Five Lines Through Twelve Dots

Date: 11/07/2000 at 14:06:05
From: Gary White
Subject: Rectangles and lines

We have been given a picture of a rectangle made of dots. It is 3x4 
with 12 dots overall. We have to draw 5 straight lines through all the 
dots. There is only one horizontal line and the lines must start where 
they finished. 

Please help.

Date: 11/07/2000 at 16:29:31
From: Doctor Ian
Subject: Re: Rectangles and lines

Hi Gary,

These pages should give you some ideas:

   Mind Games: Join the Dots - Gihan Perera
   http://www.q-net.net.au/~gihan/mindgames/dots9.html   

   Join the Dots - Math Forum
   http://mathforum.org/k12/k12puzzles/join.dots.html   

I hope this helps. Write back if you have more questions, about this 
or anything else.

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/   

Date: 11/07/2000 at 17:32:11
From: Gary White
Subject: Rectangles and lines

I asked you for help on a question, but you gave me the answer for 
dots in a 3x3 pattern.

I asked how to draw 5 straight lines through 12 dots in a 3x4 pattern. 
The line must start and end in the same place. The answer has one 
horizontal line and no vertical lines but four diagonal lines.

Date: 11/08/2000 at 10:46:07
From: Doctor Ian
Subject: Re: Dots

Hi Gary,

The point of sending you that was to give you some _hints_ for how to 
go about solving your own problem, not to simply give you an answer.

The specific solution to your specific problem will depend on the 
distances between the dots, and the sizes of the dots. If the dots are 
large enough and close enough together, almost _any_ set of connected 
diagonal lines will do. If the dots are small enough, _no_ set of five 
lines will pass through all of them.

(It also depends on how wide the line can be. With a wide enough line 
-- think 'paintbrush' -- a single line segment will pass through all 
of the dots.)

Since I don't know how your dots are arranged, or how big they are, 
there is no way for me to simply tell you how to draw those lines.  

But look again at the illustration at the top of the answer to the 
problem I showed you. Now imagine that there is a fourth row of dots 
above the other three. That would be the format you're working with, 
right?

Now look at the line segment that ends at the upper right corner of 
the illustration. If you were to extend that straight back to the 
left, couldn't it pass through that extra row of dots? And wouldn't it 
end up where you started?

I understand that this solution doesn't meet _all_ the constraints of 
the problem you were given (since it obviously contains both 
horizontal and vertical lines), but the point that it makes is an 
essential one, which is that you're almost certainly going to have to 
let your lines extend beyond the edges of the rectangle.

The other solutions on the page may also be applicable to your 
problem, depending on how flexible you want to be, and what you think 
you can get away with.

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/