 = 313764467313
138. 1012101 * 13031 = 13188688131
139. 13031 * 2010102 = 26193639162
140. 2010102 * 12421 = 24967476942
141. 12421 * 2011102 = 24979897942
142. 2011102 * 22022 = 44288488244
143. 22022 * 400004 = 8808888088
144. 400004 * 22122 = 8848888488
145. 22122 * 1020201 = 22568886522
146. 1020201 * 32023 = 32669896623
147. 32023 * 1102011 = 35289698253
148. 1102011 * 22222 = 24488888442
149. 22222 * 2100012 = 46666466664
150. 2100012 * 13131 = 27575257572
151. 13131 * 2101012 = 27588388572
152. 2101012 * 13231 = 27798489772
153. 13231 * 3000003 = 39693039693
154. 3000003 * 13331 = 39993039993
155. 13331 * 10200201 = 135978879531
156. 10200201 * 13431 = 136998899631
157. 13431 * 20100102 = 269964469962
158. 20100102 * 21312 = 428373373824
159. 21312 * 2003002 = 42687978624
160. 2003002 * 30203 = 60496669406
161. 30203 * 1013101 = 30598689503
162. 1013101 * 30303 = 30699999603
163. 30303 * 3001003 = 90939393909
164. 3001003 * 22322 = 66988388966
165. 22322 * 10111101 = 225699996522
166. 10111101 * 10701 = 108198891801
167. 10701 * 11100111 = 118782287811
168. 11100111 * 4224 = 46886868864
169. 4224 * 100010001 = 422442244224
170. 100010001 * 373 = 37303730373
171. 373 * 100101001 = 37337673373
172. 100101001 * 383 = 38338683383
173. 383 * 101000101 = 38683038683
174. 101000101 * 393 = 39693039693
175. 393 * 101010101 = 39696969693
176. 101010101 * 464 = 46868686864
177. 464 * 1000000001 = 464000000464
178. 1000000001 * 474 = 474000000474
179. 474 * 1001001001 = 474474474474
180. 1001001001 * 484 = 484484484484
181. 484 * 1010000101 = 488840048884
182. 1010000101 * 494 = 498940049894
183. 494 * 10000000001 = 4940000000494
184. 10000000001 * 505 = 5050000000505
185. 505 * 110000011 = 55550005555
186. 110000011 * 515 = 56650005665
187. 515 * 110010011 = 56655155665
188. 110010011 * 525 = 57755255775
189. 525 * 1000110001 = 525057750525
190. 1000110001 * 535 = 535058850535
191. 535 * 1100000011 = 588500005885
192. 1100000011 * 545 = 599500005995
193. 545 * 923757329 = 503447744305
194. 923757329 * 5450000545 = 5034477946497744305
195. 5450000545 * 100000000001 = 545000054505450000545
196. 100000000001 * 565 = 56500000000565
197. 565 * 95359 = 53877835
198. 95359 * 565000565 = 53877888877835
199. 565000565 * 938839 = 530444565444035
200. 938839 * 565565 = 530974479035
201. 565565 * 1000000000001 = 565565000000565565
202. 1000000000001 * 575 = 575000000000575
203. 575 * 9119 = 5243425
204. 9119 * 575575 = 5248668425
205. 575575 * 911080119 = 524394939493425
206. 911080119 * 10000100001 = 9110892298922980119
207. 10000100001 * 585 = 5850058500585
208. 585 * 10010001001 = 5855850585585
209. 10010001001 * 595 = 5955950595595
210. 595 * 92829 = 55233255
211. 92829 * 54145 = 5026226205
212. 54145 * 9282992829 = 502627646726205
213. 9282992829 * 546808645 = 5076020730370206705
214. 546808645 * 10000000000001 = 5468086450000546808645
215. 10000000000001 * 606 = 6060000000000606
216. 606 * 1100110011 = 666666666666
217. 1100110011 * 616 = 677667766776
218. 616 * 11000000011 = 6776000006776
219. 11000000011 * 626 = 6886000006886
220. 626 * 11000100011 = 6886062606886
221. 11000100011 * 636 = 6996063606996
222. 636 * 100001100001 = 63600699600636
223. 100001100001 * 707 = 70700777700707
224. 707 * 858 = 606606
225. 858 * 777 = 666666
226. 777 * 858000858 = 666666666666
227. 858000858 * 70007 = 60066066066006
228. 70007 * 88088 = 6166776616
229. 88088 * 7000007 = 616616616616
230. 7000007 * 8580858 = 60066066066006
231. 8580858 * 700007 = 6006660666006
232. 700007 * 880088 = 616067760616
233. 880088 * 7007 = 6166776616
234. 7007 * 88000088 = 616616616616
235. 88000088 * 70000007 = 6160006776000616
236. 70000007 * 85800858 = 6006060660606006
237. 85800858 * 7000000007 = 600606006600606006
238. 7000000007 * 8800088 = 61600616061600616
239. 8800088 * 700000007 = 6160061661600616
240. 700000007 * 11111111 = 7777777777777777
241. 11111111 * 11511 = 127899998721
242. 11511 * 100111001 = 1152377732511
243. 100111001 * 10801 = 1081298921801
244. 10801 * 101101101 = 1091992991901
245. 101101101 * 11611 = 1173884883711
246. 11611 * 110101011 = 1278382838721
247. 110101011 * 11711 = 1289392939821
248. 11711 * 111000111 = 1299922299921
249. 111000111 * 5005 = 555555555555
250. 5005 * 1110000111 = 5555550555555
251. 1110000111 * 5115 = 5677650567765
252. 5115 * 10001010001 = 51155166155115
253. 10001010001 * 1551 = 15511566511551
254. 1551 * 10010101001 = 15525666652551
255. 10010101001 * 1661 = 16626777762661
256. 1661 * 10100000101 = 16776100167761
257. 10100000101 * 1771 = 17887100178871
258. 1771 * 10100100101 = 17887277278871
259. 10100100101 * 1881 = 18998288289981
260. 1881 * 10101010101 = 18999999999981
261. 10101010101 * 2552 = 25777777777752
262. 2552 * 100010010001 = 255225545522552
263. 100010010001 * 646 = 64606466460646
264. 646 * 100100001001 = 64664600646646
265. 100100001001 * 656 = 65665600656656
266. 656 * 1000001000001 = 656000656000656
267. 1000001000001 * 666 = 666000666000666
268. 666 * 1000100010001 = 666066606660666
269. 1000100010001 * 676 = 676067606760676
270. 676 * 1001000001001 = 676676000676676
271. 1001000001001 * 686 = 686686000686686
272. 686 * 1001001001001 = 686686686686686
273. 1001001001001 * 696 = 696696696696696
274. 696 * 10000100100001 = 6960069669600696
275. 10000100100001 * 717 = 7170071771700717
276. 717 * 874478 = 627000726
277. 874478 * 7887 = 6897007986
278. 7887 * 87470107478 = 689876737678986
279. 87470107478 * 100000000000001 = 8747010747800087470107478
280. 100000000000001 * 727 = 72700000000000727
281. 727 * 100110011001 = 72779977997727
282. 100110011001 * 808 = 80888888888808
283. 808 * 110000000011 = 88880000008888
284. 110000000011 * 818 = 89980000008998
285. 818 * 7580990857 = 6201250521026
286. 7580990857 * 8998 = 68213755731286
287. 8998 * 7587447857 = 68271855817286
288. 7587447857 * 1000000000000001 = 7587447857000007587447857
289. 1000000000000001 * 737 = 737000000000000737
290. 737 * 888 = 654456
291. 888 * 737000737 = 654456654456
292. 737000737 * 888000000000888 = 654456654456654456654456
293. 888000000000888 * 1000001001000001 = 888000888888888888888888000888
294. 1000001001000001 * 747 = 747000747747000747
295. 747 * 8274728 = 6181221816
296. 8274728 * 7470747 = 61818399381816
297. 7470747 * 82747255274728 = 618183809101908381816
298. 82747255274728 * 74700747 = 6181281781221871821816
299. 74700747 * 10000000000000001 = 747007470000000074700747
300. 10000000000000001 * 757 = 7570000000000000757
301. 757 * 8421248 = 6374884736
302. 8421248 * 75700000757 = 637488479974884736

Note that there are a few duplicate products. I count 292 distinct entries, nine of which have duplicates: eight with one duplicate and one with two duplicates.

616: 2
8888: 2
666666: 2
44888844: 3
6166776616: 2
39693039693: 2
616616616616: 2
666666666666: 2
60066066066006: 2

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
292131111111  108   85340586078947427654321  24930640234106850806012082150260631
311111133141  108   96790137164277028525881  30112489250051042810723218755322221
311111213214  144   96790186987486968209796  30112512500886986516764283239444344
311111314131  108   96790249780317760285161  30112541804222392461075080008910091
311113141122  144   96791386578797487418884  30113072312083479476410277419747848
311113411122  144   96791554579966593298884  30113150713174648766021468115787848
311114112312  144   96791990879683749985344  30113354321444009871981641569955328
311121241311  144   96796426794897492998721  30115424458897849222764009255363231
311212114113  144   96852979970682933776769  30141820654820280581077456096440897
311212141311  144   96852996899377832798721  30141828557443018948936490472063231
311221131114  144   96858592451877578880996  30144440700983282712399333518909544
311221141131  144   96858598686881819959161  30144443611680938167828178237351091
312211111911  108   97475778400702966071921  30433021158873710341350163405751031
312211411113  144   97475965229170699898769  30433108653801106646148515741619897
312341111112  144   97556969690678729876544  30471052309906301439064663546556928
314111112231  144   98665790826995877797361  30992021295818872475381222146622391
321121111911  108  103118768514956986071921  33113613604416005601752611735751031
332111112114  144  110297790789597877548996  36631121962850657295651054084137544

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.