Duh, you're all correct. I was looking at the different orderings as Identity, then swap pairs. This works under the swap rule, but doesn't count all the orders. It's a straight permutation.
I was looking at the whole problem as a set closed under the operations defined in Treehouse, rather than a straight combinatorial problem. Whoops.
-Diane
On 8/21/07, kerry_and_ryan@xxxxxxx <kerry_and_ryan@xxxxxxx> wrote:
Andy:
> Hi Diane! Am I right in thinking you're the one who has at our booth
> calculating that there are 96 ways you can arrange your pieces in
> Treehouse?
Wooop wooop Geek Alert wooop wooop
Huh, I get 204. I think one issue is that orientation of the entire row matters. 1<, 2<, 3< is distinct from 3>, 2>, 1>, even though they have a large pointing at the medium, which points at the small.
Specifically...
There are four "classes" of arrangements:
A. Three separate pyramids.
There are 6 possible orders for the pyramids: 1 2 3, 1 3 2, 2 1 3, 2 3 1, 3 1 2, 3 2 1
For each pyramids there are three directions it could point: <, ^, >
That makes 6*3*3*3 = 162 arrangements.
B. A stack of two pyramids to the left of a single pyramid.
There are 6 possible orders for the pyramids: 1/2 3, 1/3 2, 2/1 3, 2/3 1, 3/1 2, 3/2 1
The stack is always upright, but there are three possible orientations for the lone 'mid.
That makes 6*3 = 18 arrangements.
C. A single pyramid to the left of a stack of two.
As with class B, there are 18 arrangements.
D. All three pyramids in a stack.
There are 6: 1/2/3 (a tree), 1/3/2, 2/1/3, 2/3/1, 3/1/2, 3/2/1 (a nest)
162 + 18 + 18 + 6 = 204
Ryan
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