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From: ansok@spp3.UUCP (Gary Ansok)
Newsgroups: net.puzzle
Subject: Re: 5 boxes (**SPOILER**)
Message-ID: <261@spp3.UUCP>
Date: Mon, 3-Feb-86 17:42:17 EST
Article-I.D.: spp3.261
Posted: Mon Feb  3 17:42:17 1986
Date-Received: Fri, 7-Feb-86 08:30:38 EST
References: <1146@ecsvax.UUCP>
Distribution: net
Organization: TRW, Redondo Beach  CA
Lines: 33

>    _____1____________2____
>   |           |           |
>  3|          4|          5|
>   |___6____7__|__8_____9__|
>   |      |         |      |
> 10|    11|       12|    13|
>   |______|_________|______|
>      14       15      16
> 
> The object is to draw a single line that crosses all of the 16 lines
> in the figure once and only once.  The line may start inside or outside
> the figure, and it may not cross itself.  I suppose the best way to 
> post an answer is to indicate whether the solution starts inside or outside
> the figure, and then list the lines as they are crossed as indicated in
> the figure above.

It's impossible.  It's related to the famous 'Seven Bridges' problem
(you should be able to look this up in most puzzle books).

Consider that if we have a region with N boundary lines, and N is odd, then
we must either begin or end, but not both, inside that region.  If N is even,
then we must both begin and end, or neither, inside that region.  From this
it can be shown that if there are no regions with an odd number of boundary
lines, then the problem is solvable with a closed loop.  If there are
exactly two regions with an odd number of boundary lines, then the problem
is solvable with an open curve which must begin and end in those two
regions.  If there are more than two regions with an odd number of boundary
lines, then the problem is not solvable with one curve.

In the given problem, we have four regions with an odd number of sides:
top left, top right, bottom center (5 each) and the outside (7) -- which
must count as a region.  Therefore, there is no single curve which will
cross all boundary lines.