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+--- Thread: Insane Math Questions. Those With Weak Hearts Beware. (/showthread.php?tid=24775)
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Insane Math Questions. Those With Weak Hearts Beware. - Dusk - 2010-04-19
The third way you mention to fill the Ls, the two blocks stacked on top of each other, are "lying flat" setups. Don't count them twice.
Still trying to work out the rest of your post.
Edit: Okay,
If you divide the blocks into two Ls as you said, there are 3 distinct, asymmetrical ways to position them.
Taking rotation into consideration, that's still only 6 possibilities. Once you place one block in the L, the other block only has one place to go. P(3,2) = 6.
+2 for the possibilty that the four blocks don't fill the assumed Ls, so the answer is 8.
Thanks for the justification on E.
Insane Math Questions. Those With Weak Hearts Beware. - Stereo - 2010-04-19
Ok, I did get a few extras in there.
But there are 3 distinct ways to place the first big L - sitting up, and clockwise/counterclockwise.
![[Image: blocks.gif]](http://img405.imageshack.us/img405/8454/blocks.gif)
If you take black as "top" and grey as "side" these are definitely distinct ways of laying them out.
P(3,2) only applies to the clock/counterclockwise ones - the sitting up version is not rotationally symmetric so it has 9 options since either of the 2 Ls can take all 3 subsets and give a distinct arrangement.
Describing this is easier if I just number the boxes:
![[Image: boxnumbers.gif]](http://img341.imageshack.us/img341/9389/boxnumbers.gif)
1 2 3
4 5 6
7 8 9
10 11 12
Assume it is reconstructed by "sliding" the top down to cover the bottom so 1 is above 7, 2 is above 8, etc.
124 356 and 145 236 are distinct ways of filling the top layer. To avoid being Ls the bottom is then 7 10 11, 8 9 12, and 7 8 10, 9 11 12.
The other 2 "flats" are
1 7 8, 2 3 9, 4 5 10, 6 11 12.
1 2 7, 3 8 9, 4 10 11, 5 6 12.
These are also not rotationally identical so there are definitely 4 possibilities there.
So my current answer is: 6*2 + 9 + 4 = 25.
Insane Math Questions. Those With Weak Hearts Beware. - Dusk - 2010-04-19
They don't have to be rotationally symmetric to be identical to one another. "1 and 2" is the same as "2 and 1" if you rotate it, even if they aren't symmetrical.
Also, two of the flats are already counted in the first part.
Insane Math Questions. Those With Weak Hearts Beware. - Stereo - 2010-04-19
Because the top & bottom of the bin are distinct, they're not identical though.
And no none of the 4 flats I mentioned have the big Ls in them at all, so they're all distinct.
Insane Math Questions. Those With Weak Hearts Beware. - Dusk - 2010-04-19
Oh, I see. The second set of flats are vertical.
Stereo Wrote:Because the top & bottom of the bin are distinct, they're not identical though.
Your two Ls are arranged side by side. How does this matter?