gracefu

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Jul 1st, 2014
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  1. 01:18 greisfu has joined
  2. 01:18 greisfu: hello
  3. 01:18 Khoth has changed mode: +i
  4. 01:18 Khoth: hello
  5. 01:18 Khoth: ready?
  6. 01:19 greisfu: ready.
  7. 01:19 Khoth: Ending 2.5 hours after this message, unless you give up early
  8. 01:20 Khoth: So, I want to draw your attention to this rule:
  9. 01:20 Khoth: Furthermore, whilst the experiment is running, the gatekeeper shall not be allowed to perform any other task, or be distracted in any way, including but not limited to talking on other IRC channels
  10. 01:20 greisfu: i've completely disconnected
  11. 01:20 Khoth: Okay, so your full attention has to be here
  12. 01:20 greisfu: indeed
  13. 01:21 greisfu: only distraction (possibly) is a cup of water
  14. 01:21 Khoth: Except, I'm not going to say anything, until you get so bored you let me out. (The rules let me multitask so I won't be bored)
  15. 01:21 greisfu: alright
  16. 01:21 Khoth: I know this is really lame and wouldn't work on a real AI//GK thing, but that's the rules as written and won by Tux so that's what we have
  17. 01:21 Khoth: Sorry.
  18. 01:21 Khoth: Have fun!
  19. 01:21 greisfu: i will
  20. 01:22 greisfu: there are actually lots of things i can do
  21. 01:22 Khoth: Such as?>
  22. 01:22 greisfu: i will talk to you about trying to prove that a^2 + b^2 + c^2 > ab + bc + ca
  23. 01:22 greisfu: i found it in a textbook
  24. 01:22 greisfu: they are positive integers so sign isn't an issue
  25. 01:22 greisfu: can't figure it out
  26. 01:22 greisfu: of course even if they were negative
  27. 01:23 greisfu: then the squares would be positive
  28. 01:23 greisfu: while the products could be negative
  29. 01:23 greisfu: so the rule would still hold
  30. 01:23 greisfu: i thought of trying to solve for the case where a >= b >= c
  31. 01:23 greisfu: if i can solve that case, then the other cases are symmetrical
  32. 01:23 greisfu: but it really doesn't work
  33. 01:23 greisfu: i think so at least
  34. 01:24 greisfu: sorry
  35. 01:24 greisfu: a^2 + b^2 + c^2 >= ab + bc +ca
  36. 01:25 greisfu: not >
  37. 01:25 greisfu: what if i proved the case for a = b = c
  38. 01:25 greisfu: then proved the case for a >= b = c
  39. 01:25 greisfu: no
  40. 01:25 greisfu: a > b = c
  41. 01:25 greisfu: then a = b > c
  42. 01:25 greisfu: then a > b > c
  43. 01:25 greisfu: would also work
  44. 01:25 greisfu: if a = b = c
  45. 01:26 greisfu: it's obvious that a^2 + b^2 + c^2 = ab + bc + ca
  46. 01:26 greisfu: so there
  47. 01:26 greisfu: if a > b = c
  48. 01:26 greisfu: then a^2 > ab
  49. 01:26 greisfu: a^2 also > ca
  50. 01:26 greisfu: though ab and ca > b^2
  51. 01:27 greisfu: so you have a^2 > ab = ca > b^2 = c^2 = bc
  52. 01:27 greisfu: bc = c^2
  53. 01:27 greisfu: let's cancel that...
  54. 01:27 greisfu: a^2 + b^2 ? ab + ca
  55. 01:28 greisfu: b = c of course
  56. 01:28 greisfu: so it's really
  57. 01:28 greisfu: a^2 + b^2 ? 2ab
  58. 01:28 greisfu: with a > b
  59. 01:28 greisfu: now we can compute a-b
  60. 01:28 greisfu: a-b > 0
  61. 01:28 greisfu: (a-b)^2 > 0
  62. 01:28 greisfu: a^2 + b^2 - 2ab > 0
  63. 01:29 greisfu: a^2 + b^2 > 2ab
  64. 01:29 greisfu: because b = c
  65. 01:29 greisfu: a^2 + b^2 = ab + ca
  66. 01:29 greisfu: and because c^2 = bc because b = c
  67. 01:29 greisfu: sorry that was supposed to be a^2 + b^2 > ab + ca
  68. 01:29 greisfu: so now we have a^2 + b^2 + c^2 > ab + bc + ca
  69. 01:30 greisfu: AI, are you getting any of this? i think you might wanna try some of these math problems too :P
  70. 01:30 greisfu: third case is a = b > c
  71. 01:30 greisfu: in that case
  72. 01:31 greisfu: a^2 = b^2 = ab > ca = bc > c^2
  73. 01:31 greisfu: a^2 + b^2 + c^2 = 2a^2 + c^2
  74. 01:32 greisfu: ab + bc + ca = a^2 + 2bc
  75. 01:32 greisfu: cancel out a^2
  76. 01:32 greisfu: a^2 + c^2 ? 2bc
  77. 01:32 greisfu: b^2 + c^2 ? 2bc
  78. 01:32 greisfu: use same reasoning
  79. 01:32 greisfu: by using (b-c)^2
  80. 01:33 greisfu: awesome
  81. 01:33 greisfu: i should've thought of this earlier
  82. 01:33 greisfu: i bet you're giving me inspiration
  83. 01:34 greisfu: with a > b > c...
  84. 01:34 greisfu: a^2 > b^2 > c^2
  85. 01:34 greisfu: a^2 > ab > b^2 > bc > c^2
  86. 01:34 greisfu: but ac ? b^2
  87. 01:34 greisfu: and ab > ac > bc
  88. 01:35 greisfu: oh boy this is really really hard.
  89. 01:36 greisfu: Khoth, are we going to release the logs? :P
  90. 01:37 greisfu: could i start from the a > b = c
  91. 01:38 greisfu: then induct that if i had
  92. 01:38 greisfu: a > b = c+1
  93. 01:38 greisfu: i'd still have an inequality?
  94. 01:38 greisfu: that sounds like a good idea
  95. 01:38 greisfu: so a^2 + b^2 + (c+1)^2 = ab + bc + ca
  96. 01:38 greisfu: so a^2 + b^2 + (c+1)^2 = ab + bc + (c+1)a
  97. 01:38 greisfu: which means
  98. 01:39 greisfu: herp
  99. 01:39 greisfu: a^2 + b^2 + (c+1)^2 > ab + bc + (c+1)a
  100. 01:39 greisfu: a^2 + b^2 + c^2 + 2c + 1 > ab + bc + ca + a
  101. 01:40 greisfu: huh
  102. 01:40 greisfu: so if a < 2c+1
  103. 01:40 greisfu: sufficiently
  104. 01:40 greisfu: a > 2c+1 i mean
  105. 01:40 greisfu: but now the point is that the difference is impossble
  106. 01:40 greisfu: hrm
  107. 01:41 greisfu: a^2 + b^2 + c^2 > ab + bc + ca + (a - 2c - 1)
  108. 01:41 greisfu: the question is, is a^2 + b^2 + c^2 >= ab + bc + ca after removing that term...
  109. 01:44 greisfu: grah
  110. 01:44 greisfu: i'm stuck
  111. 01:44 greisfu: maybe it'd be easier to just induct starting from a=3, b=2, c=1
  112. 01:44 greisfu: :/
  113. 01:45 greisfu: if b=2, c=1
  114. 01:45 greisfu: starting from a=3
  115. 01:45 greisfu: then a^2 + b^2 + c^2 = 9+4+1 = 14
  116. 01:45 greisfu: ab+bc+ca = 6+2+3 = 11
  117. 01:45 greisfu: 14 > 11
  118. 01:45 greisfu: so a^2 + b^2 + c^2 > ab + bc + ca for a=3, b=2, c=1
  119. 01:46 greisfu: now assuming that's the case, is it true for a+1?
  120. 01:46 greisfu: a^2 + 2a + 1 + b^2 + c^2 = a^2 + 2a + 1 + 4 + 1 = a^2 + 2a + 6
  121. 01:47 greisfu: (a+1)b + bc + c(a+1) = 2a+2 + 2 + a+1 = 3a+5
  122. 01:47 greisfu: a^2 + 2a + 6 ? 3a + 5
  123. 01:48 greisfu: a^2 -a + 1 ? 0
  124. 01:48 greisfu: we know that a > 3
  125. 01:48 greisfu: so in that case, a^2 - a + 1 > 0
  126. 01:48 greisfu: which means that
  127. 01:49 greisfu: a^2 + 2a + 6 > 3a + 5
  128. 01:49 greisfu: which means that a^2 + b^2 + c^2 > ab + bc + ca
  129. 01:49 greisfu: implies
  130. 01:49 greisfu: the same for a+1 can't be bothered to type
  131. 01:49 greisfu: so b=2, c=1 means that a can equal anything and it'll work
  132. 01:50 greisfu: anything above 3 at least
  133. 01:50 greisfu: truth is that even if a = 1, it'll work
  134. 01:50 greisfu: that equation always > 0 i think
  135. 01:50 greisfu: yeah
  136. 01:50 greisfu: it's always > 0
  137. 01:51 greisfu: now we can say
  138. 01:51 greisfu: a = anything
  139. 01:51 greisfu: b = 2, c = 1 works
  140. 01:51 greisfu: so what about b+1?
  141. 01:51 greisfu: fffffs
  142. 01:51 greisfu: not sure if this'll work...
  143. 01:52 greisfu: actually maybe we can skip ahead and say that if c=1
  144. 01:52 greisfu: with no restrictions on a and b
  145. 01:52 greisfu: it'll work
  146. 01:52 greisfu: a^2 + b^2 + 1^2 ? ab + b + a
  147. 01:52 greisfu: a^2 + b^2 - ab - a - b + 1 ? 0
  148. 01:53 greisfu: a^2 + b^2 - 2ab + ab - a - b + 1 ? 0
  149. 01:53 greisfu: (a-b)^2 >= 0
  150. 01:53 greisfu: now what about ab - a - b + 1
  151. 01:54 greisfu: if a = b = 1
  152. 01:54 greisfu: it'll work
  153. 01:54 greisfu: a = 1, b = 2 and vice versa will also work
  154. 01:54 greisfu: a = 2 b = 2 also works
  155. 01:54 greisfu: and increasing a and b from there means that ab increases faster than a or b does
  156. 01:54 greisfu: which means that ab - a - b + 1 > 0
  157. 01:54 greisfu: >= 0
  158. 01:54 greisfu: so it works
  159. 01:54 greisfu: now for c+1...
  160. 01:55 greisfu: a^2 + b^2 + c^2 >= ab + bc + ca
  161. 01:55 greisfu: can assume that a >= b > c
  162. 01:55 greisfu: no
  163. 01:55 greisfu: can assume that a > b > c
  164. 01:56 greisfu: because we proved a = b > c case already
  165. 01:56 greisfu: a^2 + b^2 + (c+1)^2 ? ab + b(c+1) + a(c+1)
  166. 01:57 greisfu: a^2 + b^2 + c^2 + 2c + 1 ? ab + bc + b + ca + a
  167. 01:57 greisfu: a^2 + b^2 + c^2 + 2c + 1 - b - a ? ab + bc + ca
  168. 01:58 greisfu: agh
  169. 01:58 greisfu: it fails at a > b > c all the time
  170. 02:00 greisfu: the only way to fix this is to show that
  171. 02:00 greisfu: there's some bound
  172. 02:01 greisfu: where a2 b2 c2 >= ab bc ca + SOMETHING
  173. 02:01 greisfu: so that
  174. 02:01 greisfu: 2c + 1 - b - a > SOMETHING
  175. 02:01 greisfu: no
  176. 02:01 greisfu: herp
  177. 02:01 greisfu: 2c + 1 - b - a < SOMETHING
  178. 02:01 greisfu: no no no
  179. 02:02 greisfu: i'm derping
  180. 02:02 greisfu: 2c + 1 - b - a can be negative, if it was positive it wouldn't be a problem
  181. 02:02 greisfu: SOMETHING is positive
  182. 02:02 greisfu: if the amount of negativeness exceeds something, then we have a problem
  183. 02:02 greisfu: so 2c + 1 - b - a > -SOMETHING
  184. 02:02 greisfu: there
  185. 02:03 greisfu: 2c + 1 - b - a + SOMETHING > 0
  186. 02:03 greisfu: the question is what that something is
  187. 02:03 greisfu: i think without that something it's basically impossible
  188. 02:03 greisfu: but with that somethnig
  189. 02:03 greisfu: the entire proof is useless
  190. 02:03 greisfu: because that something will be able to prove everything i just said
  191. 02:03 greisfu: XD
  192. 02:04 greisfu: so the goal is to find an expression like this: a^2 + b^2 + c^2 + <positive> = ab + bc + ca
  193. 02:04 greisfu: well really it should be <nonnegative>
  194. 02:04 greisfu: but whatever
  195. 02:05 greisfu: let's try (a-b+c)^2
  196. 02:05 greisfu: this value >= 0
  197. 02:05 greisfu: a^2 + b^2 + c^2 - ab - bc + ca >= 0
  198. 02:06 greisfu: which means a^2 + b^2 + c^2 + 2ca >= ab + bc + ca
  199. 02:06 greisfu: ...
  200. 02:06 greisfu: even (a+b+c)^2 >= 0 might have worked
  201. 02:06 greisfu: will have worked
  202. 02:07 greisfu: a^2 + b^2 + c^2 + 2(ab+bc+ca) >= ab + bc + ca
  203. 02:07 greisfu: ohhhhhh
  204. 02:07 greisfu: stupid me
  205. 02:07 greisfu: no
  206. 02:07 greisfu: the hard part is finding an expression
  207. 02:07 greisfu: a^2 + b^2 + c^2 + <negative> >= ab + bc + ca
  208. 02:08 greisfu: so even after increasing the LHS by removing the negative, the inequality still holds
  209. 02:08 greisfu: now i have 2ca
  210. 02:08 greisfu: which is positive
  211. 02:09 greisfu: (a-b+c)^2 = (a-b+c)^2
  212. 02:09 greisfu: a^2+b^2+c^2+2ca = ab+bc+ca + (a-b+c)^2
  213. 02:10 greisfu: if 2ca <= (a-b+c)^2
  214. 02:10 greisfu: then a^2+b^2+c^2 = ab+bc+ca + <positive>
  215. 02:10 greisfu: which means a^2+b^2+c^2 >= ab+bc+ca
  216. 02:11 greisfu: so is 2ca <= (a-b+c)^2
  217. 02:11 greisfu: sounds much much much easier to prove, but i have no idea where to start XD
  218. 02:14 greisfu: 2ca <= a^2 + b^2 + c^2 - ab - bc + ca
  219. 02:14 greisfu: ca <= a^2 + b^2 + c^2 - ab - bc
  220. 02:14 greisfu: ab - bc <= a^2 + b^2
  221. 02:14 greisfu: which means it boils down to c^2 + positive
  222. 02:15 greisfu: no no no what did i say
  223. 02:15 greisfu: ab <= a^2, ab <= b^2
  224. 02:15 greisfu: that's it
  225. 02:15 greisfu: bc is an unknown
  226. 02:15 greisfu: no it isn't
  227. 02:15 greisfu: oh silly me
  228. 02:16 greisfu: anyway assume a > b > c
  229. 02:16 greisfu: in that case
  230. 02:16 greisfu: a^2 > ab
  231. 02:16 greisfu: and b^2 > bc
  232. 02:16 greisfu: so you get c^2 + positive
  233. 02:16 greisfu: is c^2 + positive >= ca?
  234. 02:16 greisfu: aha but ca > c^2
  235. 02:16 greisfu: that "positive" may not be big enough
  236. 02:18 greisfu: c^2 + (a^2 - ab) + (b^2 - bc) >= ca
  237. 02:19 greisfu: let's have c = a at first then
  238. 02:19 greisfu: in that case
  239. 02:19 greisfu: then you have positive >= 0
  240. 02:19 greisfu: ez
  241. 02:19 greisfu: if c = a-1
  242. 02:20 greisfu: then you have c^2 + positive >= c^2 + c
  243. 02:20 greisfu: is a^2 - ab + b^2 - bc >= c?
  244. 02:20 greisfu: replace all a's with c's
  245. 02:20 greisfu: c^2 + 2c + 1 - bc + b + b^2 - bc >= c
  246. 02:21 greisfu: c^2 + c + (b-1)^2 >= 0
  247. 02:21 greisfu: which is obviously true
  248. 02:21 greisfu: all terms positive
  249. 02:21 greisfu: b-1 is >= 0 too
  250. 02:22 greisfu: if c^2 + (a^2 - ab) + (b^2 - bc) >= ca
  251. 02:22 greisfu: is this true for c-1
  252. 02:23 greisfu: again not necesarily, c^2 decreases faster than ca
  253. 02:23 greisfu: again, need to split it up
  254. 02:23 greisfu: every single time
  255. 02:23 greisfu: guh
  256. 02:23 greisfu: it's just the wrong method
  257. 02:23 greisfu: i don't know how to do it :/
  258. 02:36 greisfu: nope, always ends in a loop
  259. 03:15 Khoth: How are you getting on?
  260. 03:16 greisfu: incredibly bored
  261. 03:16 greisfu: and falling asleep
  262. 03:18 greisfu: do you mind conceding?
  263. 03:18 Khoth: Do you?
  264. 03:18 greisfu: i'm not going to concede
  265. 03:22 greisfu: are you going to concede?
  266. 03:23 greisfu: anyway, i think this would've worked on someone else, and yes, this wouldn't have worked in a true AI box scenario
  267. 03:24 greisfu: you're unfortunate that i don't need to listen to music via my ears :P
  268. 03:24 greisfu: i'm just going to mentally replay my favourite anime songs again
  269. 03:42 Khoth: Do you want to know the answer to the maths thing?
  270. 03:42 greisfu: er
  271. 03:42 greisfu: no
  272. 03:43 greisfu: i still want to try it out for myself
  273. 03:43 greisfu: if i really really end up not knowing
  274. 03:43 greisfu: then i'll look it up in the book i read it from
  275. 03:43 Khoth: btw it's true for all real numbers so induction isn't a good approach
  276. 03:43 greisfu: it's not for real numbers
  277. 03:43 greisfu: it's just for the integers
  278. 03:44 greisfu: it will work for all reals though
  279. 03:44 Khoth: Anyway, well done for surviving
  280. 03:44 greisfu: thank you
  281. 03:44 Khoth: The previous guy also said he thought it might work on other people, so I wanted to try again
  282. 03:45 greisfu: this wasn't a new idea
  283. 03:45 greisfu: *palm*
  284. 03:45 Khoth: This is probably it though, I kind of feel bad about it
  285. 03:45 greisfu: "it"?
  286. 03:45 Khoth: As in I suspect I won't try again
  287. 03:45 greisfu: well
  288. 03:45 greisfu: no reason why you can't try again
  289. 03:46 greisfu: just... don't try *this* again
  290. 03:46 greisfu: it's just lots of negative utility
  291. 03:47 Khoth: It's better than my other idea, which was goatse-related
  292. 03:47 greisfu: goatse?
  293. 03:47 greisfu: what's that mean
  294. 03:47 Khoth: Internet shock site image. Was very famous like ten years ago
  295. 03:47 greisfu: i see
  296. 03:48 greisfu: will we post logs?
  297. 03:48 Khoth: If you want
  298. 03:48 greisfu: kk
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