Another puzzle

[needimprovement]

This is an old one, so if you've seen it before, let the others think for a while before replying...

 

You have nine dots on a paper, as seen below. Your task is to connect them with four straight lines without lifting the pencil:

 

o o o

o o o

o o o

 

.... and when you're done with that: Tell me how to do it with *one* straight line...

[Sisyphus]

 

 

Dots labeled:

 

123

456

789

 

Draw a line through 159, through 369, through 24, and through 789, or any rotated or flipped variation. You could also do each one in reverse order, so that's 16 possible solutions that I see.

 

With one line, it can't be done in Euclidian space, unless you can give the line non-zero width, i.e. wider than the whole grid, i.e. "use a huge pencil." Or you could make the dots bigger. Make them big enough so that they all overlap in one region, and draw a line through that region.

 

[CaptainPanic]

Sisyphus, I believe you missed the "without lifting the pencil" part. Here's my solution:

 

Numbering the dots:

 

123

456

789

 

Draw a line 1-2-3 (line 1)

Draw a line 3-6-9-6-3 (down and back up, you've drawn one extra line, 2 in total so far, never lifted the pencil)

Go back 3-2 (you've drawn no line at all, still haven't lifted the pencil)

Draw a line 2-5-8-5-2 (third line, still haven't lifted the pencil)

Go back 2-1

Draw a line 1-4-7 (4th line, and done)

 

I know many will argue that I have drawn more lines than I was allowed...

 

I have no clue how to connect them with 1 line, unless you have either a big pen, or some scissors to arrange them in a single line first.

 

[Sisyphus]

Sisyphus, I believe you missed the "without lifting the pencil" part.

 

No I didn't:

 

 

 

Start upper left. Go 159. Go up from 9 through 6 and 3, until you intersect the line through 2 and 4, then draw that line until you intersect the line through 789, and draw it. You just have to go beyond the edges of the grid.

 

[DJBruce]

Sisyphus, I believe you missed the "without lifting the pencil" part.

 

Sisyphus' solution works. Bellow is a visual representation of his solution.

 

 

 

As for doing it with one line I will to agree with Sisypus that you can't do it in Euclidean Space how you have depicted the problem. You can however do it with three lines.

]

 

Edited August 26, 2010 by DJBruce

[Enigm4]

with 4, its simple, with 3, even more simple ( you can zigue-zague up and down or left and right if you make the straight lines not parallell to the one that's formed by the circles center's ) , with one line a big line with big size that crosses all of them, you can even discuss the fact of the circles are disposed ,or not, in a sphere ( earth like ) witch makes one single line go around the earth and go back to the same point or not if you dont make it parallell to the meridians (or whatever the horizontal things are called , i probably should just example by the equator xD ), you can think of placing the circles the way you want so you can cross them with various lines... or... even with just one dot... if you place all the circles upon eachother, and you simply dot in...if that's not how its done with zero lines if you imagine the circles in 3D. haha... but i think that lower than 3 lines its called cheating on the actual question ...

Edited September 25, 2010 by Enigm4