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by Greg Brown

This weeks quiz is based off of a mathematical proof we were taught in my
discrete mathematics class a few weeks ago. Though I can already hear some of
you running for the hills at the mention of the four letter word that is math, I
can assure you that this is more of a thinking problem than it is number
crunching. The basic idea is pretty simple.

You're going to tile L-trominos on a 2**n by 2**n board that is missing a single
square. What's an L-tromino? It's simply a 2 by 2 square with a corner missing
that looks like an L.

For further clarification, this is what they look like:

#
# #

Well, I guess it's time to stop being vague and give you the problem definition.

For any 2**n by 2**n board with a single square missing in a random location,
you must write a program that will tile L-trominos in such a way that the grid
is completely filled in.

Your program should take the value for n as the input, and somehow display the
board with the trominos properly laid out as the output. It should place the
missing square at random each time the board is generated. (Otherwise this would
be too easy!)

For example, the empty board of n = 2 would be something like this:

_ _ _ _
_ _ X _
_ _ _ _
_ _ _ _

The solution for that might look like:

1 1 2 2
1 5 X 2
3 5 5 4
3 3 4 4

As you can see, all squares are completely filled with the exception of the
empty square which was randomly placed.

It may look simple on a 4 by 4 board, but once you get up to a 128 by 128 or
even 4096 by 4096, it becomes more and more obvious that guess and check is just
not going to work.

The level of difficulty of this problem depends entirely on how much you already
know about it. If you want an easy ride, look for the proof and just implement
it.

If you want more of a challenge, avoid Googling the topic and try to find clever
ways to actually show how your program solves the problem.

Hope you have fun with this quiz, and if you write a really good one, you can
help me tile my bathroom floor next week as a prize.  :)

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