Seven Lines and Seven Points

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Halleck
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Joined: Thu Dec 16, 2010 12:40 am UTC

Seven Lines and Seven Points

You are given seven lines and seven points. Place the lines and points in such a way that all three of the following rules are obeyed. You are allowed to manipulate the line's length, (not every line need be the same length).

* Arrange 7 points and 7 lines on a plane.
* For each pair of points, there must be a line that goes through both points.
* A line can go through multiple points, if they're collinear. It'll then satisfy the above item for all pairs of points on that line.
* No line can go through all 7 points.

Good luck and have fun.

Edit:Removed incorrect rule.
Edit2: I changed the spoiler to contain what I believe are the only 2 families of solutions. All other solutions that I can see are spinoffs of these two branches of thinking. Also copied phlip's explanation of the rules. He was much more elegant in explaining than I was.

SOLUTIONS:
Spoiler:
6 collinear points and 1 non collinear point with lines connecting to other six points.
3 points on each line "Fano Plane"
Last edited by Halleck on Tue Nov 15, 2011 10:18 pm UTC, edited 2 times in total.

douglasm
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Re: Seven Lines and Seven Points

If one line with 3+ points on it counts as a line per distinct pair of points, and every pair of points in the whole set has to have a line, then this is impossible. No matter how you arrange it, connecting every pair of points will require 7*6/2 = 21 lines. Reducing this count absolutely requires combining some of the lines, but your lengthier explanation of rule 1 prevents that.

skullturf
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Re: Seven Lines and Seven Points

This is reminiscent of finite geometry and the Fano plane. In what context did this question arise?

jestingrabbit
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Re: Seven Lines and Seven Points

It feels like the answer to this was meant to be the Fano plane,

http://en.wikipedia.org/wiki/Fano_plane

but even that doesn't satisfy the crazy making reading of rule 1 that has been forced on you.

edit: ninja'd
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

Halleck
Posts: 32
Joined: Thu Dec 16, 2010 12:40 am UTC

Re: Seven Lines and Seven Points

This was given to us in APCalc. It was a riddle given to our student teacher by his 400-level Geometry teacher. I will ask him to clarify rule 1 in class today.

HonoreDB
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Re: Seven Lines and Seven Points

Well, if the rules were repeated incorrectly to create a seemingly impossible problem, there's no shame in a 169-ish solution.

Groan-inducing hint:
Spoiler:
It's never specified that the lines and points must be distinct.

Halleck
Posts: 32
Joined: Thu Dec 16, 2010 12:40 am UTC

Re: Seven Lines and Seven Points

Turns out that the explanation of rule 1 is incorrect. 3 collinear points can form 3 lines or be one collinear line. I'll edit the original post when I get home. I've got three solutions that I'll post when I get home as well.

Danny Uncanny7
Posts: 74
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Re: Seven Lines and Seven Points

I still am not clear on rule one. "If there are two points..." Of course there are two points, in fact you just said there were seven points. How could you only have two points? What if there was one point or three points? Maybe there are some words missing after this statement, like "If there are two points unconnected to anything..." or "If there are two points not connected to each other" It sounds like it is saying that every point must be connected to every other point by a line. Of course that would make the puzzle impossible. Or is it just saying that you can't have any floating points unconnected to lines?

Halleck
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Re: Seven Lines and Seven Points

The rule basically means each point must be connected to every other point. Does that make more sense? I can break out the scanner if I need to show examples.

Danny Uncanny7
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Re: Seven Lines and Seven Points

Halleck wrote:The rule basically means each point must be connected to every other point. Does that make more sense? I can break out the scanner if I need to show examples.

Then you solution doesn't work. If there is a line connecting each dot, you have 21 lines, 15 of which are collinear.

douglasm
Posts: 630
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Re: Seven Lines and Seven Points

Danny Uncanny7 wrote:
Halleck wrote:The rule basically means each point must be connected to every other point. Does that make more sense? I can break out the scanner if I need to show examples.

Then you solution doesn't work. If there is a line connecting each dot, you have 21 lines, 15 of which are collinear.

Ah, but collinear lines count as the same line.

ConMan
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Re: Seven Lines and Seven Points

Am I reading the instructions wrong, or is the solution just a heptogram?
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jestingrabbit
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Re: Seven Lines and Seven Points

You're reading it wrong. Nearly opposite points have no train joining them, contradicting 1 for instance.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

phlip
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Re: Seven Lines and Seven Points

OK, so what I think the OP is trying to say, based on the example solution they give, is:

* Arrange 7 points and 7 lines on a plane.
* For each pair of points, there must be a line that goes through both points.
* A line can go through multiple points, if they're co-linear. It'll then satisfy the above item for all pairs of points on that line.
* No line can go through all 7 points.

Given that, I think the solution in the OP is the only solution that works... haven't proved it, but I can't think of another solution.

Code: Select all

`enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};void ┻━┻︵​╰(ಠ_ಠ ⚠) {exit((int)⚠);}`
[he/him/his]

taggedjc
Posts: 12
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Re: Seven Lines and Seven Points

Since he also includes the Fano plane as a solution, I'm not exactly sure what he's intending by "Line", since the Fano plane is one circular "line" connecting the three middle points of the triangle to each of the others. It also implies that "colinear" means something other than its usual meaning, since the Fano plane certainly doesn't have those three middle points of the triangle being "colinear" to each other.

If this is allowed, then:
Spoiler:
Six points arranged in a circle with one center point will also work. Each outer point is connected to each other with the one outer circle line, and then there are six inner lines connecting the inner point to all outer points.

t1mm01994
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Re: Seven Lines and Seven Points

@ tagged,
Spoiler:
with that definition of line, that is equivalent to the solution of 6 points on a line and 1 elsewhere, with all connected to the other point, given in the OP.

Adacore
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Re: Seven Lines and Seven Points

If you're allowed circles/curves, as in the Fano plane solution, I believe there are a number of alternatives. For example:
Spoiler:
Place three points (A, B, C) equidistantly in a straight line and draw a circle through the two end points (A, C), centred on the middle point (B). Place another point (D) on the circumference of this circle, perpendicular to the straight line at B. Draw another circle of the same radius centred on this point, draw another straight line parallel to the first through D, and place two more points (E, F) where this line intersects with the new circle's circumference. You should now have two circles, one intersecting (A, D, C) and the other intersecting (E, B, F), along with two lines, one intersecting (A, B, C) and the other intersecting (E, D, F). Now, draw another larger circle intersecting (A, C, F, E), and a third straight line intersecting (B, D). Place the final point (G) at either of the two locations where this line intersects with the (A, C, F, E) circle's perimeter, giving you a circle (A, G, C, F, E) and a line (G, B, D).

I believe that uses six lines to adequately connect all seven points. Draw the seventh line wherever you want, if you need to have it to complete the solution.

@t1mm01994
Spoiler:
Strictly, the point at the centre of a circle solution is not necessarily identical to the six points in a line and one not solution, as while the latter requires all seven lines, you can complete the former with just four.

taggedjc
Posts: 12
Joined: Sun Aug 01, 2010 3:13 am UTC

Re: Seven Lines and Seven Points

You're right, with that definition of a line, at its most basic form, it's equivalent to six points on one line and another point connected to all the others.

Keep in mind that you do actually need seven lines, so if you can complete the rest of the requirements with fewer, you can easily create a solution by adding extraneous lines between two points that are already connected via some other line.

However,
Spoiler:
you can complete the figure using only four lines in total (assuming curves can be considered a line), meaning you can then add "extra" arcs or curves from the center point to any outer point in order to make a multitude of different solutions depending on which outer points you connect.

Here is one such example:

Let's name the points starting at the one in the upper right as 1, 2, 3, 4, 5, 6, and the center point as 7.
Line 6-7 also continues through as one of the two lines from 7-3.
Line 5-7 also continues through as the line from 7-2.
Line 4-7 also continues through as one of the three lines from 7-1.
There are two extra lines going from 7-1.
There is one extra line going from 7-3.

Thus, the actual lines are:
6-3
5-2
4-1
1-1 (the outer circle)
7-1
7-1
7-3

If those bent-line shapes wouldn't be considered lines and only circle arcs would be considered lines, this is still possible, though a bit stricter, as the core lines (6-3, 5-2, and 4-1) will have to be positioned with the points in such a way as to ensure that point 7 will meet at their intersection; then the extraneous lines would simply be circle arcs between point 7 and any of the other points.

eculc
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Re: Seven Lines and Seven Points

Spoiler:
place 6 points collinear, and place a line through them. then, a seventh point is placed seperate of the first line, and a line is drawn from that point to each of the other points. seven points, seven lines, each pair is connected to each other.
Um, this post feels devoid of content. Good luck?
For comparison, that means that if the cabbage guy from Avatar: The Last Airbender filled up his cart with lettuce instead, it would be about a quarter of a lethal dose.

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