# Dissecting A Square

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## Dissecting a Square (Part 1) - 2011/06/01

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. What is the next question you should look at?

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