HDTenEightyP/NetNews huggingface dataset:
From: leue@crd.ge.com (Bill Leue)
Newsgroups: rec.puzzles
Subject: Re: Tiling a plane
Date: 28 Jul 92 18:04:35 GMT
Reply-To: leue@crd.ge.com
Organization: General Electric Research & Development

In article 712246849@ro, dfs@doe.carleton.ca (David F. Skoll) writes:
>In article <711892126.F00001@csource.oz.au> joe@csource.oz.au (Joe Slater) writes:
>>A plane can be tiled with three regular geometric shapes: squares, triangles and
>>hexagons. It can also be tiled with pairs of shapes: hexagons and triangles, or
>>octagons and squares.
>
>>Are there any other pairs that will tile a plane? What about triplets? Answers
>>requiring fractal-like series to fill up gaps aren't counted.
>
>What about Penrose's kites and darts?  I don't have the exact specs for
>building them, but I know they tile the plane aperiodically.
>
>--
>David F. SKoll

	The specs are as follows:  both the kite and dart are quadrilaterals.
	The kite can be constructed from two isosceles triangles with side lengths
	rho, rho, and 1 (rho being the golden number = (sqrt(5) + 1)/2 =
	1.618... with two of the rho-length sides abutting.  The dart can
	be constructed from two isosceles triangles with side lengths, rho,
	1, and 1, with two of the 1-length sides abutting.

	FYI, the areas of the kite and dart are also in the ratio rho to 1, 
	and for any perfect tiling (to infinity) the ratio to the number
	of kites to the number of darts is also rho.  There are many other
	interesting connections between the kites and darts and the golden
	ratio.  The other Penrose tilings (P1 and P3 ['rhombs'] also have
	strong connections to the golden ratio.  For more information,
	see Martin Gardener's Book "Penrose Tilings to Trapdoor Ciphers"
	(Freeman, ppb) or Grunwald & Shepard's encyclopedic "Tilings and
	Patterns."  (also pub. by Freeman)

-Bill Leue
leue@crd.ge.com
From: t9114145@phillip.edu.au
Newsgroups: rec.puzzles
Subject: Re: Ten equals 9.9999....
Date: 29 Jul 92 16:41:01 GMT
 <1992Jul28.114531.12524@phillip.edu.au> <Bs3LuG.L1M@watdragon.uwaterloo.ca>
Organization: RMIT (Bundoora Campus)

In article <Bs3LuG.L1M@watdragon.uwaterloo.ca>, deghare@daisy.uwaterloo.ca (Dave Hare) writes:
> In article <1992Jul28.114531.12524@phillip.edu.au> t9114145@phillip.edu.au writes:
>>In article <1992Jul27.150032.12515@phillip.edu.au>, t9114145@phillip.edu.au writes:
>>> In article <Bry202.4BF@watdragon.waterloo.edu>, deghare@daisy.waterloo.edu (Dave Hare) writes:
>>>> In article <1992Jul25.145154.12498@phillip.edu.au> t9114145@phillip.edu.au writes:
>>>>>In article <1992Jul24.102553.17411@pasteur.Berkeley.EDU>, tjoa@cory.Berkeley.EDU (Richard Tjoa) writes:

(stuff missing)
> 
>>My teachers used to love saying "don't believe calculators ever! They will
>>tell you that things like 9.999... = 10!" :-)
> 
> Then you should have taken your teachers to task on this.  It is certainly 
> true that you should always question the results of finite precision machines,
> because, as I have frequently told my students in numerical analysis, when
> it comes to floating point arithmetic, computers are always *wrong*.
> 

Well, I was about 12 at the time and not that interested, just disappointed
that I still had to use fractions.

> However, a calculator can never tell you that 9.999... = 10 .  The most
> it can tell you is that the result of a computation is 10 plus or minus
> some error, whose magnitude depends on the details of the computation.
> 
> Dave Hare
-- 

Adam Eberbach      		| The clever men at Oxford
t9114145@phillip.edu.au         | Know all that there is to be knowed.
Computer Science student, 	| But they none of them know one half as much
Phillip Institute of Technology.| As intelligent Mr. Toad!
From: miklas@remus.rutgers.edu (Keith Miklas)
Newsgroups: rec.puzzles
Subject: Alien problem
Date: 29 Jul 92 18:42:42 GMT
Organization: Rutgers Univ., New Brunswick, N.J.

Don't even read this one unless you have a few hours on your hands.

Two scientists are walking together discussing how much better an
editor emacs is than vi.

An alien appears, and says "hi guys, I'm
thinking of 2 numbers between 3 and 97 (inclusive)."  He then says,
"The sum of these two numbers is..." and whispers it into the first
professor's ear.  He then continues, "The product of these two numbers
is..." and whispers it into the second professor's ear.  The alien then
vanishes in a cool puff of smoke.

The first professor says to the second, "It is impossible for you to
know what the two numbers are."

The second arrogantly replies, "Well, now that you've said that, I do
know what they are."

The first smiles and says, "Well, now I know what they are too."

And they continue their conversation.



Fellahs, The question is, What are the two numbers?
Newsgroups: rec.puzzles
From: radcliff@csd4.csd.uwm.edu (David G Radcliffe)
Subject: Re: Proof of -1 = 1
Organization: University of Wisconsin - Milwaukee
Date: Fri, 24 Jul 1992 06:21:32 GMT

Another proof of -1 = 1.  I hope this hasn't been posted before.

  1  =  3 - 2
     =  (2 + 1) - (3 - 1)
     =  2 + 1 - 3 - 1
     =  2 - 3
     =  -1     QED

--
David Radcliffe                                
radcliff@csd4.csd.uwm.edu
From: clong@remus.rutgers.edu (Chris Long)
Newsgroups: rec.puzzles
Subject: Gummy Drop-bears
Date: 29 Jul 92 07:30:41 GMT
Organization: Rutgers Univ., New Brunswick, N.J.


Real gummy drop-bears mass 10 grams, while imitation gummy
drop-bears mass 9 grams.  Spike has 7 cartons of gummy drop-bears,
4 of which contain real gummy drop-bears, the others imitation.
Using a scale only once and the minimum number of drop-bears, how
can Spike determine which cartons contain real gummy drop-bears?
-- 
Chris Long, 265 Old York Rd., Bridgewater, NJ  08807-2618

"In a study of schoolboys, an educator discovered a correlation between size
 of feet and quality of handwriting.  The boys with the larger feet were,
 on the average, older."        Wallis & Roberts, _The Nature of Statistics_
From: imc@prg.ox.ac.uk (Ian Collier)
Newsgroups: rec.puzzles
Subject: Re: Plane Division
Date: 24 Jul 92 12:44:55 GMT
Organization: Oxford University Computing Laboratory, UK
X-Local-Date: Friday, 24th July 1992 at 1:43pm BST

In article <2387.imc@uk.ac.ox.prg>, I wrote:
[If a million points are selected randomly in a 2D closed bounded ares, can
the points and the area be bisected by a single line?]

>                                                Assuming that lines may pass
>through points, if no three of the points are collinear then the answer is
>"yes". [and]
>restricting the points so that no three are collinear with the centroid will
>work.  If lines may not pass through points, then restricting the points so
>that no two are collinear with the centroid will work.

I received the following interesting response from Martin.Ward@uk.ac.durham,
who has given me permission to quote his note:

|The puzzle stated that "1 million random points are chosen."
|What is the probability that a random point chosen within a closed and bounded
|area lies on a given line? Zero of course! So the nth chosen point (n>1)
|has zero probability of lying on each of the lines joining the centroid
|and the ith point (1<=i<n). So the restriction is referring to an impossible
|situation and so is not needed!
|
|It's amazing how many people don't read the question properly!
|
|                Martin.
|
|JANET: Martin.Ward@uk.ac.durham    Internet (eg US): Martin.Ward@durham.ac.uk
|or if that fails:  Martin.Ward%uk.ac.durham@nsfnet-relay.ac.uk
|or even: Martin.Ward%DURHAM.AC.UK@CUNYVM.CUNY.EDU
|BITNET: Martin.Ward%durham.ac.uk@UKACRL UUCP:...!uknet!durham!Martin.Ward

Ian Collier
Ian.Collier@prg.ox.ac.uk | imc@ecs.ox.ac.uk
Newsgroups: rec.puzzles
From: deghare@daisy.uwaterloo.ca (Dave Hare)
Subject: Re: Ten equals 9.9999....
Organization: University of Waterloo
Date: Tue, 28 Jul 1992 12:11:03 GMT

In article <1992Jul28.114531.12524@phillip.edu.au> t9114145@phillip.edu.au writes:
>In article <1992Jul27.150032.12515@phillip.edu.au>, t9114145@phillip.edu.au writes:
>> In article <Bry202.4BF@watdragon.waterloo.edu>, deghare@daisy.waterloo.edu (Dave Hare) writes:
>>> In article <1992Jul25.145154.12498@phillip.edu.au> t9114145@phillip.edu.au writes:
>>>>In article <1992Jul24.102553.17411@pasteur.Berkeley.EDU>, tjoa@cory.Berkeley.EDU (Richard Tjoa) writes:
>>>>
>>>>> Ok, here's something interesting...
>>>>> 9.9999...  (nine point nine repeating)  =   10  (ten)
>>>>
>>>>There's the error. 9.9999... does not equal 10, it approaches 10.
>>> 
>>> No, it equals 10.  This type of question has subtleties that too often
>>> are too easily dismissed.  Consider:
>> 
>> Alright, if 9.999... = 10, then why isn't .33333... exactly 1/3?
>
>After consulting a maths lecturer I'll admit I'm wrong. It still seems
>right though, after working with devices accurate to a finite number
>of decimal places.

To any finite number of places, 9.9999...999 is not equal to 10.  But
9.9999.... is equal to 10.  "Devices accurate to a finite number of
decimal places" will not help to resolve this problem.

>My teachers used to love saying "don't believe calculators ever! They will
>tell you that things like 9.999... = 10!" :-)

Then you should have taken your teachers to task on this.  It is certainly 
true that you should always question the results of finite precision machines,
because, as I have frequently told my students in numerical analysis, when
it comes to floating point arithmetic, computers are always *wrong*.

However, a calculator can never tell you that 9.999... = 10 .  The most
it can tell you is that the result of a computation is 10 plus or minus
some error, whose magnitude depends on the details of the computation.

Dave Hare
Newsgroups: rec.puzzles
From: phaedrus@cs.washington.edu (Mark Phaedrus)
Subject: Re: How Many Composites?
Organization: University of Washington Computer Science
Date: Thu, 30 Jul 92 06:51:28 GMT

In article <12435@ntdd-1> grayt@Software.Mitel.COM (Tom Gray) writes:
>Consider a prime number - p

>How many composite numbers with p as a factor that are:

>less than p^2

     p-2.  Reasoning:  Any composite number with prime number p as a factor
must be a multiple of p, so the acceptable choices are 2p,3p,...,(p-2)p,(p-1)p.
There are p-2 terms in this series, so there are p-2 suitable composites.

>less than p!

     (p-1)!-2.  Reasoning:  Again, the only acceptable choices are multiples 
of p, so we have 2p,3p,...((p-1)!-2)p,((p-1)!-1)p.  (p! can also be expressed
as (p-1)!p, which is where the funny-looking upper-bound terms come from).
There are (p-1)!-2 terms in this series.
-- 
\ Mark Phaedrus, Computer Science Major, Univ. of Washington, Seattle, WA \*\
*\ Work: phaedrus@cs.washington.edu        Play: phaedrus@u.washington.edu \*\
\*\ Cute .sig quote still under alpha-testing; planned for release in 1993. \*\
 - Hi!  I'm an anti-virus utility!  Install me in your .signature right away! -
Newsgroups: rec.puzzles
From: rcutler@banach.WPI.EDU (Robert Cutler)
Subject: Re: Proof of -1 = 1
Originator: rcutler@banach.WPI.EDU
Organization: Worcester Polytechnic Institute
Date: Fri, 24 Jul 1992 05:04:58 GMT


In article <chan.711948973@s1.elec.uq.oz.au>, chan@elec.uq.oz.au (Chuen Wen Chan) writes:
|> 
|> I hope this problem hasn't been posted before, but here it is anyway:
|> 
|> 	sqrt(-1) = sqrt(-1)
|> 
|> 	sqrt( -1 / 1 ) = sqrt( 1 / -1 )
|> 
|> 	sqrt(-1)    sqrt(1)
|> 	-------- = ---------
|> 	sqrt(1)     sqrt(-1)
|> 
|> 	now cross multiply and we get:
|> 
|> 	-1 = 1  QED
|> 
|> ie:  Everything we learnt in classical maths is wrong
|>    => no more maths subjects in future
|>    => all mathematicians lose their jobs. Yippe!
|> 
|> PS:  If no one gets the solution by Monday, it will be posted.
|> 
|> --
|>  ,-_|\  Chuen Chan (chan@elec.uq.oz.au)
|> /     * Department of Electrical Engineering
|> \_,--_/ The University of Queensland, St Lucia
|>      v  Brisbane, Australia 4072


	Sorry only one little problem, sqrt(-1)^2 = i^2 = -1.
From: althof5805@iscsvax.uni.edu
Newsgroups: rec.puzzles
Subject: Re: Rift in the fabric of the watch!
Date: 30 Jul 92 17:13:12 -0500
Organization: University of Northern Iowa

In response to the question of what screwed up the digital watch--

You set your watch 12 hours behind.  This means that on Sunday at midnight,
the date didn't change (because the watch thought that it was noon).  But, at
noon on Monday (after you'd set your date ahead one day), the clock thought it
was midnight and advanced the date.

I know this is the case because I do it all the time!  Annoying, isn't it?

--Stephanie
Newsgroups: rec.puzzles
From: mmokhtar@attmail.com
Subject: Re: 3 puzzles
Originator: mmdh@cbnewsf.cb.att.com
Organization: AT&T
Date: Mon, 27 Jul 1992 14:55:17 GMT

In article <1992Jul26.134105.13917@ens.fr> courtes@trimaran.ens.fr (Francois Courtes) writes:
>>>>	What is the first number, when spelled out (e.g. O-N-E) is
>>>>in alphabetical order?
>>>40  (F-O-R-T-Y)
>In French the first one is 2 (D-E-U-X).
>Is there a language in which 1 is so ?

  Eins in German.

Mazen Mokhtar
Newsgroups: rec.puzzles
From: dougc@bert.cs.byu.edu (Douglas R. Cannon)
Subject: Re: Babble Sort (restatment)
Organization: BYU, Provo, Utah
Date: 28 Jul 92 13:01:33


>>  Here is a problem for the programmers out there :)
>>  Line up the numbers from 1 to 20 in the usual way.
>>  Now  imagine sliding a box, which can contain two numbers,
>>  along the sequence. If the numbers in the box have two or more
>>  digits in common, switch the numbers.
>>
>>  After a number of passes with the box, whitch number is last and 
>>  whitch is first.
>
>   The nombers should be switched if they have ONE or more digits in
>   common, ie  12&20  or 1&11

It seems to me that 12 will sometimes be last, 10 will sometimes be
last, and 20 will be last most of the time.
1 will always be first, in fact, the numbers 1-8 will never change
positions, and 9 will only switch with 19 sometimes, being switched
back with the next pass of the box.

After 77 passes of the box, the numbers return to the original order.

Doug

Newsgroups: rec.puzzles
From: s290@brems.ii.uib.no ( Per Ivar Roervik)
Subject: Babble Sort
Organization: DevNull A/S
Date: Tue, 28 Jul 92 16:43:09 GMT

  Here is a problem for the programmers out there :)
  Line up the numbers from 1 to 20 in the usual way.
  Now  imagine sliding a box, which can contain two numbers,
  along the sequence. If the numbers in the box have two or more
  digits in common, switch the numbers.

  After a number of passes with the box, whitch number is last and 
  whitch is first.
Newsgroups: rec.puzzles
From: mack@isl.Stanford.EDU (mack)
Subject: Easy prob. question
Organization: Stanford University
Date: Fri, 24 Jul 92 22:57:45 GMT


This looks to be an easy one:

Q: Two cards are picked at random from a complete deck of cards. What
   is the probability that they are of the same suit?

Have fun.   :-)

    --mack

--
For as long as I know how to love, I know I'll stay alive.
I've got all my life to live, I've got all my love to give
and I'll survive, I will survive.          --Gloria Gaynor
Newsgroups: rec.puzzles
From: jkenton@world.std.com (Jeff Kenton)
Subject: Re: Crossing a river
Organization: Kenton Systems Corporation, Weston MA
Date: Fri, 31 Jul 1992 14:01:18 GMT

In article <1992Jul30.173556.5883@iscsvax.uni.edu> althof5805@iscsvax.uni.edu writes:
>Three kings are travelling with three thieves.  The kings know that if, at any
>time, thieves outnumber kings (e.g. if one king leaves to take a leak behind an
>accacia tree), the thieves will kill this outnumbered kings.  The kings may
>outnumber the thieves, and the number may be equal for both sides (one king and
>one thief may go to take a leak together), but the thieves must *never*
>outnumber the kings.
>

I've always seen this puzzle stated in terms of 3 missionaries and 3
cannibals, where the cannibals try to avoid being outnumbered by the
missionaries.  It even works if only 1 of the missionaries can row.



-- 
-------------------------------------------------------------------------
=			Jeff Kenton	(617) 894-4508			=
=			    jkenton@world.std.com			=
-------------------------------------------------------------------------
Newsgroups: rec.puzzles
Subject: Re: preposition proposition
From: wald2@husc9.harvard.edu (Kevin Wald)
Date: 29 Jul 92 13:07:11 EDT
Organization: Harvard University Science Center

In article <1992Jul28.000455.4263@iscnvx.lmsc.lockheed.com> marshall@force.ssd.lmsc.lockheed.com writes:
>In article <1992Jul24.124459.1280@newstand.syr.edu>, kanad@top.cis.syr.edu (Kanad Chakraborty) writes:
>>In article <2113@tymix.Tymnet.COM> filfeit@rael.Tymnet.COM writes:
>>>Everybody knows that a preposition is a word you should not
>>>end a sentence with B-).
>>>
>>>Can someone post a sentence that ends with FIVE prepostions?
>
>I remember this one :
>
>What did you bring that book that I didn't want to be read to out of up for?
>
>OK, so maybe these aren't prepositions as they're used here, but at times
>they *ARE* prepositions. I can't believe there is really a valid sentence
>ending in five prepositions that are really being used as prepositions.
>
>=============================================================================
>Bob Marshall                     \\     "It ain't age
>Lockheed Missiles & Space Co.     \\     That makes me look this way
>Sunnyvale, CA                      \\    It ain't the years, boys,
>marshall@force.ssd.lmsc.lockheed.com\\   It's the miles"
>"I tell the truth 'cept when I lie"  \\     - Chris LeDoux
>=============================================================================

Actually, in English it is possible to get sentences ending in abitrary
numbers of prepositions which actually function as prepositions(of course,
if they have to be *different* prepositions, eventually you run out of
vocabulary). All you have to do is take the rule that a noun phrase
can be of the form "the OBJECT_OF_PREPOSITION SUBJECT VERB DIRECT_OBJECT
PREPOSITION," and note that since the DIRECT_OBJECT is itself a noun 
phrase, we can expand recursively, adding one preposition with each 
iteration. Thus, starting with the simple sentence "Which bed is the
mouse under?" we can get

  Which bed is the mouse you threw the wrench at under?

  Which bed is the mouse you threw the wrench I fixed the truck 
  with at under?

  Which bed is the mouse you threw the wrench I fixed the truck
  the vicar left the piano in with at under?

and, finally,

  Which bed is the mouse you threw the wrench I fixed the truck
  the vicar left the piano we wrote a song about in with at under?

or, equivalently

  Which bed is the mouse which you threw the wrench which I fixed
  the truck which the vicar left the piano which we wrote a song
  about in with at under?

This sentence, if it were rewritten to avoid having prepositions
at the end, would come out as

  "Under which bed is the mouse at which you threw the wrench with
   which I fixed the truck in which the vicar left the piano about
   which we wrote a song?"

which would be somewhat clearer--the problem with sentences with multiple
prepositions at the end is that it rapidly becomes impossible to parse
them without pencil and paper.

Kevin
wald2@husc.harvard.edu

Newsgroups: rec.puzzles
From: bhayes@Xenon.Stanford.EDU (Barry Hayes)
Subject: Re: Ten equals 9.9999....
Followup-To: sci.math
Keywords: nine ten repeating digits
Organization: Computer Science Department, Stanford University.
Date: Wed, 29 Jul 1992 16:50:39 GMT

Since this has gone well beyond being a puzzle, I suggest that
discussion continue in sci.math -- the followup-to of this message is
so set.

What it means for a number to be a well-defined real, for one real to
be equal to another, and so on all follow from what we might mean by
"real number" and "equal to".  Until there is an agreement as to what
those terms mean, let alone such ill-defined terms as "9.999...", we
cannot expect to come to agreement.

For example, I will define the following sequence:
P[n] is the nth digit in the decimal expansion of Pi: P[1] = 1, P[2] = 4. . .
A[n] is 0 if P[n] through P[100+n] are all ones.
A[n] is 1/10^n if P[n] through P[n+100] are all twos.
A[n] is P[n]/10^n

Note that since any P[n] can be computed, any A[n] can be computed.  

Now define A as the infinite sum of all the A[n]'s.  If you believe in
Pi, you have to believe in A.  Well, what's the order between A and
Pi?  Depends a lot on Pi, doesn't it?

Some would deny both A and Pi, by the way.  "Since constructing Pi
requires an infinite number of calculations, it is not reasonable to
discuss it," they would say.

In short, tell me what you mean by "A is a real number", "A and B are
equal", "A and B are not equal", "10", and "9.999..." and maybe I can
tell you if 10=9.9999...
Newsgroups: rec.puzzles
From: mrw@mpx1.lampf.lanl.gov (Wennersten, Matthew R.)
Subject: Crystal Balls
Organization: Los Alamos National Lab
Date: Thu, 23 Jul 1992 19:27:00 GMT

with a vacuum, such that weight shape etc. gave no clues, it would
be possible to figure out which was which by shining light (lasar for ex.)
through both balls and seeing which ball "bent" the light more.  That
ball would be the one filled with helium.  
					-MRW
From: warwick@cs.uq.oz.au (Warwick Allison)
Newsgroups: rec.puzzles
Subject: Re: Frog and Turtle play 20
Date: 31 Jul 92 01:53:50 GMT
Reply-To: warwick@cs.uq.oz.au

tolman%asylum.cs.utah.edu@cs.utah.edu (Kenneth Tolman) writes:

>Frog:(eager for retribution) Lets play twenty questions!

>Frog:  Anything?  Alright, lets go.  It will be impossible for you to get it
>       now in 20 questions.

>Turtle: Uh Oh.  That is not possible to do, is it?

>Frog: Yes, 19 left.

>Turtle: It is not my 21st question is it that I have to guess?

>Frog: No.  18 left.


>What was it that Frog picked?  

We cannot know... since if we did, Turtle would also know, then we would
be wrong.

Or Frog could be wrong.

Or it could be "The concatenation of questions 1 to 20"... but I would
consider any thing that Frog could not write down literally (ie. not
a *reference* to a thing) to be cheating.


Frog picked a turtle to play against, that's what she picked.


--
Warwick
--
  _-_|\      warwick@cs.uq.oz.au            /Disclaimer:
 /     * <-- Computer Science Department,  /  
 \_.-._/     University of Queensland,    /      void (if removed)
      v      Brisbane, Australia.        /
From: srao@brunansky.Prime.com (Shalini Rao)
Newsgroups: rec.puzzles
Subject: Regrouping letter order in a sentence
Date: 21 Jul 92 20:13:50 GMT
Reply-To: Re:Regrouping letter order in a sentence
Organization: Computer Vision

How about ...

    " What ! And you say this is true too ! "

    1. Uh, Yes Sir ! Was it not a hot day out ? 
    2. Oh ! Yes, this was at our unit today.
    3. Oh ! Yes, I saw that sour unit today.
    4. Oh ! yes, that "saw" is our unit today !
    5. I said "Yo ! sure that's not thou way ? "
    6. Oh yes ! Was it that, on your audits ?
    7. O Yes ! What is that on you audits ? 
    8. You Daisy ! What is that on our set ? 
    9. Why you Stu ? that is a stone radio !! 

I can't help feeling there are some more hidden in there !! 

- shalini



**************************************************************************
Shalini Rao                                      srao@brunansky.Prime.Com
Computer Vision                                  (617)-275-1800
Bedford, Ma.                                          x 2546        
***************************************************************************
Newsgroups: rec.puzzles
From: s290@brems.ii.uib.no ( Per Ivar Roervik)
Subject:  Plane Division
Organization: DevNull A/S
Date: Wed, 22 Jul 92 15:40:53 GMT

  This is adopted from an oldie given by Gardener.
  
  In a closed and bounded area of any 2-D shape, 1 million random
  points are chosen.
  Is it always possible to bisect the area AND the points into two
  eqal parts with a straight line?

  Proof or disproof needed.  
    
Newsgroups: rec.puzzles