Apr. 23rd, 2008 02:54 pm
gipsieee: (Default)
[personal profile] gipsieee
A handful of us, four to be precise, got together to watch a movie on Saturday. Other gatherings with that group have been games nights and we haven't played any games. Saturday it was a movie night and the movie was not watched, we played games instead.

And we got silly.
We put together this blokus board trying to legally get all the pieces on the board in such a manner that except for a rotation of 90 degrees, each colors pattern was identical to all of the others.

Anyone want to try to do better? Better being defined two ways, either you get all the pieces on the board, or you wind up with a smaller piece left over. More's the better if you can keep the singletons arranged at the very center of the board.

I really want to get the triangle version, my brain likes it much better.

Date: 2008-04-23 07:15 pm (UTC)
katybeth: (Default)
From: [personal profile] katybeth
Nice! I've tried to do similar with a Blokus board, but never gotten it this far.

Date: 2008-04-23 08:04 pm (UTC)
reedrover: (Default)
From: [personal profile] reedrover

We don't have nearly enough game nights either.

Date: 2008-04-23 09:17 pm (UTC)
katybeth: (Default)
From: [personal profile] katybeth
Oh, interesting. I just noticed that your board has reflection symmetry on two axes, rather than rotational symmetry.

Date: 2008-04-23 09:29 pm (UTC)
From: [identity profile]
~grin~ It does have rotational symmetry, but only at pi radians 180 degrees...

Edit: Which leads me to wonder whether doing rotational symmetry all around would better lend itself to a solution.
Edited Date: 2008-04-23 09:31 pm (UTC)

Date: 2008-04-23 09:48 pm (UTC)
katybeth: two interlocked squares (square dance symbol) (square dance)
From: [personal profile] katybeth
Well, OK, yes. :)

Date: 2008-04-23 10:31 pm (UTC)
From: [identity profile]
Actually I think you were right the first time. Rotation of 180deg != reflection. Unless we're talking about 3-d rotation of the planar board... ;)

Date: 2008-04-23 10:33 pm (UTC)
katybeth: (Default)
From: [personal profile] katybeth
No, they're not equal, but in this case both apply.

Date: 2008-04-23 10:52 pm (UTC)
From: [identity profile]
Heh. I suppose I was interpreting "rotation" in the original problem spec too narrowly. My bad!

Date: 2008-04-23 11:01 pm (UTC)
katybeth: (Default)
From: [personal profile] katybeth
Hm, *are* they not equal? If I have reflection symmetry about the X and Y axis, is that the same as rotational symmetry 180 degrees about the origin?

Date: 2008-04-23 11:17 pm (UTC)
From: [identity profile]
Yes, reflecting through *both* axes is equivalent to a 180deg rotation. A single reflection is not equivalent to any rotation, though, and from the picture above it looks like a single reflection is needed to map red or blue onto green or yellow. I initially took the word "rotation" quite literally, and figured single reflection was not allowed.

Date: 2008-04-23 11:28 pm (UTC)
katybeth: (Default)
From: [personal profile] katybeth
reflection over the Y axis: red<->green, yellow<->blue
reflection over the X axis: red<->yellow, green<->blue
rotation by 180deg around the origin: red<->blue, green<->yellow

rotation by 180deg out of the plane of the board: all the pieces fall out :)

Date: 2008-04-23 11:58 pm (UTC)
From: [identity profile]
rotation by 180deg out of the plane of the board: all the pieces fall out :)

lol! Can we call that a gravitational reboot? :)

Date: 2008-04-25 02:16 am (UTC)
From: [identity profile]
It seems likely. Rotational symmetry lets pieces cross the center line if that makes things work better.

I don't have a Blokus board to try this with, or I'd probably get sucked in in fairly short order.

Date: 2008-04-25 05:33 pm (UTC)
katybeth: (Default)
From: [personal profile] katybeth
You'll have to come over and play with mine. :)

Date: 2008-04-24 11:35 am (UTC)
From: [identity profile]
That was an awesome game.


gipsieee: (Default)

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