Posers and Puzzles
29 Mar 09
29 Mar 09
4 edits
Ok, we know how to do it with 4 straight lines. But what about three...
Draw nine dots so that they are arranged in a 3x3 square.
Can you connect all nine dots with 3 STRAIGHT lines without taking pen (or pencil or chalk or other marking-tool) from paper?
I got the task from my teacher in high school. But the question was how to connect the spot with 4 lines..
I had an idea of doing it with three straight lines (i projected my problem on a sphere). For example... sphere meridianes are perfectly straight if you look at them from the right angle. So you pick 3 evenly distant meridianes and place your dots. When I presented my anwser to my teacher in high school, he told me that's wrong and other students made fun of me 😞
http://img2.blogcu.com/images/s/a/k/sakligezegen/25487747jz1.jpg
look, it seems pretty natural to plant your 9 spots here.. And connect them with 3 straight lines (you change the directions of your pen on north and south pole). So three lines are : north pole-south pole, south pole-north pole and again north pole-south pole.
They are perfectly straight, you just have to keep in mind that the problem is translated in 3D.
Does that have any sense to you ?
29 Mar 09
Originally posted by ivan2908This old problem!
Ok, we know how to do it with 4 straight lines. But what about three...
Draw nine dots so that they are arranged in a 3x3 square.
Can you connect all nine dots with 3 STRAIGHT lines without taking pen (or pencil or chalk or other marking-tool) from paper?
I got the task from my teacher in high school. But the question was how to connect the spot wi ...[text shortened]... have to keep in mind that the problem is translated in 3D.
Does that have any sense to you ?
You can do it with three straight lines on a plane. No need for a sphere.
30 Mar 09
1 edit
Originally posted by ivan2908Curiously, you use the phrase "project your problem onto a sphere" - but the thing is, the reason your problem works is because there is no decent projection from a flat plane onto a sphere! (and everyone is busy thinking in flat space.)
Ok, we know how to do it with 4 straight lines. But what about three...
Draw nine dots so that they are arranged in a 3x3 square.
Can you connect all nine dots with 3 STRAIGHT lines without taking pen (or pencil or chalk or other marking-tool) from paper?
I got the task from my teacher in high school. But the question was how to connect the spot wi have to keep in mind that the problem is translated in 3D.
Does that have any sense to you ?
Anyway, your solution is certainly correct, especially since we live on a sphere. Good job on thinking outside the box! However, I came across this problem when I was in school, and there is a very neat solution which is also "outside the box".
'twas probably the first genuinely neat/beautiful solution I ever saw...
30 Mar 09
Originally posted by ivan2908... or do we mean "arranged in a *perfect* 3x3 square."?
Draw nine dots so that they are arranged in a 3x3 square.
Let's call the points
1..2..3
4..5..6
7..8..9
In a flat plane there are the same distance between 1 and 2, as between 5 and 8, right? It's easy to see that there are 12 pairs that have the same distance between them.
What about putting them on a sphere? Can we put 9 dots in a 'square' on the surface of a square, so you can find 12 pairs with equal distance between them? Is it still a perfect square?
And if we find a perfect square with the properties above, is the solution of Ivan's problem still valid?
Is it important to use the word 'perfect square' and use the property '12 pairs of exactly the same distance in between'? Yes!
If you let this property go, then we can draw any sloppy squares, and we can find a solution very easy.
(Like design the positions of the dots as a capital Z kind of shape.)
The solution of Ivan is neat, and outside the box, but ... is it correct?
31 Mar 09
1 edit
Originally posted by geepamoogleI don't really understand what your saying - parallel lines don't exist on spheres.
This solution does in fact rely on the fact that "dots" have positive thickness rather than being merely point. Thus a line of an angle sufficiently close to parallel given the point thickness and spacing can cross all 3.
31 Mar 09
Originally posted by DejectionTrue, and point taken. I was a tad confused as I was thinking of spherical geometry where we take the "lines" to be the great circles. The axioms of this geometry are identical to those of Euclid, however we remove the one about parallel lines and replace it with an axioms (I believe) which states that there are no parallel lines (see my mistake 😛). A sphere, with the great circles for lines, is a good example of this.
Yes they do. I can find two lines on a sphere that never intesect.
Question: are all straight lines on a sphere also circles?
Curiously, spherical geometry is analogously defined to be a geometry where the the angles of any triangle sum to greater than 180 degrees.
31 Mar 09
Originally posted by DejectionIf by "straight line on a sphere" you mean an intersection of a given plane with that sphere, then yes, all "straight lines" on spheres are circles.
Yes they do. I can find two lines on a sphere that never intesect.
Question: are all straight lines on a sphere also circles?
31 Mar 09
Originally posted by forkedknightIs the reverse also true? Are all circles on a sphere also straight lines?
If by "straight line on a sphere" you mean an intersection of a given plane with that sphere, then yes, all "straight lines" on spheres are circles.
Answer: No. Only great circles.
So the problem of nine spots on a sphere drawn into a square cannot be intersected with three straight lines.
01 Apr 09
Originally posted by SwlabrMy response was based on the basis that the problem was solved on an infinite square grid, rather than a sphere. The first line would cut across the top row and extend some distance outward, until it is in line with a point in each of the three dots in the middle row. The rest is simple.
I don't really understand what your saying - parallel lines don't exist on spheres.
On a sphere, one might use a line which is slightly off the great arc, and if it travels like I think it might, it could form a series of parallelish lines close enough together to hit each of the dots, and maybe even each of the points..
However, of this latter case, I'm not sure. I haven't made a study of 3D spherical geometry though.
01 Apr 09
Originally posted by geepamoogleI cannot think that all circles of a sphere parallell to eachother all the way round can be concidered straight on the surface.
My response was based on the basis that the problem was solved on an infinite square grid, rather than a sphere. The first line would cut across the top row and extend some distance outward, until it is in line with a point in each of the three dots in the middle row. The rest is simple.
On a sphere, one might use a line which is slightly off the gr ...[text shortened]... r, of this latter case, I'm not sure. I haven't made a study of 3D spherical geometry though.
The line of equator is of course straight line, bcause it is a great circle, but think of the latitude 80 degrees north circle, it's certainly not straight! If you don't believe me, suppose you stand 10 meters sout of the North Pole and go straight west 63 meters or so, is a straight line the first you will think of? Of course no.
Two different great circles are only parallell at to places. You cannot place a square 9 point dot grid at any places and expect it to be crossed by three straight lines.
01 Apr 09
Originally posted by geepamoogleOkay - I was unsure on the geometry your solution was using.
My response was based on the basis that the problem was solved on an infinite square grid, rather than a sphere. The first line would cut across the top row and extend some distance outward, until it is in line with a point in each of the three dots in the middle row. The rest is simple.
On a sphere, one might use a line which is slightly off the gr ...[text shortened]... r, of this latter case, I'm not sure. I haven't made a study of 3D spherical geometry though.