Some time ago, mid-2009 I think, I was given a challenge that I found
fascinating. You might choose to have a think about it, and here is
the way I introduce it to people:
Given a square, you can dissect it into congruent pieces
such that they all touch the centre point.
I suggest you have a quick go at that to make sure
Congruent means the same size and shape - you can't
tell them apart (and you're allowed to turn them over)
Equally, given a square, you can dissect it into congruent pieces
such that one of them contains the centre point.
I suggest you have a go at that as well. It's not too hard.
Finally, given a square, you can dissect it into congruent pieces
such that some (more than one) touch the centre point, and some don't.
And, you guessed it, you should have a go at this too.
OK, so there are three possibilities.
Exactly one piece touches the centre point
There's more than one piece and they all touch the centre point
Two or more pieces touch the centre point, but some don't.
Right, so we've done that for a square. Now do it for:
An equilateral triangle
A regular pentagon
A regular hexagon
I'm not going to say any more than that all three possibilities
can be achieved with the triangle and the square, and that you
might have to think really hard about the other cases.
Now, the challenge I was given way back in 2009 was whether all
three possibilities can be achieved for a circle.
All
touch
Some do
some don't
One
contains
Triangle
???
???
???
Square
Yes
Yes
Yes
Pentagon
???
???
???
Hexagon
???
???
???
...
Circle?
???
???
???
And I'm going to leave it there. Next time we'll look at the
very first of these questions, and see that there's more to it
than it might first appear. In the meantime, you've got lots
of challenges. The most interesting is this:
Suppose you can answer all
the questions I've asked.