The solution to this classic puzzle is that it's impossible to solve. But can you explain *why* it's impossible?

## Sunday Puzzle #25: Solving The Unsolvable Puzzle

This classic puzzle goes by many names, but the challenge it poses is always some variation of the following:

*Without lifting your pen from the page, draw one continuous line that crosses each of the sixteen line segments in this figure one time and one time only. You can start anywhere you'd like and end anywhere you'd like. The line you trace can be sharp and it can be smooth, but it must be continuous. *

It is a particularly devious puzzle, because it is unsolvable. I encourage you to try it for yourself, but after enough time (for some people it's minutes, for others its weeks or longer), you'll arrive at the conclusion that it is impossible to pass through all sixteen line segments without crossing through at least one of them more than once. In the figure below, for example, I adhere to the one-cross-and-one-cross-only rule, but I also fail to cross two line segments altogether:

Similarly, in the figure below, I manage to cross through all 16 line segments, but not without crossing the top middle segment two times. I also pass through the righthand bottom segment twice:

There are plenty of other ways to draw one, continuous line and exactly zero ways to do so while satisfying the conditions of the puzzle. Sure, there are ways to bend the rules. You can draw your line *along* one of the segments, for example, or through a corner where two or more segments meet. Another lateral solution involves solving the puzzle on a torus, rather than a flat plane. While certainly creative, these solutions do not count, and so the puzzle is unsolvable.

But the point of this week's puzzle is not to show that it can be solved. It's to prove it *can't* be solved. The solution to this problem is to prove it has no solution. Are you up for it?

We'll be back next week with the solution – and a new puzzle! Got a great brainteaser, original or otherwise, that you'd like to see featured? E-mail me with your recommendations. As always, be sure to include "Sunday Puzzle" in the subject line!

*Art by Tara Jacoby*

## SOLUTION To Sunday Puzzle #24: Timing Trains

Last week, I asked you to explain why a man with two girlfriends spent significantly more time with one, in Brooklyn, than the other, in the Bronx.

The man likes both girlfriends equally, reaches the subway platform at a random moment each Saturday, and the Brooklyn and Bronx trains arrive at the station equally often—every 10 minutes. Yet he still finds himself visiting Brooklyn 9 times out of 10. Why?

The solution, which several readers correctly identified both in the comments and over e-mail, is surprisingly straightforward, even if the reasoning behind it is not immediately apparent:

**The Bronx train, we can conclude, leaves exactly 1 minute after the Brooklyn train.**

If it is not immediately obvious to you why this would result in our young man visiting Brooklyn nine times for every one trip he makes to the Bronx, don't worry. It can take some time to wrap your head around. Give yourself a chance to ruminate on it, and if you're still stumped, go read the explanation posted by commenter Testifye in last week's comments.

## Previous Weeks' Puzzles

- This Week's Puzzle Has A Very Simple Solution. Can You Find it?
- Can You Solve This Extremely Difficult
*Star Trek*Puzzle? - You Either Solve This Riddle, Or You Die
- Can You Solve 'The Hardest Logic Puzzle In the World'?
- You'll Need All 3 Clues To Solve This Puzzle
- Think You Know The Solution To This Classic Riddle? Think Again.
- 100 Lives Are On The Line In This Week's Puzzle. How Many Can You Save?
- Can You Figure Out This Parking Lot's Numbering System?
- To Solve This Riddle, Look To Your Family
- Solving This Puzzle Will Help You Grasp The True Nature Of Puzzles
- Can You Guess The Next Number In This Sequence?

## DISCUSSION

First, convert every region of space into a node.

Next, convert every line segment that you have to cross into a path between two nodes.

Now you have an Eulerian Trail problem, in which you're trying to see whether you can visit every path in the graph exactly once.

http://en.wikipedia.org/wiki/Eulerian_…

One necessary (but not sufficient) rule for all Eulerian Trails is that there must be exactly zero or two nodes with odd "degree" (degree being defined as the number of paths connecting to that node).

In this representation, the nodes have the following degrees:

A - 9, B - 4, C - 5, D - 4, E - 5, F - 5

As you can see, four of the six nodes have odd degrees. Therefore, per Euler's proof, this graph has no Eulerian Trail. Likewise, there can be no path in the original diagram that crosses each line segment exactly once.