Consider a square. There are all sorts of interesting
questions about cutting up a square. For example, there
is a way of cutting a square into four pieces that can
then be reassembled to form an equilateral triangle.
But that's not where I'm going with this.
| Technical term: "identical in size and shape" is
called "congruent, and you are allowed to turn them
over. |
|
There are many ways of cutting up a square, but we want
to consider only those that have all the pieces identical
in size and shape.
So dissect a square into identical pieces, and ask - where is
the centre point? Is it contained within one of the pieces?
Is it on a corner? Is it on an edge? Does every piece touch
the centre? Or do some touch, and some not touch?
| There are technical questions about what we really
mean by "cut up", or what's the technical definition of a
"dissection." The generally accepted definition is that a
"piece" is defined by its "interior," and while that might
not help much, there is a technical defined for that, and
so on. All this boils down to this:
- Use a finite number of pieces,
- The pieces must be "solid,"
- Don't try to cheat!
- If it seems dodgy, it probably is.
|
|
So here are the challenges. Dissect a square into identical
pieces such that:
- the centre point is contained within a piece,
- every piece touches the centre point,
- some pieces touch the centre point, and some don't.
One of the things that marks the difference between mathematicians
and normal people is that for mathematicians it's not enough just
to solve the puzzle - they then ask in how many different ways can
it be done?
So in how many ways can each of these things be done? Can you
characterise them all? In particular, can you describe every
solution to each of those questions? There might be more than
you initially suspect.
So having done all that, the next thing is - why a square? What
about an equilateral triangle?
So, dissect a triangle into identical pieces such that:
- the centre point is contained within a piece,
- every piece touches the centre point,
- some pieces touch the centre point, and some don't.
Are all these possible? And in how many ways? Here's a hint,
they are all possible - see if you can solve them all.
So triangles, check. Squares, check. Nothing really challenging,
so perhaps this is easy. But perhaps not.
- Can you dissect a pentagon into identical pieces
with the centre point contained within the interior
of a piece?
Hmm. Maybe this game isn't trivial. If you can't do it, can
you prove that it's impossible? Because that's the other
thing that distinguishes mathematicians from normal people:
when something seems impossible, they set out to prove
that's impossible,and not just that they haven't succeeded
yet.
And so to finish ...
It's (surprisingly!) possible to dissect a circle such that
some pieces touch the centre point, and some don't. Can you
find one?
There's more than one ...
Part of the Farrago of Fragments.
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