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Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

Posted May 10, 2009 5:01 PM

This week's Challenge Question:

Nearly everyone has seen the problem where you have to draw 4 straight lines through a matrix of 9 (3x3 dots) without taking the pencil off the paper and without retracing any lines. This is the solution:

This one a bit trickier; a three by four square matrix of dots: five lines PLUS you have to finish where you started. All the other obvious rules apply (the dots are infinitely small, and the lines are not infinitely long):


And the Answer is....

Here is the solution:

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#1

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/11/2009 12:50 PM

I see this has worked its way through. I didn't solve it yet.

Does the pencil crossing a fold count as staying on the paper?

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#2

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/11/2009 1:02 PM

I've found at least two different ways to connect all 12 dots with just five lines, but not to start & finish at the same point.

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#3
In reply to #2

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/11/2009 1:08 PM

Quite so!

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#4

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/11/2009 1:19 PM

I do have a little insight. I think there are only thirty five possible lines that connect the dots and intersect somewhere. If I made no mistakes, these lines are shown below. Maybe someone can find the five that satisfy the challenge. I've got to get back to work.

Thanks,

Jim

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#5

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/11/2009 3:17 PM

Ireka (= I need a bath?). You need four slanted lines that deal with the centre row and either the top or bottom. The fifth line joins them using the other row. Now to download that drawing package again...

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#6

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/11/2009 3:45 PM

For me that was a struggle. But then, would I have preferred to do it using four lines followed by a fold?

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#7
In reply to #6

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/11/2009 4:40 PM

QA. An interesting feature of the solution is that none of the points are at the ends of the line segments. Not what I would have guessed.

Jim

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#9
In reply to #7

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/11/2009 5:15 PM

The odd thing is that this method would have crossed some of the points twice - except that the lines had to be truncated to cover the "return-to-go" requirement.

I see that the regular 4x4 matrix of points can be done with six lines. I suspect (don't know) that this is as small as it gets.

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#8
In reply to #6

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/11/2009 4:56 PM

Bravo!

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#16
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Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/12/2009 6:43 AM

Well done Fyz: I always knew that the biggest problem with this challenge was that once someone had solved it the the game was up.

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#18
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Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/12/2009 6:52 AM

Shhh!
I was in two minds whether to leave post #5 as a clue and allow others to put up the solution - plus I haven't told anyone that (inversions apart) I have demonstrated that the solution is unique (whoops).

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#25
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Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/12/2009 1:08 PM

Whilst tinkering with this (very nice solution you drew, btw), I have succeeded beyond even my own expectations in losing the plot; Remember the goat that needed enough rope to reach half a field ? Well, the picture below, with the half-field enclosed by green, is very close to the points on the circle.

The green circle doesn't quite cross points m and p, but it's very close. They'd better strap my jacket extra tight this evening ! Origami is hard with just my ears, but I'll try

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#27
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Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/12/2009 1:33 PM

looks like a snowman with some parts missing.

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#33
In reply to #27

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/13/2009 2:29 AM

Customers must buy accessories separately.....

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#29
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Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/12/2009 1:37 PM

Must be because I didn't quite keep a square matrix.

Fyz

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#34
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Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/13/2009 2:33 AM

I know, unit .

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#32
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Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/13/2009 2:21 AM

But then, would I have preferred to do it using four lines followed by a fold?

I'd like to go round the bend first ;

Roll the paper into a cylinder before drawing......

For simplicity sake, the above picture only shows 1 line, drawn in red (with the paper bent along it's horizontal axis to make a cylinder). From the end point "L", I can screw back down to the other end, etc etc. The 'pitch' of the line could be altered, but if I drew it all completely I'm reasonably sure that 4 lines would suffice using this approach. There are some minor details that could be pointed out to clarify why 4 lines would be enough for this (and a 4x4 matrix), but that's enough to go on since people might cry foul play over a cylinder. Shame, because the next method is even better.......

1 line to connect the dots ();

The above picture represent a square of card with a chunk cut out at the top. Lay a straight edge along the 3 dots on the left. When the ruler pivots over the top left edge, you can draw (light blue) across the back of the card, and then pivot back up to draw across the second line of dots.

I can't see any way to improve on the answer given by Fyz, but since nobody objected to the mention of folding earlier, I'm going to roll with a cylinder and/or infinite line. Feel free to argue the wording of the question.............

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#37
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Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/13/2009 3:52 AM

You'd need to be a doughnut to to think any of those are straight lines:-

But, I suppose I have to admit they're worth a Good Answer.

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#38
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Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/13/2009 4:28 AM

LOL - and ta for the GA.

That doughnut is doh'ing my nut in ! Having the wire wrapped at angle looks like a dodgy bit of deception to suggest 1 loop. Your drawing is clearly better than mine - if you add to that picture, there's a good chance of getting a GA from me.

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#39
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Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/13/2009 5:24 AM

Non-Euclidean geometry is a neat idea - though I'm not certain that a square matrix is possible in the geometry you chose. But a square matrix with a single "straight" line would be possible on a cylinder with suitably domed ends. => GA.

The reason that I kept to four lines was so that the lines could stay straight in standard space (i.e. not cross the fold). The dots would also stay on a square matrix - albeit pairs would become coincident. Of course, if we assume infinitely thin (or perfectly stretchable) paper it can be done with two (coincident) lines (the second one is needed to return to the start point).

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#10

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/12/2009 12:39 AM

Made you laugh!

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#11
In reply to #10

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/12/2009 5:00 AM

Where's the joke? (and that illustration looks like an exact inverted copy of #6)

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#15
In reply to #11

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/12/2009 6:36 AM

It Is! but the dots are the not inverted.

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#17
In reply to #15

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/12/2009 6:48 AM

He he (there are no dots)

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#28
In reply to #17

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/12/2009 1:34 PM

Oh.... Circles!!

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#30
In reply to #17

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/12/2009 1:45 PM

The dots inside the circles are to small to see. He said they were real small.

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#12

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/12/2009 5:44 AM

How about this :)

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#13
In reply to #12

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/12/2009 6:12 AM

Doesn't this fail the requirement: "PLUS you have to finish where you started"?

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#14
In reply to #13

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/12/2009 6:21 AM

oops..didnt read question properly. cheers

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#49
In reply to #12

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

11/24/2013 7:52 AM

Dear Mr. sal man,

I also thought and got the same way as you did. w When I was about to post, I saw your posting. Hence I did not put the sketch here.

Your thinking and my thinking is the same.

DHAYANANDHAN.S

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#50
In reply to #49

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

11/26/2013 5:06 AM

See replies #13 and #14

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#19

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/12/2009 9:47 AM

Aren't all of these answers wrong? as you stated "(the dots are infinitely small, and the lines are not infinitely long)". If the lines are not infinitely long then that implies that they end at points and cannot continue past them.

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#20
In reply to #19

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/12/2009 9:56 AM

Of course finite lines end at points - but not in this case at one of the predefined twelve.

It pays to correlate logic with observation - you can see the (finite) lengths of the lines and that they would pass through all twelve points.

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#23
In reply to #20

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/12/2009 10:38 AM

ok, i'll agree just this once, lol

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#21
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Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/12/2009 10:16 AM

The "(the dots are infinitely small, and the lines are not infinitely long)" rule is to stop people "cheating" like this:-

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#22

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/12/2009 10:38 AM

This is a rectangle, not a square.

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#24
In reply to #22

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/12/2009 10:47 AM

It's a square MATRIX

.....Neo

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#26

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/12/2009 1:15 PM

As I was solving the easy puzzle, I created a vortex and transported to another dimension... Can someone erase a few of my lines?

Find my address online, show up to my house and go upstairs into my office...

Oh, and please throw a peair of clean socks and underwear into the portal before you erase the lines and bring me home.

JL Mealer

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#31

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/12/2009 5:00 PM

can't draw.... STAR shape

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#35
In reply to #31

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/13/2009 2:57 AM

Try TurboCad LE (TurboCad Learning Edition): this free download (just click on the bulleted item at the top of the linked URL) includes a 271 page book. Although this is only version 4 (TurboCAD is now up to version 16), it includes everything that a casual user would use in a 2D cad package.

IMSI (who make TurboCad) also now do a much more comprehensive 2D CAD called DoubleCAD. It's got the look and feel of AutoCAD lite, but with some of the better features of TurboCAD: you can get it free from CADcourse (it's a dollar with a dollar discount; you don't need to enter your credit card details or anything and they haven't bombarded me with spam).

For a more mathematical package try GeoGebra (sorry to give away one of your secrets Kris), also free.

When you've done your drawing: do a screen capture; save as a .jpg file, and, insert into the CR4 editor using the little green camera above the editing screen.

Click on browse and find the picture on your computer.

Sparky did a great little tutorial on inserting graphics some time ago.

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#36
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Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/13/2009 3:16 AM

he he....no big secret, I've mentioned it enough. Not as fancy as the others, but cool for Euclidean play.

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#40

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/14/2009 1:07 AM

it is not over. I have found a second solution. I wonder if there are more....

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#41
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Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/14/2009 1:31 AM

This looks very like Fyz's solution at post # 6:-

http://cr4.globalspec.com/comment/390230/Re-Five-Lines-Twelve-Dots-CR4-Challenge-05-12-09

Are you saying you've found a different one?

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#42
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Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/14/2009 1:51 AM

its the same - i hadnt realsied that ll the shuffling had got back to the same solution......

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#43

Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/14/2009 9:54 PM

Tricky, but doable.

Here's one that plagued us during those dark days pre-WWII in general physics lab. Mr. Lariton promised a final grade of 100% without taking the final exam to anyone who could solve the puzzle.

There are three blocks representing three houses. In front of each house is a utility hook-up.

The problem is to run a line from each utility hook-up to each house without crossing any lines.

No one could do it, but no one flunked his course either.

He then revealed to all of us at the end of the semester that there is no solution the way the problem is presented.

Is there a solution?

Ken Leigh

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#44
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Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/15/2009 2:51 AM

It's an oldie, but good. That site is a neat one, though I first saw the problem some place I've long forgot. "Riddles In Mathematics" (Eugene P Northrop) is a fun read, though I'm not sure if it's still in print.

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#45
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Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/15/2009 4:25 AM

You can do it by taking one of the lines through one of the houses or through one of the utilities. You can also do it by placing all the buildings on the surface of a doughnut, but, that really only amounts to taking one utility under or over one of the others, which is of course what happens in the real world.

To prove that it can't be done without cheating, you need to look at the way loops divide up a simple 2D space. I know that Fyz will say that this is not rigorous, but it "works for me".

First lets have a stab at the problem:-

I've got in all the lines except 2 to B. (I could "cheat" at this point and take a line through A). Let's take out some of the lines and simplify the drawing a bit:-

First look at the loop 1,A,3,C,1

It divides "the world" into two areas: an inside, and, an outside.

It doesn't matter whether we put B inside or outside, once we connect up 1,B,3 we have sub divided one of those two areas into two more distinct areas. In this case I've put B inside the original "loop", but I think you can see that it doesn't matter which I chose.

Now the world is divided into three areas. If we put 2 outside the original area it is isolated from B; if we put 2 in the loop 1,A,3,B,1 it is isolated from C, and, if we put 2 in the loop 1,B,3,C,1 it is isolated from A.

As I said not rigorous, but, a basis on which a "reasonable" person could argue that the original problem is impossible.

I see that Kris has beaten me to this with a link to a more more formal solution.

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#46
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Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/15/2009 6:20 AM

I think your proof is a well-chosen subset of the full proof. The missing bits probably aren't worth knowing about (except for those who want all the i's crossed as well as dotted)

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#47
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Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/15/2009 7:10 AM

To: Physicist

Makes sense(?) to me. As I recall, Mr. Lariton used a similar concept when he showed us that a solution was impossible.

That did not mean that thirty-three students (of which I was one) didn't try that whole spring to solve it.

Of course, none of us did, but Barb H and myself determined early on that it was impossible to solve, for which Mr. Lariton gave us a "passing" grade of 95% (after we took the final exam!), the lowest grade I received that year. It brought my average down enough so that I was merely third in a class of eighty-six instead of first!

Ah, yes, I was a smart ass then, and haven't changed one iota since!

You know what they say, gals and guys, once a smart ass, always a smart ass, or, everyone likes ass, but no one likes a smart ass, or, you can always tell a smart ass, but you can't tell them much!

Oh, BTW, what do you call a retarded lawyer?

Ken Leigh

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#48
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Re: Five Lines, Twelve Dots: CR4 Challenge (05/12/09)

05/15/2009 7:18 AM

The old ones are still good.
It depends on the level of retardation. Full-scale (in the UK) would be "My Lord".

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