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From: DrMajorBob on 4 Jan 2010 05:59 The simplest algorithm for determining "what's the next term" is simply "fill in the first bubble". Bobby On Sun, 03 Jan 2010 02:42:21 -0600, Andrzej Kozlowski <akoz(a)mimuw.edu.pl> wrote: > > On 2 Jan 2010, at 19:05, DrMajorBob wrote: > >> If one had the Encyclopedia of Integer Sequences handy, those SAT >> questions could be interesting. But they'd still be nonsense. > > I think only their wording should be changed to something like: "find > the simplest formula generating the sequence ... where simplest means > requiring the least number of standard mathematical symbols to write > down". These kind of questions, of course, should probably allow the > possibility that the person answering the question finds an answer that > is simpler than the expected one. > > -- DrMajorBob(a)yahoo.com
From: DrMajorBob on 4 Jan 2010 06:00 > The issue here is > whether the student has enough common culture with the test writer to > find the same answer. And that's *always* an issue. So those are cultural conformity questions?!? That's even worse than I thought! Bobby On Sun, 03 Jan 2010 02:40:36 -0600, Noqsi <jpd(a)noqsi.com> wrote: > On Jan 2, 3:05 am, DrMajorBob <btre...(a)austin.rr.com> wrote: >> When I clicked on the link below, the search field was already filled >> with >> the sequence >> >> target = {1, 2, 3, 6, 11, 23, 47, 106, 235}; >> >> Searching yielded "A000055 Number of trees with n unlabeled >> nodes." >> >> I tried a few Mathematica functions on it: >> >> FindLinearRecurrence(a)target >> >> FindLinearRecurrence[{1, 2, 3, 6, 11, 23, 47, 106, 235}] >> >> (fail) >> >> FindSequenceFunction(a)target >> >> FindSequenceFunction[{1, 2, 3, 6, 11, 23, 47, 106, 235}] >> >> (fail) >> >> f[x_] = InterpolatingPolynomial[target, x] >> >> 1 + (1 + (1/ >> 3 + (-(1/ >> 12) + (7/ >> 120 + (-(1/ >> 60) + (1/144 - (41 (-8 + x))/20160) (-7 + x)) (-6 + >> x)) (-5 + x)) (-4 + x)) (-3 + x) (-2 + x)) (-1 + x) >> >> and now the next term: >> >> Array[f, 1 + Length(a)target] >> >> {1, 2, 3, 6, 11, 23, 47, 106, 235, 322} >> >> But, unsurprisingly, the next term in A000055 is 551, not 322. >> >> A000055 actually starts with another three 1s, but that doesn't change >> things much: >> >> target = {1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235}; >> >> FindLinearRecurrence(a)target >> >> FindLinearRecurrence[{1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235}] >> >> (fail) >> >> FindSequenceFunction(a)target >> >> FindSequenceFunction[{1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235}] >> >> (fail) >> >> f[x_] = InterpolatingPolynomial[target, x] >> >> 1 + (1/24 + (-(1/ >> 40) + (1/ >> 90 + (-(1/ >> 280) + (1/ >> 1008 + (-(43/ >> 181440) + (191/3628800 - (437 (-11 + x))/ >> 39916800) (-10 + x)) (-9 + x)) (-8 + x)) (-7 + >> x)) (-6 + x)) (-5 + x)) (-4 + x) (-3 + x) (-2 + x) (-1 + >> x) >> >> Array[f, 1 + Length(a)target] >> >> {1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235, -502} >> >> So I ask you, from the data alone: what's the next term? > > It's the sort of question where one might expect a specialist to > recognize a familiar sequence. It's all context. > > Consider that in a narrow mathematical sense, spectroscopy is an > utterly ambiguous, "ill conditioned" problem. But show me a gigagauss > cyclotron spectrum, and I'll recognize it as such (see the > acknowledgment at the end of arxiv.org/pdf/astro-ph/0306189: the > authors were struggling to contrive an interpretation from atomic > physics before one of them showed the spectrum to me). But I expect > very few could do this, since few have the background. > >> >> If one had the Encyclopedia of Integer Sequences handy, those SAT >> questions could be interesting. But they'd still be nonsense. > > No they are not. Remember that the SAT isn't about the ability of a > student to function in some ideal abstract world of infinite > possibility. In the real world of academia, every single question they > will encounter will be ambiguous in some sense. The issue here is > whether the student has enough common culture with the test writer to > find the same answer. And that's *always* an issue. > >> >> Bobby >> >> >> >> On Fri, 01 Jan 2010 04:32:58 -0600, Noqsi <j...(a)noqsi.com> wrote: >> > On Dec 31, 1:16 am, DrMajorBob <btre...(a)austin.rr.com> wrote: >> >> >> This is a little like those idiotic SAT and GRE questions that ask >> >> "What's >> >> the next number in the following series?"... where any number will >> do. >> >> Test writers don't seem to know there's an interpolating polynomial >> (for >> >> instance) to fit the given series with ANY next element. >> >> > Explanations in terms of epicycles may be mathematically adequate in a >> > narrow sense, but an explanation in terms of a single principle >> > applied repeatedly is to be preferred in science. The ability to >> > recognize such a principle is important. >> >> > And my mathematical logician son (who's looking over my shoulder) >> > directed me tohttp://www.research.att.com/~njas/sequences/for >> > research on this topic. When he encounters such a sequence in his >> > research, he finds that knowledge of a simple genesis for the sequence >> > can lead to further insight. >> >> -- >> DrMajor...(a)yahoo.com > > -- DrMajorBob(a)yahoo.com
From: Andrzej Kozlowski on 4 Jan 2010 06:00 I am not sure I you can call that a "forumla". Besides StringLength["fill in the first bubble"] 24 So, if we also include them in the simplicity test, this one is probably not going to pass. On 4 Jan 2010, at 07:38, DrMajorBob wrote: > The simplest algorithm for determining "what's the next term" is simply "fill in the first bubble". > > Bobby > > On Sun, 03 Jan 2010 02:42:21 -0600, Andrzej Kozlowski <akoz(a)mimuw.edu.pl> wrote: > >> >> On 2 Jan 2010, at 19:05, DrMajorBob wrote: >> >>> If one had the Encyclopedia of Integer Sequences handy, those SAT >>> questions could be interesting. But they'd still be nonsense. >> >> I think only their wording should be changed to something like: "find >> the simplest formula generating the sequence ... where simplest means >> requiring the least number of standard mathematical symbols to write >> down". These kind of questions, of course, should probably allow the >> possibility that the person answering the question finds an answer that >> is simpler than the expected one. >> >> > > > -- > DrMajorBob(a)yahoo.com
From: Vince Virgilio on 4 Jan 2010 06:02 On Jan 3, 3:38 am, Andrzej Kozlowski <a...(a)mimuw.edu.pl> wrote: SNIP > Philosophically speaking, we cannot be sure that that is not exactly > what we do in our own minds when we thing of "real numbers" - our brains > are quite possibly only finite state automata and our reals are also > approximations to "reals" most of whose digits always remain unknown. So > I am not convinced that there is in this respect any fundamental > difference between humans and computers. SNIP On the other hand, perhaps our intellect does work with the Reals. (If the mind can handle some precision, why not all precision?) Which would suggest, at least, a different order of computation. Not merely Super, but Hyper. ( In debt to Stephen Wolfram's latest novelty essay for this concept. http://www.stephenwolfram.com/publications/recent/fqxi09/ ) Vince Virgilio
From: Noqsi on 4 Jan 2010 06:03
On Jan 3, 1:37 am, DrMajorBob <btre...(a)austin.rr.com> wrote: > Mathematica Reals may not be Rational, but computer reals certainly are. > (I shouldn't have capitalized "reals" in the second case.) Only in the shallow sense that there is a low entropy mapping between computer "reals" and rational numbers in the intervals they represent. But computer "reals" don't behave arithmetically like rationals or the abstract "reals" of traditional mathematics. This fact often causes confusion. |