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The background of 'Friends':
If we take one square, it's the only possible shape. And it fills a rectangle.
If we use two squares, they can only be arranged in a queue. That's one shape and it fills the rectangle.
Taking three squares, they can arranged in two ways: in a queue, and with an edge. You can't a fill a square with both of them.
Four squares gives us 5 shapes. It's no rectangle possible, too.
Beginning with 5 squares there's a "critical mass". We get 12 shapes and a few thousand ways to fill the rectangle.
With more than 5 squares you will soon get a few hundred of shapes...
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A numerical remark to 'Friends'
Imaging the systematical search - like a machine has to do... You got 12 shapes, they can be mirrored and rotated
8 times. And must be placed: the first in 60 ways, the second in 55 ways, and so on.
The poor machine has to try
(60*55*50* ... * 10*5) * 812 = 512 * 12! * 812 = 8.04 * 1027
combinations.
Well, the Big Bang was 15 billions years ago, before 4.73*1017
seconds. If our machine find a combination per second - never mind, if it's a solution or a wrong one - she will need
17 billion universes like ours here and now.
The good player remarks: the cross won't change himself by mirror or rotate.
Same to other shapes. Some are 8 times, but others are 4, 2 or 1 times. We inform the machine ... and there's only a need
for
512 * 12! * (1 * 2* 45
* 85) /4 = 1.96*1024 combinations .
4 millionen time the existing of our universe.
Of course: it's a rough approximation. But for a better one you must tell the machine
- ignore overlaps
- don't go over the border, and really hard
- think like a human. |
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