Suggest a change

These fields are all optional and need only
be supplied if you would like a direct reply.

Subject
Your email address
Your real name

You must answer this!

If you don't, my spam filtering will
ensure that I never see your email.

What's 8 plus five (in digits only)?

Please make your changes here and then
Editing tips and layout rules.

File: ParallelogramPuzzle

''' <link rel="alternate" type="application/rss+xml"
'''       href="/rss.xml" title="RSS Feed">

********> width="25%"
|>>
''' <a title="Subscribe to my feed"
'''    rel="alternate"
'''    href="https://www.solipsys.co.uk/rss.xml">
'''    <img style="border-width: 0px;"
'''         src="https://www.feedburner.com/fb/images/pub/feed-icon32x32.png"
'''         align="middle"
'''         alt="" />Subscribe!</a>
_
''' <a href="https://twitter.com/ColinTheMathmo">
''' <img src="https://www.solipsys.co.uk/new/images/TwitterButton.png"
'''      title="By: TwitterButtons.net"
'''      width="212" height="69"
'''      alt="@ColinTheMathmo"
'''     /></a>
<<|
----
My latest posts can be
found here:
  * ColinsBlog
----
Previous blog posts:
  * BackOfTheEnvelopeCOVID19
  * APointAgainstTheAxiomOfChoice
  * InDefenseOfTheAxiomOfChoice
  * JourneyingHomeThroughStormDennis
  * EarthRadiusRefined
  * VolumeOfASphere
  * BigOhAndRelations
  * MathematicalRelations
  * IntroducingBigOh
  * ConstantDifferences
  * AlgorithmsAndSizesOfInstances
  * IntroducingTimeComplexity
  * TheLinearFrog
  * SeventyVersusOneHundredRevisited
  * HowTheFarragoWorks
  * SeventyVersusOneHundred
  * PowersOfTwoInLexOrder
  * EmergingEExpanded
  * RageInducingSystemImplementation
  * TheBookIsNotAlwaysRight
  * EmergingE
  * ImpossibleToTranslate
  * WaitingInVain
  * NonRepeatingDecimals
  * RationalRepeats
  * WhyIsItLovely
  * CompilingCryptoConnections
  * ExploringConnectionsBetweenCryptoSystems
  * ElwynBerlekampHasLeftUs
  * RootCauseAnalysisAndThePhotocopierQuestion
  * TheUpDownTides
  * TheForeAftTide
  * TheSidewaysTide
  * WrappingUpWrappingUpTheEarth
  * TheOtherWrappingTheEarthProblem
  * WrappingTheEarth
  * TheRingOfSteel
  * RoundingUpTheRopes
  * OtherOtherOtherRopeAroundTheEarth
  * RopeAroundTheEarthRefined
  * TheOtherRopeAroundTheEarth
  * ElementaryEstimates
  * LatitudeCorrection
  * JustGiveMeTheAnswer
  * MoreMusingOnPollardRho
  * IdleThoughtsAboutPollardRho
  * WhenOptimisingCodeMeasure
  * ADogCalledMixture
  * AnotherPayPalScam
  * WhyTopPostingHasWon
  * UnexpectedInteractionOfFeatures
  * ArchimedesHatBoxTheorem
  * ConsideringASphere
  * ToLinkOrNotToLink
  * GenericAdviceForWritingAThesis
  * JustTeachMyChildTheMaths
  * NotASpectatorSport
  * LeftTruncatablePrime
  * TheDoctorAndTheLawyer
  * FourPointsTwoDistancesProof
  * MeetingRonGraham
  * NapkinRingVersusSphericalCap
  * TheFourPointsPuzzle
  * RadiusOfTheEarthPartTwo
  * GrepTimingAnomaly
  * TheRadiusOfTheEarth
  * ThisWorksToCureMyHiccoughs
  * PerhapsWeSavedOne
  * ThinkingAboutMastodon
  * DisappearingTrainsOnVirgin
  * TheIndependenceGame
  * OneOfMyFavouritePuzzles
  * ThinkingAboutRecursion
  * MemorisingTheTube
  * SpikeySpheres
  * SurprisinglyQuick
  * AnUnexpectedFraction
  * YouHaveToAdmireTheirOptimism
  * RepresentativesMatter
  * PythagorasByIncircle
  * APuzzleAboutPuzzles
  * HowNotToDoTwitter
  * Calculating52FactorialByHand
  * SmallThingsMightNotBeSoSmall
  * NotIfYouHurry
  * FactoringViaGraphThreeColouring
  * AnotherProofOfTheDoodleTheorem
  * WhenObviousIsNotObvious
  * GraphThreeColouring
  * TheDoodleTheorem
  * BeCarefulWhatYouSay
  * TheMutilatedChessboardRevisited
  * AMirrorCopied
  * TheOtherOtherRopeAroundTheEarth
  * PhotocopyAMirror
  * ThePointOfTheBanachTarskiTheorem
  * SieveOfEratosthenesInPython
  * FastPerrinTest
  * RussianPeasantMultiplication
  * FindingPerrinPseudoPrimes_Part2
  * FindingPerrinPseudoPrimes_Part1
  * TheUnwiseUpdate
  * MilesPerGallon
  * TrackingAnItemOnHackerNews
  * HackerNewsUserAges
  * PokingTheDustyCorners
  * ThereIsNoTimeForThis
  * PublicallySharingLinks
  * LearningTimesTables
  * GracefulDegradation
  * DiagrammingMathsTopics
  * OnTheRack
  * SquareRootByLongDivision
  * BeyondTheBoundary
  * FillInTheGaps
  * SoftwareChecklist
  * NASASpaceCrews
  * TheBirthdayParadox
  * TheTrapeziumConundrum
  * RevisitingTheAnt
  * TheAntAndTheRubberBand
  * IrrationalsExist
  * MultipleChoiceProbabilityPuzzle
  * RandomEratosthenes
  * WrappingUpSquareDissection
  * DissectingASquarePart2
  * DissectingACircle
  * DissectingASquare
  * AnOddityInTennis
  * DecisionTreeForTennis
  * DecisionTreesInGames
  * AMatterOfConvention
  * DoYouNourishOrTarnish
  * BinarySearchReconsidered
  * TwoEqualsFour
  * TheLostPropertyOffice
  * TheForgivingUserInterface
  * SettingUpRSS
  * WithdrawingFromHackerNews
----
Additionally, some earlier writings:
  * RandomWritings.
  * ColinsBlog2010
  * ColinsBlog2009
  * ColinsBlog2008
  * ColinsBlog2007
  * ColinsBlogBefore2007
********
[[[> TaggedAsPuzzles ]]]
!! A Parallelogram Puzzle

A while ago (March 2nd at 16:28 to be reasonably precise)
''' <a href="https://twitter.com/edsouthall">Ed Southall</a>
asked
''' <a href="https://twitter.com/edsouthall/status/1234515949086220288">a great question</a>
on Twitter:

|>> [[[ |>>
Can you add a further point _
such that joining them up _
can make a parallelogram?
<<| ]]] <<|

Here's the setup:

[[[> |>>
https://www.solipsys.co.uk/images/ParallelogramPuzzle.png
----
The setup ...
<<| ]]]
Several people chimed in quite quickly, others took a little
longer.  Some were happy to have found a solution, others were
happy to have found two solutions, and then quite a few people
started to claim that there were, in fact, exactly two solutions.

That's a pretty bold claim, and of late I've become less sure of
myself about these sorts of things.  More than once I've made a
claim like that, only to have an underlying assumption exposed,
leaving me to think more carefully.

Which is good, though sometimes embarrassing, and sometimes even
a little painful.

So I wondered, are there /really/ exactly two solutions?

I invite you to go and read the thread, which I've extracted and
''' <a href="https://www.solipsys.co.uk/Chitter/Parallogram.svg">drawn in chart form.</a>

There you will see some of the solutions, and some of the reasoning.
But I promised I'd write more about my thinking, so here it is ...

!! The first solutions

Here are two solutions:

[[[>
https://www.solipsys.co.uk/images/ParallelogramPuzzle_0.png
----
https://www.solipsys.co.uk/images/ParallelogramPuzzle_1.png
]]]
We can see that we've got our four original points, one extra, and
then taken what is technically called the "convex hull".  Imagine
the points are pins in a pinboard, and stretch a rubber band around
them.  Now all five points lie on the perimeter of the resulting
parallelogram, and I'm pretty sure these are the solutions people
had when they said there are exactly two.

I'm pretty sure there's no other way to:

* Add one point;
  * Take the convex hull;
  * Have all five points on the perimeter.

Proving that doesn't seem too complicated, but it would be interesting
to nail *all* the details.  That might be harder that you think.  Then
again, maybe not.  I haven't done it.

!! Starting to wonder ...

But then I thought of this puzzle.

[[[> |>>
https://www.solipsys.co.uk/images/CountTheTriangles.png
----
Count the _ Triangles
<<| ]]]
The whole point here is that there are triangles with internal lines,
and not just the triangles that surround blank space.  In particular,
we don't insist on using or involving every part of the diagram.

So let's go back and look at the original wording:

* Can you add a further point such that
      joining them up can make a parallelogram?

It's easy to assume that "joining them up" means joining *all* of them.
But if you join every dot to every other dot, you don't get just the
perimeter, you also get diagonals, and crossing lines.

So maybe we *don't* have to involve all the dots, maybe we only need
to have three of our original dots plus one extra, and require that
they are the corners/vertices of our parallelogram.  So maybe the
question is:

* Add one dot;
  * Take the convex hull of four of them;
  * How many ways does this give a parallelogram?

Here's one solution:

|>> [[[ |>>
https://www.solipsys.co.uk/images/ParallelogramPuzzle_a.png
----
Convex hull of four points
<<| ]]] <<|

So how many of /these/ are there?

OK, take any three points and label them A, B, and C.  Assume that you
are going around the parallelogram cyclically, and where does your next
point have to be?

[[[>50 I'm not going to give you a complete list here. Firstly, you
probably wouldn't check them, and secondly, it would be a good test
of my explanation to see if you can find a few to convince yourself.
]]]
Think of B->A as a vector, and add that to C.  The point you get to is
the fourth point of the parallelogram ... yes, you should check that.
Try a few examples.

There are 24 ways to choose 3 points in order from 4, so there should
be 24 solutions, assuming no repeats.

!! But wait!  There's more!

So now, what if our parallelogram just needs to "go through" the
points, and the points don't necessarily have to be at the corners?

For example, like this:

|>> [[[ |>>
https://www.solipsys.co.uk/images/ParallelogramPuzzle_b.png
----
Reaching beyond ...
<<| ]]] <<|

As you start to explore this you will find infinitely many solutions,
but they will, perhaps, come in families, and perhaps you can start
to describe those families and get a handle on what's going on.

----
!! Feedback

Colin Beveridge
''' (<a href="https://www.flyingcoloursmaths.co.uk/">Web site</a>)
has got in touch and sent me this diagram:

|>> [[[ |>>
https://www.solipsys.co.uk/images/ParallelogramPuzzle_CB.png
----
Infinitely many solutions ...
<<| ]]] <<|

If point E is put anywhere in the green region(s) then we can construct
a parallelogram that has all 5 points on the perimeter.  It's not the
convex hull, but it *is* a parallelogram that joins all the dots.

----

|>>
| |>> <<<< Prev <<<< ---- BackOfTheEnvelopeCOVID19 <<| | : | |>> >>>> Next >>>> ---- PerceptionOfSpace ... <<| |
----
********>
''' <a href="https://mathstodon.xyz/@ColinTheMathmo">
''' <img src="https://www.solipsys.co.uk/images/Mastodon_Mascot.png"
'''      width="256" height="280"
'''      alt="https://mathstodon.xyz/@ColinTheMathmo"
'''     /></a>
********
''' <a href="https://mathstodon.xyz/@ColinTheMathmo/">You can follow me on Mathstodon.</a>
_ _ _ _
[[[>
''' <a href="https://twitter.com/ColinTheMathmo">Of course, you can also<br>follow me on twitter:</a>

''' <a href="https://twitter.com/ColinTheMathmo">
''' <img src="https://www.solipsys.co.uk/new/images/TwitterButton.png"
'''      title="By: TwitterButtons.net"
'''      width="212" height="69"
'''      alt="@ColinTheMathmo"
'''     /></a>
''' <img src="/cgi-bin/CountHits.py?ParallelogramPuzzle" alt="" />
]]]
********<

----

!! Send us a comment ...

''' <form action="https://www.solipsys.co.uk/cgi-bin/FormMail.pl" method=post>
''' <input type=hidden name="recipient"               value="colinsblogcomment@solipsys.co.uk" >
''' <input type=hidden name="subject"                 value="Blog comment : ParallelogramPuzzle"         >
''' <input type=hidden name="redirect"                value="https://www.solipsys.co.uk/new/ThankYouForYourComment.html" >
''' <input type=hidden name="missing_fields_redirect" value="https://www.solipsys.co.uk/RequestError.html">
''' <input type=hidden name="env_report"              value="REMOTE_HOST, REMOTE_ADDR, HTTP_USER_AGENT"  >
''' <input type=hidden name="print_blank_fields"      value="1"                                          >

********> width="47%"
You can send us a message here. It doesn't get published,
it just sends us an email, and is an easy way to ask any
questions, or make any comments, without having to send a
separate email. So just fill in the boxes and then
''' <font size="+4"><INPUT TYPE="submit" VALUE="CLICK HERE TO SEND"></font>
******** width="53%"
********<

''' <table cellpadding="5">
'''   <tr>
'''     <td valign="top">Your name </td> <td valign="top">:</td>
'''     <td> <input type=text name="realname" size="48"> </td>
'''   <tr>
'''     <td valign="top">Email </td> <td valign="top">:</td>
'''     <td> <input type=text name="email" size="48"> </td>
'''     </tr>
'''   <tr>
'''     <td valign="top">Message </td> <td valign="top">:</td>
'''     <td> <TEXTAREA NAME="Message" ROWS=10 COLS=64></TEXTAREA> </td>
'''     </tr>
'''   </table>

''' <center>
'''   <font size="+4">
'''     <INPUT TYPE="submit" VALUE="CLICK HERE TO SEND">
'''     </font>
'''   </center>
''' </form>

********<