Puzzle - Is there a math solution...

that shows that this is impossible to solve?


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The above was a puzzle I spotted in /b/.  The instructions were to draw the image you see on a sheet of paper and using one continuous line, without crossing itself or the same line segment twice, pass through all 16 segments at least once.  I believe it is impossible - is math available to prove me right?

The top half of the box, to be solved in the same manner as the puzzle, requires 7 segments to be crossed (since the two boxes share a common side).  The lower half requires 10 segments to be crossed (since the middle box shares two of the would-be 12) giving us a total of 17 segments to cross IN ORDER to solve using one unbroken, uncrossed line, however there are only 16 segments available for the whole puzzle.

On any 15th move there is one unreachable segment left, so essentially you must use the 16th "move" to cross a segment twice, and the 17th move is the one which would complete the puzzle - illegally.

Is there anything you all would add to this or disagree with?  I'd like to see a simple math solution as to why this won't work.  Thanks for your assistance.

>>5 here.

I'll take a crack at a proof.

We have three boxes with 5 segments to pass through.  For each of these boxes, ee can either go I/O/I/O/I or O/I/O/I/O (I = "intersection that goes in"; O = "intersection that goes out").

Case 1: I/O/I/O/I
You better be finished becuase you can't get out of this box.  The last "in" stroke intersected the last segment in the box.

Case 2: O/I/O/I/O
Feel free to continue (or stop, if you're finished).  You're outside the box, and you've finished the segments for it.

Since there's three boxes of 5 segments each, you can start inside one of them, and make O/I/O/I/O strokes for it and finish that box.  But that leaves two boxes that require the I/O/I/O/I pattern.  If you complete this for one of the boxes, you can't get out to do the other.  Therefore, it's impossible to cover both of the remaining boxes, even if you conquer the first box of five segments.  QED?