Re: Tortured Algebra by an amateur

[Rick Parkany <rparkany@BORG.COM>]

Manifestly the most elegant solution I've seen, Allen. I'm glad
you enjoyed the Snickers. I'm chuckling now, myself. Thanks for
the levity.

It is a well-fitting model, I must admit. In order to capture the
sense I urged on our friend Victor, though, we need to start the
numeration @ 0 and run it through 11 (i.e., 12 solutions). In
this case, your model shows the augmentation of the angular
offset to be 360 degrees (i.e., 12). My point is that *both hands
are moving*. the clock face is not a static entity.

When I *run* this model in real gedanken Zeit, I see the *start*
from 12am (the minute hand begins to run *away* from the hour
hand) to be quite a different *thing* than the minute hand
chasing the hour hand ever so closely to 11:59:59.59.59... pm.

Please note my aggressive demand that I place on this
interpretation by the am/pm conversion...and we haven't even
talked about the 24 hour clock, or the magnificent Chronometer
that adorns the Cathedral in Strasbourg, France, have we? but I
won't stoop that low to make a point...instead, consider things
*in motion* and the model Victor implies (i.e., 11 static
*snapshots* as if motion and gearing in the hands weren't a
factor) is indefensible...

...I'm impressed Allen, thanQ for taking things *to the extreme*,
one of my favourite rhetorical ploys: but, that's why I loved
engineering and physics, too--for the modeling that occurs...
;-} rap.

"Allen Flanigan." wrote:

>  Warning:  What follows contains algebra and other mathematical
> expressions which may be unsuitable for those having weak
> mathematical constitutions.  Feel free to delete and get back
> to the real world pronto.OK, Rick, I'll admit that given my
> background I have no qualms about raising the white flag where
> complicated algebra is concerned and falling back on my old pal
> calculus to solve a problem, but my guess is the clock-hands
> overlap problem can be solved with algebra.  Here's my crack at
> it:I propose that we can express an equation (in terms of
> degrees, naturally) which relates the angular position of the
> hour hand H as a function of the angular position of the minute
> hand M.  How about if we express the possible values of hour
> hand position as ranging from zero to (12X360) 4320 degrees.Now
> a table of values:Minute    Hour0            0360
> 30720        60So our equation is M=H/12 or H=12MAnd our other
> stipulation is if N is an integer from one to twelve, and a
> potental angular overlap position of the hour hand is given
> byH-360Nand N=M/30 minus whatever partial remainder is left
> (i.e. if M = 50, N = 1, if M = 70, N = 2)Then for what possible
> values do H-360N and M coincide?For N = 1, 12M - 360 = M(12M -
> 360)/M = 1And I admit at this point that I'd forgotten my
> expansion and distribution tricks, so I had to check out
> http://members.aol.com/algbrahlp/simpdist.htmYep, I can
> distribute 1/M into the parentheses to get:12 - 360/M = 112 - 1
> = 360/M11 = 360/MM = 360/11M = 32.7272727272727273 (approx.)And
> for values of N greater than one, we just multiply the above
> answer by N.Do I get full credit?  How many snickers bars do I
> get, perfesser?Is there a more elegant solution using
> algebra?Allen F. Physicist and erstwhile dabbler in
> algebra"Veni, Duci, Vesci succarum" (I came, I algebrated, I
> ate candy)
> -----Original Message-----
> From: rparkany@BORG.COM [mailto:rparkany@BORG.COM]
> Sent: Tuesday, February 26, 2002 7:59 PM
> To: ARN-L@listsrva.CUA.EDU
> Subject: Re: [eddra] Algebra
>
>
>      Yes, my friend, Allen:   this is a job for
>      SU-PERmath...The Calculus.
>
>      The solutions for this and other cognate problems of
>      *motion* (and other such *Antimonies of Pure Reason*,
>      as Kant called them) implying the notion of *limit*,
>      such as Zeno's paradox, were impossible for the
>      classisists until Leibnitz invented the *derivative*
>      or, in his words, *the ghost of a departed
>      quantity*--one of my favourite Leibnitz quotes...
>
>      BTW, Victor...that would be 12, according to
>      Leibnitz...ask the Ghost of the departing quantity
>      approaching 11:59:59.59....
>
>      ...keep Tight!        ;-} rap.
>
>      "Allen Flanigan." wrote:
>
>     > I can vaguely recall seeing some clever question
>     > asking students to figure out the exact positions
>     > of the 12 places where the hands of the clock
>     > overlap, 12 o'clock, Just past 6 minutes after one,
>     > etc.  Maybe it got you admission into mensa or a
>     > date with a member of Mathematically Correct, or
>     > something like that.
>     >
>     >      -----Original Message-----
>     >      From: Allen.Flanigan@USPTO.GOV
>     >      [mailto:Allen.Flanigan@USPTO.GOV]
>     >      Sent: Tuesday, February 26, 2002 6:10 PM
>     >      To: ARN-L@listsrva.CUA.EDU
>     >      Subject: Re: [eddra] Algebra
>     >      Well, Rick, if you want to be a real
>     >      *academic* father, you must figure out
>     >      how she came by her deduction about "no
>     >      time".  Was it visual pattern recognition
>     >      (having never seen the particular angular
>     >      relationship of overlapping hands at the
>     >      6) or does she implicitly understand the
>     >      fact that one rotation of the big hand (
>     >      360 degrees, 12 to 12) corresponds to one
>     >      tick of the small hand between numbers
>     >      (30 degrees), and that the fractional
>     >      representations of these angular
>     >      dispositions must correspond exactly
>     >      (e.g. a quarter of the way from 12
>     >      corresponds to a quarter of the way from
>     >      6, etc.)?The answer to this question will
>     >      go a long way towards determining whether
>     >      your daughter is merely observant or a
>     >      bona fide genius ; > )No Lite Te
>     >      Bastardes Carborundum
>     >
>     >           -----Original Message-----
>     >           From: rparkany@BORG.COM
>     >           [mailto:rparkany@BORG.COM]
>     >           Sent: Tuesday, February 26,
>     >           2002 5:51 PM
>     >           To: ARN-L@listsrva.CUA.EDU
>     >           Subject: Re: [eddra] Algebra
>     >           Notes From the Crypt:
>     >           Ruminations of a Harried Math
>     >           Teacher under SBE stress...
>     >
>     >           wrt: NYS MST Standard 3.5.C
>     >
>     >           ...my Rachael Margaret (just
>     >           8yrs. old, Sunday) came home
>     >           from school today w/some
>     >           homework from a *workbook* that
>     >           showed various clock faces and
>     >           she was to indicate the time.
>     >           She was toddling along wiggling
>     >           all over the place w/one foot
>     >           on her chair, standing on the
>     >           other w/her tongue hangin' out
>     >           humming when she suddenly
>     >           exclaimed: *Hey! that's no
>     >           time!* Theresa & I popped over
>     >           and behold! she points to the
>     >           clock face w/BOTH hands on the
>     >           six and spouts out: *The small
>     >           hand needs to be half way to
>     >           the 7! THAT's no time!*
>     >
>     >           How did we handle this? of
>     >           course we praised her. We then
>     >           wrote a smiley faced memo to
>     >           her teacher relating our story.
>     >           Theresa goes up tomorrow for
>     >           the *February Birthdays*
>     >           w/cookies and she'll probably
>     >           share some small talk about
>     >           this w/the teacher. No
>     >           problem--we all know the
>     >           resources stink for the price
>     >           ya pay, but...Go figure (aka
>     >           *You do the math...*, *You do
>     >           the textbook marketing*)
>     >
>     >           I've got a poor young soul in
>     >           one of my AIS Math sections in
>     >           the HS. He was out sick one
>     >           week for his GrandPa's funeral,
>     >           wake and all; he was out a week
>     >           w/the strep. BOTH times he came
>     >           back, there was a unit test
>     >           from his Math teacher, the head
>     >           of the Department, Mr. *Drill &
>     >           Kill*. Now this fellow is
>     >           failing the Math A. He tries
>     >           real hard and does well w/help.
>     >           He's a mild mannered fellow,
>     >           docile, and easy going, and is
>     >           struggling with perceptual
>     >           problems (taking notes in class
>     >           during lecture--9th grader).
>     >
>     >           Now, I happen to be untenured.
>     >           I'm gonna call the mother when
>     >           I get through here and cite the
>     >           student handbook that says he
>     >           is entitled to two weeks to
>     >           make up excused work and
>     >           instigate her to call and get
>     >           what is his by right. Here goes
>     >           one more career option...I can
>     >           see it comin' down Highway 61
>     >           w/the brights on...
>     >
>     >           ...this teacher and folks like
>     >           him remind me of that Laurel &
>     >           Hardy routine where they've
>     >           just aggravated the ex-con and
>     >           he glares at Stan and says: *If
>     >           ya do that again, I'll hit ya
>     >           so hard HE'LL feel it*,
>     >           pointing to Ollie. Then, of
>     >           course, it happens again, he
>     >           hits Stan, and Ollie falls to
>     >           the floor in pain. The teachers
>     >           rant and complain about the
>     >           problems that *non-standard*
>     >           behaviors bring to their
>     >           desks--and sometimes, it's
>     >           warranted in the extreme (e.g.,
>     >           gross, illegal absences from
>     >           school)--but not in most cases.
>     >           These folks just don't want to
>     >           be bothered making up an
>     >           *equivalent form* and try to
>     >           *go over the test* in class as
>     >           soon as possible, yet feel that
>     >           will give these folks and
>     >           *additional advantage*...
>     >
>     >           ...the nice thing about loosing
>     >           your job is the prospects the
>     >           next one brings.    ;-} rap.
>     >
>     >           "Dr. Leo Casey" wrote:
>     >
>     >           > In his book _Radical
>     >           > Equations_, Bob Moses makes a
>     >           > telling point about the
>     >           > different intellectual
>     >           > structures of arithmetic and
>     >           > algebra to explain why some
>     >           > students have a difficulty
>     >           > making the transition from the
>     >           > former to the latter.
>     >
>     >           ...SNIP...
>     >           --
>     >           "Dein Wachstum sei feste und
>     >           lache vor Lust!
>     >           Deines Herzens Trefflichkeit
>     >           Hat dir selbst das Feld
>     >           bereit',
>     >           Auf dem du bluehen musst." JS
>     >           Bach: Bauern Kantata
>     >           Richard A. Parkany: SUNY@Albany
>     >
>     >           Prometheus Educational Services
>     >
>     >           http://www.borg.com/~rparkany/
>     >           Upper Hudson & Mohawk Valleys;
>     >           New York State, USA
>     >
>      --
>      "Dein Wachstum sei feste und lache vor Lust!
>      Deines Herzens Trefflichkeit
>      Hat dir selbst das Feld bereit',
>      Auf dem du bluehen musst." JS Bach: Bauern Kantata
>      Richard A. Parkany: SUNY@Albany
>      Prometheus Educational Services
>      http://www.borg.com/~rparkany/
>      Upper Hudson & Mohawk Valleys; New York State, USA
>
>
--
"Dein Wachstum sei feste und lache vor Lust!
Deines Herzens Trefflichkeit
Hat dir selbst das Feld bereit',
Auf dem du bluehen musst." JS Bach: Bauern Kantata
Richard A. Parkany: SUNY@Albany
Prometheus Educational Services
http://www.borg.com/~rparkany/
Upper Hudson & Mohawk Valleys; New York State, USA