Connecting the Boxes

Date: 12/28/98 at 01:15:30
From: Danny
Subject: Box puzzle

   |-----|------|
   |     |      |
   |     |      |
   |--|--|--|---|
   |  |     |   |    
   |--|-----|---|

This may sound silly, but we have been trying to figure this out for 
almost a month now. Draw one continuous line crossing every line 
(including the outside edges) once and only once. You can not trace 
along the lines you are trying to cross or cross on the same point as 
a previous line. 

Many thanks,
Danny Mainprize, 
Brookwood High School

Date: 12/28/98 at 09:53:53
From: Doctor Rob
Subject: Re: Box puzzle

Thanks for writing to Ask Dr. Math!

Sorry, this is impossible. Do not waste any more time trying!

The upper two boxes and the middle bottom one each have five line
segments that have to be crossed by your curve. Every time you enter
and exit a box, the curve crosses two line segments. If you do that
twice to each of these three boxes, that will leave one line segment
to be crossed in each. To do that, you have to either start or end your 
curve in that box. But the curve has only two ends, and there are three 
such boxes. That means that in whichever boxes you start and end the 
curve, there will be one leftover box with a line segment not crossed.

This problem is very similar to a famous one called the Bridges of
Koenigsberg, which was solved by Leonhard Euler in the 18th century.
Here is a good place to look for information on the history of the 
Bridges of Koenigsberg:

   http://mathforum.org/~isaac/problems/bridges1.html   

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