cheating on the nine dot problem.

While procrastinating before an essay deadline, I happened across a blog post with the following challenge

In their study, 22 volunteers were faced with the 9 dots problem, a notoriously difficult puzzle. The goal here is to draw exactly four straight lines connecting all nine of these dots, without retracing any line, or lifting your pen from the page.


Can you do it?




Given that I had a relatively important deadline approaching, that was one opportunity to procrastinate I was not going to miss.

Transcranial DC stimulation


The article describes a study comparing the performance of volunteers who had been subjected to "transcranial direct current stimulation (tDCS)", and a control group.

The study reports that none of the 22 subjects in the control group were able to complete the puzzle, however 5 of 11 in the brain buzzed group got the answer right.


So "Staring hard" not such a successful problem solving technique


I stared at the dots for while. Nothing.

Drew some zig zags. Nothing less than 5 lines for me.

The post seems to imply that the answer should simply float up from your right (?) brain. There is a comment about an earlier experiment stimulated novel thinking, or insight. But whatever, the answer was not forthcoming by just staring at it. It felt like failure.


Given that I don't have access to a tDCS device, and that after staring at the dots for a while I was starting to go cross eyed, I decided that some other strategy was in order.

Lance Armstrong continues to inspire


Inspired by Lance Armstrong's glorious achievements, and eyeing a future spot on Oprah, I decided to cheat.

I decided to use some systematic method to approach the solution, however this kind of feels like cheating given that I was expecting to just "see the answer"

Hence the title of this article. ("cheating on the nine dot problem.")

Anyway, so... lets proceed with some sort of systematic deconstruction, starting with the rules.

Step 1 - define the problem


"The goal here is to draw exactly four straight lines connecting all nine of these dots, without retracing any line, or lifting your pen from the page."
So the goal;
  •  To connect all nine of the dots.

The constraints
  1. 4 straight lines
  2. no retracing
  3. and no lifting the pen

Step 2 - identify valid solutions

So the first step could be to enumerate the set of, or identify what the end state is going to look like.

I see that the rule "4 straight lines" as constraining what the solution looks like, but that 2) and 3) are about the operations to get there.

So if we relax rules 2) and 3) and just look for the goal state "connect all the dots" and "with 4 straight lines" we can generate a number of possible solutions.

After 9 whole seconds of pondering, My firstborn solution.

 After that they came thick and fast.
 this is starting to mess with my vision...

But why stick with zero and one as gradients.

 If this is a valid move, then there are a bunch more solutions;

 

 So they are still connected by straight lines...

 
So there are other valid moves such as ;

You can do things like this;

So when I drew out number 6, you would have thought it would have given me a clue.

But no, it didn't.

Number 6


So its pretty clear that the solution involves lines that extend beyond the edge of the square delimited by the dots.


And if you didn't work that one out, then the blog post drops a massive hint with

'The "L− R+" current was designed to boost the right temporal lobe while inhibiting the left, on the hypothesis that the right side of the brain helps us "think outside the box" (literally.)'

Candidate solution


So that last solution looked like the sort of thing that would fit.




brainstorming on the number 6 solution is most frustrating

but you can see it is getting close...

 





Gah!
 
A characteristic of solutions 1-5, is that each line configuration either diverges or runs parallel on extension. For example;

 

So its pretty clear from the extending the lines, that they don't converge.


 But solution 6 has the unique characteristic, of lines extending outside the box, that cross over;




So  Recap




1234
56




so the final solution is going to look something like 6, we have tried all the variations of extending lines on 6, so either we have proved the 9 dot problem impossible. Or I have missed a solution.

So while staring at this one for a bit;



Its fairly obvious, that this is a solution to the relaxed problem also;



So extending out those lines gives this;


and then the solution is obvious,


References

Neurostimulation - The Genius Machine?

http://neuroskeptic.blogspot.co.uk/2012/04/neurostimulation-genius-machine.html



9 dot image nicked from  http://neuroskeptic.blogspot.co.uk/2012/04/neurostimulation-genius-machine.html

Chi, R., and Snyder, A. (2012). Brain stimulation enables the solution of an inherently difficult problem Neuroscience Letters DOI:10.1016/j.neulet.2012.03.012