Thinking outside outside the box

Rob Eastaway joins the dots.


Image: Todd Lappin, CC BY-NC 2.0

There aren’t many puzzles that became famous as a corporate cliche, but the nine dots problem is certainly one of them.

The nine dots problem

There are nine dots arranged in a square, and your challenge is to join all the dots using only four straight line strokes of a pen, and with your pen never leaving the paper. No tricks are needed, no folding the paper and no using an ultra-fat pen. You’ll have no problem finding a solution with five lines, but in the unlikely event that you’ve never seen this puzzle before, four lines might prove to be a struggle.

The puzzle first rose to notoriety when Sam Loyd included it in his 1914 cyclopedia of puzzles. He called it the Columbus egg puzzle, after an apocryphal story about Christopher Columbus challenging his colleagues to get a boiled egg to stand on its end — a challenge that seemed impossible until Columbus crushed the base of the egg and rested it on its now flat bottom. Easy when you know how.

The solution to the nine dot puzzle led to the cliche of ‘thinking outside the box’ (that’s a massive clue to the solution, but if you still can’t find the answer, you’ll find it at the bottom of the article). The solution is a useful metaphor for the challenge of problem solving, when we often become restricted by constraints of our own making. Deliberately challenging your assumptions and allowing yourself to be ‘silly’, at least for a short period, can be a handy strategy for creative problem solving in all walks of life.

What I find interesting, however, is that the lessons of the nine dots puzzle are often quickly forgotten when the puzzle is modified.

The 16 dots problem

The sixteen dots problem

Take the 16 dots problem. It’s exactly the same idea, but this time instead of using four straight lines, you are allowed to use six straight lines.

Can you find a solution? And if you want a proper challenge, can you find a solution that nobody else is likely to find? I strongly encourage you at this point to have a go at the 16 dots problem, and allow yourself at least ten minutes on it before returning to this article, because there are spoilers aplenty coming up.

Warning: the rest of the article contains spoilers for this puzzle. Please have a go at the puzzle before reading on…

How did you get on? If you gave the puzzle a serious go, then you probably did find a solution. Take a look at your attempts. Of course, you allowed yourself to go outside the box this time, but what angles did your lines go at? The majority of solvers try lines that go horizontally, vertically, or at a $45^{\circ}$ angle. They do this because (a) that’s the pattern they’ve seen already, and (b) those are the most ‘efficient’ lines to take out dots. A diagonal line can take out up to four dots, whereas a line at a different angle will take out two dots at most.

If you took the $45^{\circ}$ approach, then most likely your solution looked like one of these two:

Both solutions are just extensions of the 9 dot solution. When it comes to categorising solutions, however, these two are in the same family. If you extend all of the lines off the page, you can see the two solutions use the same set of lines.

Turning a hat into a cathedral

Are there other solutions, that don’t belong to this family? Yes there are. But to find them, you have to overcome the self-imposed constraint of 90 and 45 degree angles. And to stand out from the crowd, you also have to allow yourself to think further outside the box. Literally.

If you allow one line to go at a more jaunty angle, you’ll get some interesting solutions (below, left). But if you really begin to let your hair down, you can find even more zany solutions (below, right):

If we call the distance between two horizontal dots one unit, notice how the right-hand solution has at one point travelled three units outside the box before turning back — that’s one whole box width beyond the boundary of the dots. Which might make you wonder, are there any solutions that push even further outside the box? And, happy days, yes there are! For example, this one to the right has gone four units, or 133%, outside the box.

Far out, man.

I believe that’s the furthest you can get out of the box when solving the $4\times4$ puzzle, but I know that for the $7 \times7$ square
there is at least one solution that is 183% out of the box. For larger $N \times N$ squares there are surely solutions that go more than
200% out of the box — but nobody (yet) knows how big $N$ has to be for this to be possible.

Symmetrical solutions

Some of the solutions to the 16 dot puzzle look a bit messy, but out of mess sometimes comes beauty. It was only by exploring messy solutions that I found these four symmetrical beauties:

To my knowledge, this is the first time the ‘Chubby pentagram’ solution has ever been published, and possibly the first time that the words chubby and pentagram have been used in the same sentence.

Inside the box

I sometimes wonder if we give too much credit to thinking outside the box, while down-playing the importance of less glamorous inside-the-box thinking. Sometimes in life we have no choice but to think inside the box we’ve been given. So let’s apply this thinking to dot puzzles. We’ve seen that for the nine dot puzzle there is no solution with four lines that are all inside the box, though a solution with five lines is possible, for example on the left here.

In fact by continuing this spiral outwards, we can see that for the $N \times N$ square there will always be a solution inside the box, as long as you are allowed $2N – 1$ lines. But what if you can only use $2N-2$ lines (so, six lines for the $4\times4$, eight lines for the $5\times5$ etc). No such inside-the-box solution is possible for the $4\times4$ dot puzzle, but if we take our old friend The Archbishop’s hat and extend it, it’s easy to find an eight line inside-box solution to the $5\times5$ puzzle, which we can then spiral outwards for larger $N \times N$.

Extending the Archbishop’s hat to spiral inside the box

So we’ve shown that for the $N \times N$ puzzle (where $N \geq 2$), not only is there always a solution that uses $2N -2$ lines, but for $N \geq 4$ there is also always at least one solution that is inside the box.

Dot puzzle theorems
For an $N \times N$ square of dots:
There exists at least one solution inside the box using $2N-1$ lines, for all $N$.
There exists at least one solution using $2N-2$ lines for all $N\geq2$.
There exists at least one $2N-2$ solution that is inside the box for all $N\geq4$.

The dot puzzle conjecture

Without bending the rules, or bending the paper, there’s no solution to the 9 dot puzzle using three lines. Nor is there any known solution to the 16 dot puzzle using five lines. In fact there is a conjecture, not yet proved as far as I know, that there does not exist any solution to the $N \times N$ puzzle that uses $2N-3$ lines. There may not be a million-dollar prize on offer, but there’s still some kudos to whoever comes up with the proof.

And finally…

There is one solution to the 16 dot puzzle that stands above all others. It’s the only solution that has two lines of symmetry and the only one with rotational symmetry. It also starts and finishes at the same point, with no dot having more than one line passing through it.

Here it is. For its elegance and the fact that it’s two intertwined zigzags, I call this solution DNA.

The solution to the nine dots problem

The famous solution to the nine dots problem

Rob Eastaway is an author and speaker best known for his popular maths books, including the bestselling ‘Why Do Buses Come In Threes?’ He is the director of Maths Inspiration, the national programme of theatre-based maths lecture shows for teenagers.

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