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My lastest posts can be
found here:
  * ColinsBlog
----
Previous blog posts:
  * DissectingACircle
  * DissectingASquare
  * AnOddityInTennis
  * DecisionTreeForTennis
  * DecisionTreesInGames
  * AMatterOfConvention
  * DoYouNourishOrTarnish
  * BinarySearchReconsidered
  * TwoEqualsFour
  * TheLostPropertyOffice
  * TheForgivingUserInterface
  * SettingUpRSS
  * WithdrawingFromHackerNews
----
Additionally, some earlier writings:
  * RandomWritings.
  * ColinsBlog2010
  * ColinsBlog2009
  * ColinsBlog2008
  * ColinsBlog2007
  * ColinsBlogBefore2007
********
!! Dissecting  a Square (Part 2) - 2011/08/08

[[[>30
If you haven't already, you might want to read:
  * DissectingACircle
  * DissectingASquare

We're talking about dissections of squares and circles
into congruent pieces.  That means the pieces are all
the same size and shape, although they may be mirror
images of each other.
]]]
So we return to the square.  It's simple enough to cut it up into
identical pieces so that all the pieces touch the centre.  One way
is just to cut it into four identical squares.  They all touch the
centre.  Another way is to cut along just the diagonal, giving two
identical (triangular) pieces, both touching the centre.

So it can be done, but in how many ways?

I rapidly got 5 (or 6, depending on a technicality), and I started
to wonder about a proof that 5 (or 6) was all of them.  I posted a
badly worded question on an internet forum, and rightly got flamed
for it, but in the answers was a shock.

There was an infinite family of solutions.

In fact there were two infinite families.

No, three!

No, wait, it's not the first or second infinity, it's an infinity
beyond that!

Oh crap.  I really, /*really*/ don't know anything at all.

But the infinite families were (and are) quite well behaved, so I
started to wonder if it wasn't so bad after all. I mean, I had three
easily described infinite families , maybe I could prove that I had
them all now!

The families have two, four or eight pieces, all of which divide the
number of symmetries of a square. There are clues there, maybe I can
work on that.

Then someone produced a "dissection" into 16 pieces.

Oh, crap.

Then someone else produced a dissection into 32 pieces, and wondered
openly about 128.

[[[>40 Oh, shoot, now someone in the comments has found another,
and I've just generalised it a little to get still another. Now
I really /really/ know that I know nothing. ]]]
Oh, crapity crap. Once again, I now know that I know nothing.

And there it stands.  I haven't told you everything yet, as you might
expect, but I hope I've made you at least a little curious.

So here's your puzzle for next time.  Dissect a square into identical
(size and shape) pieces so that all of them touch the centre point.
This can be done in infinitely many ways, but your challenge is to
find some and describe them.

Good luck!

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