🚀 go-pugleaf

dan@danberg.llnl.gov (Dan Bergmann) writes:
>[The following question is supposedly from an SAT, typically with
> questionable answers because of smart students and not-so-smart 
> question-writers]
>
>Two circles are next to each other and touching at one point (one is NOT
>inside the other). The larger circle has twice the radius of the smaller
>circle. The smaller circle rolls around the large one until it comes back
>to its original starting position. How many revolutions does it make about
>its center?

Given that this question was on the SAT, I'm not surprised.  It seems fairly
obvious to me that the larger circle has a circumference twice that of the
smaller circle.  However, the question asks for the number of "revolutions
about its center".  This raises several options:

1) Revolutions of the larger circle about the larger circle's center --
(since the larger circle is the nearest antecedent of the pronouns "it"
and "its") 0.

2) Revolutions of the smaller circle... -- 0.  "Revolution" implies motion
about a point outside a body.  The center of the smaller circle moves with
the circle, and the circle doesn't even move around the initial location of
the center point.

3) *Rotations* of the larger circle... -- 0. Again, using the nearest
antecedent, and the fact that "rotation" implies motion about a point
interior of the body in question.  Oh, and this also assumes that the frame
of reference for defining a rotation is based on the orientation of the
larger circle (i.e. the large circle doesn't move).

4) Rotations of the smaller circle... -- 3.

Consider the circles joined as: Oo, touching at the larger circle's "east"
point.  Move the smaller circle counter-clockwise.  When the smaller
circle is at "north", it is touching the larger circle with the
point originally at the smaller circle's "east".  Thus, in 1/4
revolution about the larger circle, the smaller circle has made 3/4
rotation (the "east" point has moved counter-clockwise to "south").
Completing the trip, the smaller circle does 4*3/4=3 rotations.

If the question as phrased above is accurately quoted from the SAT, I have
even less faith in the SATs than I did before.  If not, well, pick an
answer.

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In article <C1IyDB.2K5@fastrac.llnl.gov> dan@danberg.llnl.gov (Dan Bergmann) writes:
>Two circles are next to each other and touching at one point (one is NOT
>inside the other). The larger circle has twice the radius of the smaller
>circle. The smaller circle rolls around the large one until it comes back
>to its original starting position. How many revolutions does it make about
>its center?

I've certainly heard this before, and it seems to me it ought to be in
the rec.puzzles archive (I couldn't find it either).

SPOILER ahead...



The answer is 3.

The way I look at it is as you roll the circle around the larger one,
turn your head at the same time so that the small circle always appears
on top.  Suppose the small circle was supposed to roll counterclockwise
around the larger one.  Then what you will see instead is one clockwise
revolution of the larger circle and two counterclockwise revolutions
of the small circle.

But now you have seen the larger circle move when it shouldn't have.
To restore the correct number of revolutions of the large circle (i.e.
0) you must turn your head back so both circles turn counterclockwise
one time.  This is the third such revolution for the small circle.

This situation is analogous to the difference between the number of
solar days in a year (~365.25) and the number of sidereal days
(~366.25).

-- David Karr (karr@cs.cornell.edu)

Dan Bergmann writes:

> Two circles are next to each other and touching at one point (one is NOT
> inside the other). The larger circle has twice the radius of the smaller
> circle. The smaller circle rolls around the large one until it comes back
> to its original starting position. How many revolutions does it make about
> its center?

My only comment: how many days are there in a year?  365.25?  366.25?  (And
yes, I know that the tail is actually something like .242198+.  Stick to the
point.)

The smaller circle makes two "solar" rotations, three "siderial" rotations.

Re: the 13 spheres

Yes, I was able to track down an old copy of "Mathematical Circus," by
Martin Gardner of SA fame.  In it, Coxeter says that David Gregory and
Isaac Newton had been talking about the distribution of stars of various
magnitudes, and this had led to the discussion of whether thirteen unit
spheres could touch a fourteenth.  Gregory believed they could, and Newton
disagreed.  "One hundred eighty years were to pass before Newton was proved
correct."  It's close: I was able to place 11 down and two more which
intersected two of the 11 by just a sliver.

So, what's the answer for the fourth dimension?  I'd guess that it's
something around 19 or 20.

A last question:

Q.  Three circles of any radii r1, r2, and r3 can be drawn so that they
    are mutually tangent.  Also, a fourth circle can be drawn that is
    tangent to the first three circles, but its radius is determined
    by those of the other circles.  If the circles are "close enough"
    in radius, then the fourth circle surrounds the first three;
    otherwise, it is external to the first three.  Prove or disprove:
    in those cases where the fourth circle surrounds the other three,
    those three circles consume the maximum fraction of the area of
    the fourth circle when r1=r2=r3.

byron elbows
(mail to brian@cs.ucla.edu)

I wrote:

> Q.  Three circles of any radii r1, r2, and r3 can be drawn so that they
>     are mutually tangent.  Also, a fourth circle can be drawn that is
>     tangent to the first three circles, but its radius is determined
>     by those of the other circles.  If the circles are "close enough"
>     in radius, then the fourth circle surrounds the first three;
>     otherwise, it is external to the first three.  Prove or disprove:
>     in those cases where the fourth circle surrounds the other three,
>     those three circles consume the maximum fraction of the area of
>     the fourth circle when r1=r2=r3.

I should add that in all cases, TWO fourth circles can be drawn.  One is
always inside the first three.  The other one is either external to the
first three, or else it encloses the first three.

byron elbows
(mail to brian@cs.ucla.edu)