The way I would generate another solution is to think about PAGES.

What are the parts of the problem?

Well, there are some dots.

That's obviously a part of the problem.

We can think about them a little bit.

They are dots, spots on the page,

poker dots arranged in a rigid way,

little circles, dots and circles can be little.

But dots and circles can also be big.

What if we made a change and drew the dots a little bigger? What would that do?

Now, when we draw lines,

the lines might only go through the edge of

a dot instead of the middle or the entire dot.

That gives us an opportunity for a solution with just three lines.

Well, if the dots can be small or large,

what if we think about another part of the problem and change that?

What about the lines?

Lines can be thin,

lines can be thick.

If we have a really thick line,

we could draw a single line that goes through all nine dots.

What about the other parts of this problem?

We talked about the dots,

we talked about the lines, what else is there?

Well, what about the paper?

What thoughts do we have about that concept?

Perhaps, we've been assuming the paper needed to stay flat,

but we can make changes to the paper.

For example, we can roll the paper in a cylinder,

tilt it a little bit and draw a line around

and around that goes through all nine dots. That's another solution.

Any other assumptions we're making when we're thinking about the paper?

My younger daughter's favorite solution to the problem is this one.

Who said the paper had to remain whole?

What if we rip the dots out of the paper,

put them in a line,

and draw a line through them all?

At this point, we have seen changes to parts,

dots of different sizes,

lines of different sizes,

paper shape, changes to actions,

lines connecting not as dot,

ripping dots out of the paper,

and changes to our goals,

a four-line solution, a three-line solution, a one-line solution.

These changes might have seemed delightful.

But sometimes some people think these changes seem like cheating,

and this happens with creativity.

When you're changing perspectives,

sometimes you find yourself thinking that you are breaking the rules.

Worse, sometimes a change that you think is creative,

other people think is breaking the rules.

Creativity can be dangerous, disruptive, threatening,

so what might make someone upset about our discussion of this problem?

Well, I usually find that people who think these solutions are

cheating have taken a perspective drawing from geometry.

They think the event here is to solve a geometry problem,

and their self concept includes the value that it is good and

appropriate to adhere to the rules of classic euclidean geometry.

The dots are points, the line is a classic geometry line so it has no width,

and everything is set in a single plane.

In this case, the classic solution is probably still perceived as

a genuine solution but everything else probably feels like cheating.

One question is, why did we take this perspective on this puzzle?

Why was this a geometry event,

and why was there self concept so committed to that interpretation?

But, okay, let's take that perspective as a given.

Is there anything we can do?

One of our colleagues here at Illinois proposed a solution

following from an observation of Einstein's,

"All parallel lines meet at infinity."

We'll leave it to you to decide if this counts as not picking up your pen.

The nine dot puzzle is a classic problem in the history of creativity.