n-cube and symmetry
Prabhakar Reddy Gudla Venkata Siva
reddyg at Glue.umd.edu
Mon Sep 2 08:37:59 PDT 2002
Dear Listers,
Before I pose the question, I will attempt to explain the problem (with
answer) for 2d and 3d for better understandability.
2D:: Given a square, if one had to splice the 4 edges into 2 equal halves,
how many unique configurations would be possible -- including the case
when none of it's edges are spliced ?
# of edges split(k): 0 1 2 3 4 Total
# of ways : 1 4(4c1) 6(4c2) 4(4c3) 1(4c4) 2^4 = 16
# of unique ways : 1 1 2 1 1 6
The first row is the number of edges (k) which are spliced at the middle.
The second row gives the number of possible ways in which one can split
the square along k-edges. The third row shows the number of unique
configurations. The results of the 3rd row for 2D case are shown in the
attachment (GIF).
3D: Given a cube, if one had to splice the 6 faces into 4 equal halves,
how many unique configurations would be possible -- including the case
when none of it's edges are spliced ?
# of faces split: 0 1 2 3 4 5 6 Total
# of ways : 1 6(6c1) 15(6c2) 20(6c3) 15(6c4) 6(6c5) 1(6c6) 2^6=64
# of unique ways: 1 1 2 2 2 1 1 10
The question for n >= 4: nD: Given a n-cube, if one had to splice the 2*n
cells (which are (n-1)D cubes) into 2*(n-1) equal sub-cells, how many
unique configurations would be possible (after removing symmetric cases)--
including the case when none of it's edges are spliced ?
regards
--
*******************************--------------**********************************
GVS Prabhakar Reddy
Research Assistant
Model Analysis Lab (GIS Lab)
Dept. of Biological Resources Engineering
University of Maryland
College Park, MD, 20742.
tel: 301-405-0109 (o)
: 301-982-7148 (r)
fax: 301-314-9023
email: reddyg at glue.umd.edu
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